general information, conversion tables, and mathematics

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SECTION 2
GENERAL INFORMATION,
CONVERSION TABLES, AND
MATHEMATICS
2.1
GENERAL INFORMATION AND CONVERSION TABLES
Table 2.1
Fundamental Physical Constants
Table 2.2
Physical and Chemical Symbols and Definitions
Table 2.3
Mathematical Symbols and Abbreviations
Table 2.4
SI Prefixes
Table 2.5
Greek Alphabet
Table 2.6
Abbreviations and Standard Letter Symbols
Table 2.7
Conversion Factors
Table 2.8
Temperature Conversion Table
2.1.1 Conversion of Thermometer Scales
2.1.2 Density and Specific Gravity
2.1.3 Barometry and Barometric Corrections
Table 2.9
Barometer Temperature Correction— Metric Units
Table 2.10
Barometric Latitude-Gravity Table— Metric Units
Table 2.11
Barometric Correction for Gravity— Metric Units
Table 2.12
Reduction of the Barometer to Sea Level— Metric Units
Table 2.13
Viscosity Conversion Table
Table 2.14
Conversion of Weighings in Air to Weighings in Vacuo
Table 2.15
Hydrometer Conversion Table
Table 2.16
Pressure Conversion Chart
Table 2.17
Corrections to Be Added to Molar Values to Convert to Molal
Table 2.18
Molar Equivalent of One Liter of Gas at Various Temperatures and
Pressures
Table 2.19
Factors for Reducing Gas Volumes to Normal (Standard)
Temperature and Pressure (760 mmHg)
Table 2.20
Values of Absorbance for Percent Absorption
Table 2.21
Transmittance-Absorbance Conversion Table
Table 2.22
Wavenumber/Wavelength Conversion Table
2.2 MATHEMATICAL TABLES
2.2.1 Logarithms
Table 2.23
Derivatives and Differentiation
Table 2.24
Integrals
2.2.2 Surface Areas and Volumes
2.2.3 Trigonometric Functions of an Angle ␣
2.2.4 Expansion in Series
Table 2.25
Some Constants
2.3 STATISTICS IN CHEMICAL ANALYSIS
2.3.1 Introduction
2.3.2 Errors in Quantitative Analysis
2.3.3 Representation of Sets of Data
2.3.4 The Normal Distribution of Measurements
Figure 2.10 The Normal Distribution Curve
Table 2.26a Ordinates (Y) of the Normal Distribution Curve at Values of z
Table 2.26b Areas Under the Normal Distribution Curve from 0 to z
2.3.5 Standard Deviation as a Measure of Dispersion
2.1
2.3
2.3
2.6
2.23
2.24
2.25
2.26
2.35
2.54
2.66
2.66
2.70
2.72
2.75
2.77
2.78
2.82
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2.85
2.87
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2.91
2.96
2.98
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2.102
2.102
2.104
2.105
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SECTION 2
2.3.6 Student’s Distribution or t Test
Table 2.27
Percentile Values for Student t Distribution
2.3.7 Hypotheses About Means
2.3.8 The Chi-square (␹2) Distribution
Table 2.28
Percentile Values for the Chi-square (␹2) Distribution
2.3.9 The F Statistic
Table 2.29
F Distribution
2.3.10 Curve Fitting
2.3.11 Control Charts
Bibliography
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GENERAL INFORMATION AND CONVERSION
TABLES
TABLE 2.1
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Fundamental Physical Constants
E. R. Cohen and B. N. Taylor, CODATA Bull. 63:1– 49 (1986); J. Res. Nat. Bur. Standards, 92:85 (1987).
A. Defined values
Physical quantity
1. Base SI units
Amount of substance
Name of
SI unit
Symbol for
SI unit
Definition
mole
mol
Amount of substance which contains as many
specified entities as there are atoms of carbon-12 in exactly 0.012 kg of that nuclide.
The elementary entities must be specified
and may be atoms, molecules, ions, electrons, other particles, or specified groups
of such particles.
Electric current
ampere
A
Magnitude of the current that, when flowing
through each of two straight parallel conductors of infinite length, of negligible
cross-section, separated by 1 meter in a
vacuum, results in a force between the two
wires of 2 ⫻ 10⫺7 newton per meter of
length.
Length
meter
m
Distance light travels in a vacuum during
1/299 792 458 of a second.
Luminous intensity
candela
cd
Luminous intensity, in a given direction, of a
source that emits monochromatic radiation
of frequency 540 ⫻ 1012 hertz and that has
a radiant intensity in that direction of
1/683 watt per steradian.
Mass
kilogram
kg
Mass of a cylinder of platinum-iridium alloy
kept at Paris.
Temperature
kelvin
K
Defined as the fraction 1/273.16 of the thermodynamic temperature of the triple point
of water.
Time
second
s
Duration of 9 192 631 770 periods of the radiation corresponding to the transition between the two hyperfine levels of the
ground state of the cesium-133 atom.
radian
rad
The plane angle between two radii of a circle
which cut off on the circumference an arc
equal in length to the radius.
steradian
sr
The solid angle which, having its vertex in
the center of a sphere, cuts off an area of
the surface of the sphere equal to that of a
square with sides of length equal to the radius of the sphere.
2. Supplementary SI units
Plane angle
Solid angle
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standard
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TABLE 2.1
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Fundamental Physical Constants (Continued)
B. Derived SI units
Physical quantity
Absorbed dose (of radiation)
Activity (radioactive)
Capacitance (electric)
Charge (electric)
Conductance (electric)
Dose equivalent (radiation)
Energy, work, heat
Force
Frequency
Illuminance
Inductance
Luminous flux
Magnetic flux
Magnetic flux density
Potential, electric
(electromotive force)
Power, radiant flux
Pressure, stress
Resistance, electric
Temperature, Celsius
Name of
SI unit
Symbol for
SI unit
Expression in terms
of SI base units
gray
becquerel
farad
coulomb
siemens
sievert
joule
newton
hertz
lux
henry
lumen
weber
tesla
volt
Gy
Bq
F
C
S
Sv
J
N
Hz
lx
H
Lm
Wb
T
V
J · kg⫺1
s⫺1 ⫽ m2 · s⫺2
C · V⫺1 ⫽ m⫺2 · kg⫺1 · s4 · A2
A·s
⍀⫺1 ⫽ m⫺2 · kg⫺1 · s3 · A2
J · kg⫺1 ⫽ m2 · s⫺2
N · m ⫽ m2 · kg · s⫺2
m · kg · s⫺2
s⫺1
cd · sr · m⫺2
V · A⫺1 · s ⫽ m2 · kg · s⫺2 · A⫺2
cd · sr
V · s ⫽ m2 · kg · s⫺2 · A⫺1
V · s · m⫺2 ⫽ kg · s⫺2 · A⫺1
J · C⫺1 ⫽ m2 · kg · s⫺3 · A⫺1
watt
pascal
ohm
degree
Celsius
W
Pa
⍀
⬚C
J · s⫺1 ⫽ m2 · kg · s⫺3
N · m⫺2 ⫽ m⫺1 · kg · s⫺2
V · A⫺1 ⫽ m2 · kg · s⫺3 · A⫺2
⬚C ⫽ (K ⫺ 273.15)
C. Recommended consistent values of constants
Quantity
Symbol
Anomalous electron moment correction
Atomic mass constant
Avogadro constant
Bohr magneton (⫽eh/4␲me)
Bohr radius
Boltzmann constant
Charge-to-mass ratio for electron
Compton wavelength of electron
Compton wavelength of neutron
Compton wavelength of proton
Diamagnetic shielding factor, spherical water
molecule
Electron magnetic moment
Electron radius (classical)
Electron rest mass
Elementary charge
Energy equivalents:
1 electron mass
1 electronvolt
␮e ⫺ 1
mu ⫽ 1 u
L, NA
␮B
a0
k
e/me
␭c
␭c,n
␭c,p
1 ⫹ ␴(H2O)
0.001 159 615(15)
1.660 540 2(10) ⫻ 10⫺27 kg
6.022 136 7(36) ⫻ 1023 mol⫺1
9.274 015 4(31) ⫻ 10⫺24 J · T⫺1
5.291 772 49(24) ⫻ 10⫺11 m
1.380 658(12) ⫻ 10⫺23 J · K⫺1
1.758 805(5) ⫻ 10⫺11 C · kg⫺1
2.426 309(4) ⫻ 10⫺12 m
1.319 591(2) ⫻ 10⫺15 m
1.321 410(2) ⫻ 10⫺15 m
1.000 025 64(7)
␮e
re
me
e
9.284 770 1(31) ⫻ 10⫺24 J · T⫺1
2.817 938(7) ⫻ 10⫺15 m
9.109 389 7(54) ⫻ 10⫺31 kg
1.602 177 33(49) ⫻ 10⫺19 C
1 neutron mass
1 proton mass
1 eV/k
1 eV/hc
1 eV/h
Value*
0.511 003 4(14) MeV
1.160 450(36) ⫻ 104 K
8.065 479(21) ⫻ 103 cm⫺1
2.417 970(6) ⫻ 1014 Hz
939.573 1(27) MeV
938.279 6(27) MeV
* The digits in parentheses following a numerical value represent the standard deviation of that value in terms of the final
listed digits.
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Fundamental Physical Constants (Continued)
C. Recommended consistent values of constants (continued)
Quantity
Symbol
1u
Faraday constant
Fine structure constant
First radiation constant
Gas constant
g factor (Lande) for free electron
Gravitational constant
Hartree energy
Josephson frequency-voltage ratio
Magnetic flux quantum
Magnetic moment of protons in water
Molar volume, ideal gas, p ⫽ 1 bar,
␪ ⫽ 0⬚C
Neutron rest mass
Nuclear magneton
Permeability of vacuum
Permittivity of vacuum
Value*
931.501 6(26) MeV
96 485.309(29) C · mol⫺1
0.007 297 353 08(33)
137.035 989 5(61)
3.741 774 9(22) ⫻ 10⫺16 W · m2
8.314 510(70) J · K⫺1 · mol⫺1
2.002 319 304 386(20)
6.672 59(85) ⫻ 10⫺11 m3 · kg⫺1 · s⫺2
4.359 748 2(26) ⫻ 10⫺18 J
4.835 939(13) ⫻ 1014 Hz · V⫺1
2.067 851(5) ⫻ 10⫺15 Wb
1.520 993 129(17) ⫻ 10⫺3
22.711 08(19) L · mol⫺1
F
␣
␣⫺1
c1
R
ge
G
Eh
⌽0
␮p/␮B
Planck constant
Proton magnetic moment
Proton magnetogyric ratio
Proton resonance frequency per field in H2O
Proton rest mass
Quantum-charge ratio
mn
␮N
␮0
⑀0
ប ⫽ h/2␲
h
␮p
␥p
␥⬘/2
p ␲
mp
h/e
Quantum of circulation
Ratio, electron-to-proton magnetic moments
Rydberg constant
Second radiation constant
Speed of light in vacuum
Standard acceleration of free fall
Standard atmosphere
Stefan-Boltzmann constant
Thomson cross section
Wien displacement constant
Zeeman splitting constant
h/me
␮e/␮p
R⬁
c2
c0
gn
atm
␴
␴e
b
␮B/hc
1.674 928 6(10) ⫻ 10⫺27 kg
5.050 786 6(17) ⫻ 10⫺27 J · T⫺1
4␲ ⫻ 10⫺7 H · m⫺1 exactly
8.854 187 816 ⫻ 10⫺12 F · m⫺1
1.054 572 66(63) ⫻ 10⫺34 J · s
6.626 0.75 5(40) ⫻ 10⫺34 J · s
1.410 607 61(47) ⫻ 10⫺26 J · T⫺1
2.675 221 28(81) ⫻ 108 s⫺1 · T⫺1
42.576 375(13) MHz · T⫺1
1.672 623 1(10) ⫻ 10⫺27 kg
4.135 701(11) ⫻ 10⫺15
J · Hz⫺1 · C⫺1
7.273 89(1) ⫻ 10⫺4 J · s · kg⫺1
6.582 106 88(7) ⫻ 102
1.097 373 153 4(13) ⫻ 107 m⫺1
1.438 769(12) ⫻ 10⫺2 m · K
299 792 458 m · s⫺1 exactly
9.806 65 m · s⫺2 exactly
101 325 Pa exactly
5.670 51(19) ⫻ 10⫺8 W · m⫺2 · K⫺4
6.652 448(33) ⫻ 10⫺29 m2
0.289 78(4) cm · K
4.668 58(4) ⫻ 10⫺5 cm⫺1 · G⫺1
D. Units in use together with SI units
Physical quantity
Area
Energy
Length
Mass
Name of unit
barn
electronvolt
megaelectronvolt1
ångström2
tonne
Symbol for unit
b
eV (e ⫻ V)
MeV
Å
t
Value in SI units
10⫺28 m
⬇1.60218 ⫻ 10⫺19 J
10⫺10 m; 0.1 nm
103 kg; Mg
*The digits in parentheses following a numerical value represent the standard deviation of that value in terms of the final
listed digits.
1
The term million electronvolts is frequently used in place of megaelectronvolts.
2
The ångström and bar are approved for temporary use with SI units; however, they should not be introduced if not used at
present.
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Fundamental Physical Constants (Continued)
D. Units in use together with SI units (continued)
Physical quantity
Name of unit
unified atomic
mass unit
dalton3
degree
minute
second
bar2
minute
hour
day
liter (litre)
milliliter
Plane angle
Pressure
Time
Volume
Symbol for unit
Value in SI units
u[⫽ ma(12C)/12]
⬇1.66054 ⫻ 10⫺27 kg
Da
⬚
⬘
⬙
bar
min
h
d
L, l
mL, ml
(␲/180) rad
(␲/10 800) rad
(␲/648 000) rad
105 Pa ⫽ 105 N m⫺2
60 s
3600 s
86 400 s
dm3 ⫽ 10⫺3 m3
cm3 ⫽ 10⫺6 m3
2
The ångström and bar are approved for temporary use with SI units; however, they should not be introduced if not used at
present.
3
The name dalton and symbol Da have not been approved although they are often used for large molecules.
TABLE 2.2 Physical and Chemical Symbols and Definitions
Symbols separated by commas represent equivalent recommendations. Symbols for physical and chemical quantities should be printed in italic type. Subscripts and superscripts which are themselves symbols for physical
quantities should be italicized; all others should be in Roman type. Vectors and matrices should be printed in
boldface italic type, e.g., B, b. Symbols for units should be printed in Roman type and should remain unaltered
in the plural, and should not be followed by a full stop except at the end of a sentence. References: International
Union of Pure and Applied Chemistry, Quantities, Units and Symbols in Physical Chemistry, Blackwell, Oxford,
1988; “Manual of Symbols and Terminology for Physicochemical Quantities and Units,” Pure Applied Chem.
31:577– 638 (1972), 37:499– 516 (1974), 46:71– 90 (1976), 51:1– 41, 1213– 1218 (1979); 53:753– 771 (1981),
54:1239– 1250 (1982), 55:931– 941 (1983); IUPAP-SUN, “Symbols, Units and Nomenclature in Physics,” Physica 93A: 1– 60 (1978).
A. Atoms and molecules
Name
Symbol
SI unit
Activity (radioactivity)
Atomic mass constant
Bohr magneton
Bohr radius
Decay (rate) constant
Dissociation energy
From ground state
From the potential minimum
Electric dipole moment of a molecule
Electric field gradient
A
mu
␮B
a0
␭
D, Ed
D0
De
p, ␮
q
Bq
kg
J · T⫺1
m
s⫺1
J
J
J
C·m
V · m⫺2
Electric polarizability of a molecule
Electron affinity
Electron rest mass
Elementary charge, proton charge
Fine structure constant
g factor
␣
Eea
me
e
␣
g
C · m2 · V⫺1
J
kg
C
Definition
A ⫽ ⫺dNB/dt
mu ⫽ ma(12C)/12
␮B ⫽ eh/4␲me
a0 ⫽ 2⑀0h2/mee2
A ⫽ ␭NB
Ep ⫽ ⫺p ⴢ E
q␣␤ ⫽
⫺⭸2V/⭸␣⭸␤
p(induced) ⫽ ␣E
␣ ⫽ e2/2⑀0hc
short
standard
long
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TABLE 2.2 Physical and Chemical Symbols and Definitions (Continued)
A. Atoms and molecules (continued)
Name
Symbol
SI unit
Hartree energy
Ionization energy
Larmor circular frequency
Larmor frequency
Longitudinal relaxation time
Magnetogyric ratio
Magnetic dipole moment of a molecule
Magnetizability of a molecule
Mass of atom, atomic mass
Neutron number
Nuclear magneton
Nucleon number, mass number
Planck constant
Planck constant/2␲
Principal quantum number (H atom)
Proton number, atomic number
Quadrupole interaction
Quadrupole moment of a molecule
Eh
Ei
␻L
␯L
T1
␥
m, ␮
␰
m, ma
N
␮N
A
h
ប
n
Z
␹
Q; ⍜
J
J
s⫺1
Hz
s
C · kg⫺1
J · T⫺1
J · T⫺2
kg
J
C · m2
Quadrupole moment
Rydberg constant
Transverse relaxation time
eQ
R⬁
T2
C · m2
m⫺1
s
J · T⫺1
J·s
J·s
Definition
Eh ⫽ h2/4␲2mea20
␻L ⫽ (e/2m)B
␯L ⫽ ␻L /2␲
␥ ⫽ ␮/L
Ep ⫽ ⴚ m ⴢ B
m⫽␰B
N⫽A⫺Z
␮N ⫽ (me/mp)␮B
ប ⫽ h/2␲
E ⫽ ⫺hcR/n2
␹␣␤ ⫽ eQq␣␤
Ep ⫽ 0.5 Q: V⬙
⫽ 1⁄3 ⍜V ⬙
eQ ⫽ 2具⍜zz典
R⬁ ⫽ Eh/2hc
B. Chemical reactions
Name
Amount (of substance)
Atomic mass
Atomic mass constanta
Avogadro constant
Concentration, amount (concentration)
Degree of dissociation
Density (mass)
Extent of reaction, advancement
Mass (molecular or formula unit)
Mass fraction
Molality (of a solute)
Molar mass
Molar volume
Molecular weight (relative molar mass)
Mole fractionb, number fraction
Number concentration
Number of entities (e.g., molecules, atoms, ions,
formula units)
Pressure (partial)
Pressure (total)
Solubility
a
b
Symbol
SI unit
Definition
n
m, ma
mu
L, NA
c
␣
␳, ␥
␰
m, mf
w
m
M
Vm
Mr
x, y
C, n
N
mol
kg
kg
mol⫺1
mol · m⫺3
nB ⫽ NB/L
kg · m⫺3
mol
kg
␳ ⫽ mB/V
⌬␰ ⫽ ⌬nB/␯BB
pB
p, P
s
Pa
Pa
mol · m⫺3
In biochemistry this unit is called the dalton, with symbol Da.
For condensed phases x is used, and for gaseous mixtures y may be used.
mol · kg⫺1
kg · mol⫺1
m3 · mol⫺1
m⫺3
mu ⫽ ma(12C)/12
cB ⫽ nB/V
wB ⫽ mB/⌺mi
mB ⫽ nB/mA
MB ⫽ m/nB
Vm,B ⫽ V/nB
Mr,B ⫽ mB/mu
xB ⫽ nB/⌺ni
CB ⫽ NB/V
pB ⫽ yBp
sB ⫽ cB (saturated
solution)
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standard
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TABLE 2.2 Physical and Chemical Symbols and Definitions (Continued)
B. Chemical reactions (continued)
Name
Symbol
␯
⌫
␾
Stoichiometric number
Surface concentration
Volume fraction
Symbols for particles and nuclear reactions:
␣
Alpha particle
Beta particle
␤⫺, ␤⫹
Deuteron
d, 2H
Electron
e, e⫺
Helion
h
Muon, negative ␮⫺
Muon, positive
Neutron
Photon
Proton
Triton
SI unit
mol · m⫺2
⌫B ⫽ nB/A
␾B ⫽ VB/⌺Vi
␮⫹
n, n0
␥
p, p⫹
t, 3H
The meaning of the symbolic expression indicating a nuclear reaction:
initial
incoming particles outgoing particles
,
nuclide or quanta
or quanta
冉
Definition
冊
final
nuclide
Examples: 14N(␣, p)17O, 23Na(␥, 3n)20Na
States of aggregation:
am
amorphous solid
aq
aqueous solution
as, ⬁ aqueous solution at
infinite dilution
g
gas
l
liquid
s
solid
sln
solution
cd
cr
fl
lc
vit
mon
pol
ads
condensed phase (solid or liquid)
crystalline
fluid phase (gas or liquid)
liquid crystal
vitreous substance
monomeric form
polymeric form
species adsorbed on a substance
C. Chromatography
Name
Symbol
Adjusted retention time
Adjusted retention volume
Average linear gas velocity
Band variance
Bed volume
Capacity, volume
Capacity, weight
Column length
Column temperature
Column volume
Concentration at peak maximum
Concentration of solute in mobile
phase
Concentration of solute in stationary phase
Density of liquid phase
Diffusion coefficient, liquid film
Diffusion coefficient, mobile
phase
t⬘R
V⬘R
␮
␴2
Vg
Qv
Qw
L
␪
Vcol
Cmax
CM
Definition
t⬘R ⫽ tR ⫺ tM
V⬘R ⫽ VR ⫺ VM
␮ ⫽ L/tM
Vcol ⫽ ␲Dd2c /4
CS
␳L
Df
DM
short
standard
long
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TABLE 2.2 Physical and Chemical Symbols and Definitions (Continued)
C. Chromatography (continued)
Name
Diffusion coefficient, stationary
phase
Distribution ratio
Symbol
Definition
DS
Dc
⫽ [A⫹]S/[A⫹]M
⫽
amount of A per cm3 stationary phase
amount of A per cm3 of mobile phase
Dg
⫽
amount A per gram dry stationary phase
amount A per cm3 of mobile phase
Dv
⫽
amount A, stationary phase per cm3 bed volume
amount A per cm3 of mobile phase
DS
⫽
amount of A per m2 of surface
amount of A per cm3 of mobile phase
Elution volume, exclusion chromatography
Flow rate, column
Gas/liquid volume ratio
Inner column volume
Interstitial (outer) volume
Kovats retention indices
Matrix volume
Net retention volume
Obstruction factor
Packing uniformity factor
Particle diameter
Partition coefficient
Partition ratio
Peak asymmetry factor
Peak resolution
Plate height
Plate number
Porosity, column
Pressure, column inlet
Presure, column outlet
Pressure drop
Fc
␤
Vi
Vo
RI
Vg
VN
␥
␭
dp
K
k⬘
AF
Rs
H
Neff
⑀
pi
Po
⌬P
Pressure-gradient correction
j
j⫽
Recovery factor
Reduced column length
Reduced plate height
Reduced velocity
Relative retention ratio
Retardation factor c
Retention time
Retention volume
Selectivity coefficient d
Separation factor
Rn
␭
h
v
␣
Rf
tR
VR
kA,B
␣A/B
Rn ⫽ 1 ⫺ (rDc ⫹ 1)⫺n; r ⫽ Vorg/Vaq
␭ ⫽ L/dp
h ⫽ H/dp
v ⫽ ␮dp/DM ⫽ Kdp/tMDM
␣ ⫽ (k⬘/k⬘)
2 1
Rf ⫽ dsolute/dmobile phase
tR ⫽ tM(1 ⫹ k⬘) ⫽ L/␮
VR ⫽ tRFc
kA,B ⫽ [A⫹]r[B⫹]/[B⫹]r[A⫹]
␣A/B ⫽ (Dc)A/(Dc)B
c
Ve
Fc ⫽ (␲d2c /4)(⑀tot)(L/tM)
VN ⫽ jV⬘R
dp ⫽ L/Nh
K ⫽ CS /CM ⫽ (VR ⫺ VM)/VS
k⬘ ⫽ CSVS /CMVM ⫽ K(VS /VM)
Ratio of peak half-widths at 10% peak height
Rs ⫽ (tR,2 ⫺ tR,1)/0.5(W2 ⫹ W1)
H ⫽ L/Neff
Neff ⫽ L/H ⫽ 16(t⬘R /Wb)2 ⫽ 5.54(t⬘R /W1/2)2
3[(pi /po)2 ⫺ 1]
2[(pi /po)3 ⫺ 1]
The distance d corresponds to the movement of solute and mobile phase from the starting (sample spotting) line.
Subscript “r” represents an ion-exchange resin phase. Two immiscible liquid phases might be represented similarly using
subscripts “1” and “2.”
d
short
standard
long
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TABLE 2.2 Physical and Chemical Symbols and Definitions (Continued)
C. Chromatography (continued)
Name
Symbol
Specific retention volume
Thickness (effective) of stationary phase
Total bed volume
Transit time of nonretained
solute
Vapor pressure
Volume liquid phase in column
Volume mobile phase in column
Weight of liquid phase
Zone width at baseline
Zone width at 1⁄2 peak height
Definition
V⬚g ⫽ 273 R/(p⬚ MwL)
V⬚g
df
Vtot
tM, t0
p
VL
VM
wL
Wb
W1/2
Wb ⫽ 4␴
D. Colloid and surface chemistry
Name
Adsorbed amount of B
Area per molecule
Area per molecule in a
filled monolayer
Average molar masses:
Mass-average
Number-average
Z-average
Contact angle
Film tension
Film thickness
Reciprocal thickness of the
double layer
Retarded van der Waals
constant
Sedimentation coefficiente
Specific surface area
Surface coverage
Surface excess of B
Surface pressure
Surface tension, interfacial
tension
Thickness of (surface or interfacial) layer
Total surface excess concentration
van der Waals constant
van der Waals-Hamaker
constant
e
Symbol
SI unit
nsB
a, ␴
am
mol
m2
m2
Mm
Mn
MZ
␪
⌺f
t, h, ␦
␬
kg · mol⫺1
kg · mol⫺1
kg · mol⫺1
rad
N · m⫺1 ␾
m
m⫺1
␤, B
J
s
a, s, as
␪
n␴B
␲s, ␲
␥, ␴
s
m2/kg
␶, ␦, t
m
⌫
mol · m⫺2
␭
AH
J
J
mol
N · m⫺1
J · m⫺2
Definition
aB ⫽ A/N␴N
am,B ⫽ A/Nm,B
Mm ⫽ ⌺niM2i /⌺niMi
Mn ⫽ ⌺niMi /⌺ni
MZ ⫽ ⌺niM3i /⌺niM2i
⌺f ⫽ 2␥f
␬ ⫽ [2F2Ic/⑀RT]1/2
s ⫽ v/a
a ⫽ A/m
␪ ⫽ N␴B/NB
␲s ⫽ ␥0 ⫺ ␥
␥ ⫽ (⭸G/⭸As)T,p
⌫ ⫽ ⌺⌫i
v is the velocity of sedimentation and a is the acceleration of free fall or centrifugation.
short
standard
long
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TABLE 2.2 Physical and Chemical Symbols and Definitions (Continued)
E. Electricity and magnetism
Name
Symbol
SI unit
Definition
Admittance
Capacitance
Charge density
Conductance
Conductivity
Dielectric polarization (dipole moment per volume)
Electrical resistance
Electric current
Y
C
␳
G
␬
P
S
F, C · V⫺1
C · m ⫺3
S
S · m ⫺1
C · m ⫺2
Y ⫽ 1/Z
C ⫽ Q/U
␳ ⫽ Q/V
G ⫽ 1/R
␬ ⫽ 1/␳
P ⫽ D ⫺ ⑀0E
R
I
⍀
A
R ⫽ U/I ⫽ ⌬V/I
I ⫽ dQ/dt
Electric current density
j, J
A · m⫺2
I⫽
Electric dipole moment
Electric displacement
Electric field strength
p, ␮
D
E
C·m
C · m ⫺2
V · m⫺1
p ⫽ Qr
D ⫽ ⑀E
E ⫽ F/Q ⫽ ⫺grad V
Electric flux
⌿
C
⌿⫽
Electric potential
Electric potential difference
Electric susceptibility
V, ␾
U, ⌬V
␹e
V, J · C⫺1
V
V ⫽ dW/dQ
U ⫽ V2 ⫺ V1
␹e ⫽ ⑀r ⫺ 1
Electromotive force
E
V
E⫽
Impedance
Loss angle f
Magnetic dipole moment
Magnetic field strength
Z
␦
m, ␮
H
⍀
rad
A · m2
A · m⫺1
Z ⫽ R ⫹ iX
␦ ⫽ (␲/2) ⫹ ␾I ⫺ ␾U
Ep ⫽ ⫺mB
B ⫽ ␮H
Magnetic flux
⌽
Wb
⌽⫽
Magnetization (magnetic dipole moment per volume)
Magnetic susceptibility
Magnetic vector potential
Molar magnetic susceptibility
Mutual inductance
Permeability
Permeability of vacuum
Permittivity
Permittivity of vacuum
Poynting vector
Quantity of electricity, electric charge
Reactance
Relative permeability
Relative permittivityg
Resistivity
Self-inductance
Susceptance
M
A · m⫺1
M ⫽ (B/␮0) ⫺ H
␹, ␬
A
␹m
Wb · m⫺1
n3/mol
␹ ⫽ ␮r ⫺ 1
B ⫽ ⵜA
␹m ⫽ Vm ␹
M, L12
␮
␮0
⑀
⑀0
S
Q
H
H · m⫺1
H · m⫺1
F · m ⫺1
F · m ⫺1
W · m⫺2
C
E1 ⫽ L12(dI2/dt)
B ⫽ ␮H
X
␮r
⑀r
␳
L
B
⍀
X ⫽ (U/I) sin ␦
␮r ⫽ ␮/␮0
⑀r ⫽ ⑀/⑀0
␳ ⫽ E/j
E ⫽ ⫺L(dI/dt)
Y ⫽ G ⫹ iB
f
g
⍀·m
H
S
␾I and ␾U are the phases of current and potential difference.
This quantity was formerly called the dielectric constant.
冕
j dA
冕
冕
冕
D dA
(F/Q) ds
B dA
D ⫽ ⑀E
⑀0 ⫽ ␮0⫺1 c0⫺2
S⫽EⴢH
short
standard
long
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TABLE 2.2 Physical and Chemical Symbols and Definitions (Continued)
F. Electrochemistry
Name
Symbol
SI unit
Definition
␴
z
n, (z)
C · n⫺2
␴ ⫽ Q/A
zB ⫽ QB/e
␬
S · m ⫺1
␬ ⫽ j/E
Kcell
j
kd
m⫺1
A · m⫺2
m · s⫺1
Kcell ⫽ ␬R
j ⫽ I/A
kd,B ⫽ |␯B|I1,B/nFcA
I
␮
⌬V, E, U
A
m2 · V⫺1 · s⫺1
V
I ⫽ dQ/dt
␮B ⫽ vB/E
⌬V ⫽ VR ⫺ VL
␮˜
J · mol⫺1
␮˜ ␣B ⫽ (⭸G/⭸n␣B)
Electrode reaction rate constant
Electrokinetic potential
(zeta potential)
Elementary charge (proton
charge)
emf, electromotive force
k
(varies)
kox ⫽ Ia/ nFA
Charge density (surface)
Charge number of an ion
Charge number of electrochemical cell reaction
Conductivity (specific conductance)
Conductivity cell constant
Current density (electric)
Diffusion rate constant,
mass transfer coefficient
Electric current
Electric mobility
Electric potential difference
(of a galvanic cell)
Electrochemical potential
冉
兿c
冊
ni
i
i
␨
V
e
C
E
V
E ⫽ lim ⌬V
emf of the cell
Faraday constant
Galvani potential difference
Inner electrode potential
Ionic conductivity
Ionic strength
Mean ionic activity
Mean ionic activity coefficient
Mean ionic mobility
Molar conductivity (of an
electrolyte)
E
F
⌬␾
␾
␭
Ic, I
a⫾
␥⫾
V
C · mol⫺1
V
V
S · m2 · mol⫺1
mol · m⫺3
E ⫽ E⬚ ⫺ (RT/nF) ⫻ ⌺␯i ln ai
F ⫽ eL
⌬␤␣␾ ⫽ ␾␤ ⫺ ␾␣
ⵜ␾ ⫽ ⫺E
␭B ⫽ |zB|FuB
Ic ⫽ 1⁄2⌺cizc2
a⫾ ⫽ m⫾␥⫾/m0
␥ (⫾␯⫹ ⫹ ␯⫺) ⫽ (␥␯⫾⫹)(␥␯⫺⫺)
m⫾
⌳
mol · kg⫺1
S · m⫺2 mol⫺1
m(⫾␯⫹ ⫹ ␯⫺) ⫽ (m␯⫹⫹)(m␯⫺⫺)
⌳B ⫽ ␬cB
pH
pH
Outer electrode potential
Overpotential
Reciprocal radius of ionic
atmosphere
Standard emf, standard potential of electrochemical
cell reaction
Surface electric potential
Thickness diffusion layer
␺
␩
␬
V
V
m⫺1
␺ ⫽ Q/4␲⑀0r
␩ ⫽ EI ⫺ EI ⫽ 0 ⫺ IRu
␬ ⫽ (2F2I/⑀RT)1/2
E0
V
E0 ⫽ ⫺⌬rG0/nF ⫽ (RT/nF) ln K
␹
␦
V
m
␹⫽␾⫺␺
␦B ⫽ DB/kd,B
Transfer coefficient
␣
Transport number
Volta potential difference
t
⌬␺
I: 0
pH ⯝ ⫺log
␣c ⫽
V
冋
c(H⫹)
mol · dm⫺3
册
⫺|␯|RT ⭸ln|Ic|
nF
⭸E
tB ⫽ jB/⌺ji
⌬␤␣ ⫽ ␺␤ ⫺ ␤␣
short
standard
long
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GENERAL INFORMATION, CONVERSION TABLES, AND MATHEMATICS
2.13
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TABLE 2.2 Physical and Chemical Symbols and Definitions (Continued)
G. Electromagnetic radiation (continued)
Name
Symbol
Absorbance
Absorbance (decaidic)
Absorbance (napierian)
Absorption coefficient:
Linear (decaidic)
Linear (napierian)
Molar (decaidic)
Molar (napierian)
Absorption index
Angle of optical rotation
Circular frequency
Complex refractive index
Concentration, amount of
substance
Concentration, mass
Einstein transition probabilities:
Spontaneous emission
Stimulated absorption
Stimulated emission
Emittance
By blackbody
First radiation constant
Frequency
Irradiance (radiant flux received)
␣
A
B
SI unit
Definition
␣ ⫽ ⌽abs /⌽0
A ⫽ ⫺log(1 ⫺ ␣i )
B ⫽ ⫺ln(1 ⫺ ␣i )
a ⫽ A/l
␣ ⫽ B/l
⑀ ⫽ a/d ⫽ A/cl
␬ ⫽ ␣/c ⫽ B/cl
k ⫽ ␣/4␲␯˜
a, K
␣
⑀
␬
k
␣
␻
n̂
c
m⫺1
m⫺1
m2 · mol⫺1
m2 · mol⫺1
␥
kg · m3
Anm
Bmn
Bnm
⑀
Mbb
c1
␯
E, (I)
s⫺1
s · kg⫺1
s · kg⫺1
dNn /dt ⫽ ⫺Anm Nn
dNn /dt ⫽ ␳␯˜ (␯ñm)BmnNm
dNn /dt ⫽ ␳ṽ(␯ñm)BmnNm
⑀ ⫽ M/Mbb
W · m2
Hz
W · m⫺2
c1 ⫽ 2␲hc20
␯ ⫽ c/␭
E ⫽ d⌽/dA
Molar refraction
R, Rm
m3 · mol⫺1
R⫽
Path length (absorbing)
Optical rotatory power
Planck constant
Planck constant/2␲
Radiant energy
Radiant energy density
Radiant exitance, emitted
radiant flux
Radiant intensity
Radiant power, radiant energy per time
Refractive index
Reflectance
Second radiation constant
Spectral radiant energy
density:
In terms of frequency
In terms of wavelength
In terms of wavenumber
l
[␣]␪␭
h
ប
Q, W
␳, w
M
m
rad
J·s
J·s
J
J · m⫺3
W · m⫺2
I
⌽, P
W · sr⫺1
W
I ⫽ d⌽/d⍀
⌽ ⫽ dQ/dt
n
␳
c2
K·m
n ⫽ c0 /c
␳ ⫽ ⌽refl /⌽0
c2 ⫽ hc0 /k
␳␯ , w␯
␳␭ , w ␭
␳␯˜ , w␯˜
J · m⫺3 · Hz⫺1
J · m⫺4
J · m⫺2
␳␯ ⫽ d␳/d␯
␳␭ ⫽ d␳/d␭
␳␯˜ ⫽ d␳/d␯˜
rad
s⫺1, rad · s⫺1
␻ ⫽ 2␲␯
n̂ ⫽ ␩ ⫹ ik
mol · m3
(n2 ⫺ 1)
Vm
(n2 ⫹ 2)
[␣]␪␭ ⫽ ␣/␥l
ប ⫽ h/2␲
␳ ⫽ Q/V
M ⫽ d⌽/dAsource
short
standard
long
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(McGHP)
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TABLE 2.2 Physical and Chemical Symbols and Definitions (Continued)
G. Electromagnetic radiation (continued)
Name
Speed of light:
In a medium
In vacuum
Stefan-Boltzmann constant
Transmittance
Wavelength
Wavenumber:
In a medium
In vacuum
Symbol
SI unit
c
c0
␴
␶, T
␭
m · s⫺1
m · s⫺1
W · m⫺2 · K⫺4
␴
␯˜
m⫺1
m⫺1
Definition
c ⫽ c0 /n
Mbb ⫽ ␴T 4
␶ ⫽ ⌽tr /⌽0
m
␴ ⫽ 1/␭
␯˜ ⫽ ␯/c0 ⫽ 1/n␭
H. Kinetics
Name
Activation energy
Boltzmann constant
Collision cross section
Collision diameter
Collision frequency
Collision frequency factor
Collision number
Half-life
Overall order of reaction
Partial order of reaction
Pre-exponential factor
Quantum yield, photochemical yield
Rate of change of quantity
X
Rate of concentration
change (chemical reaction)
Rate constant, rate coefficient
Rate of conversion change
due to chemical reaction
Rate of reaction (based on
concentration)
Relaxation time
Standard enthalpy of activation
Standard entropy of activation
Standard Gibbs energy of
activation
Volume of activation
Symbol
SI unit
Definition
Ea ⫽ RT 2 d ln k/dT
Ea , E
k, k B
␴
d
ZA
zAB , zAA
ZAB , ZAA
t1/2
n
nB
A
␾
J · mol⫺1
J · K⫺1
m2
m
s⫺1
m3 · mol⫺1 · s⫺1
m⫺3 · s⫺1
s
(mol⫺1 · m3)n⫺1 · s⬃1
c(t1/2 ) ⫽ c0 /2
n ⫽ ⌺nB
v ⫽ k⌸cnBB
k ⫽ A exp(⫺Ea /RT)
Ẋ
(varies)
Ẋ ⫽ dX/dt
rB , vB
mol · m⫺3 · s⫺1
rB ⫽ dcB /dt
k
(mol⫺1 · m3)n⫺1 · s⫺1
v ⫽ k⌸cnBB
␨˙
mol · s⫺1
␨˙ ⫽ d␨ /dt
v
mol · m⫺3 · s⫺1
v ⫽ ␨ /V ⫽ ␯⫺B 1 dcB/dt
␶
⌬H⫹⫹
s
J · mol⫺1
␶ ⫽ 1/(k1 ⫹ k⫺1 )
⌬S⫹⫹
J · mol⫺1 · K⫺1
⌬G⫹⫹
J · mol⫺1
⌬ ⫹⫹V
m3 · mol⫺1
␴AB ⫽ ␲d2AB
dAB ⫽ rA ⫹ rB
zAB ⫽ ZAB /LcAcB
⌬ ⫹⫹V ⫽ ⫺RT (⭸ln k/⭸p)T
short
standard
long
DEAN #37261
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RIGHT
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GENERAL INFORMATION, CONVERSION TABLES, AND MATHEMATICS
2.15
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TABLE 2.2 Physical and Chemical Symbols and Definitions (Continued)
I. Mechanics
Name
Acoustic factors:
Absorption
Dissipation
Reflection
Transmission
Angular momentum
Bulk modulus, compression
modulus
Density, mass density
Energy
Fluidity, kinematic viscosity
Force
Friction coefficient
Gravitational constant
Hamilton function
Kinematic viscosity
Kinetic energy
Lagrange function
Linear strain, relative elongation
Mass
Modulus of elasticity,
Young’s modulus
Moment of inertia
Momentum
Normal stress
Symbol
SI unit
Definition
␣a
␦
␳
␶
L
K
J·s
Pa
␳
E
␾
kg · m⫺3
J
m · kg⫺1 · s
␳ ⫽ m/V
F
␮, ( f )
G
H
␯
Ek
L
⑀, e
N
F ⫽ dp/dt ⫽ ma
Ffrict ⫽ ␮ Fnorm
F ⫽ Gm1 m2 /r2
H(q, p) ⫽ T(q, p) ⫹ V(q)
␯ ⫽ ␩ /␳
Ek ⫽ 1⁄2 mv2
L(q, q̇) ⫽ T(q, q̇) ⫺ V(q)
⑀ ⫽ ⌬l/l
m
E
kg
Pa
E ⫽ ␴/⑀
I, J
p
␴
kg · m2
kg · m · s⫺1
Pa
I ⫽ ⌺m iri2
p ⫽ m␷
␴ ⫽ F/A
Potential energy
Ep
J
Ep ⫽
Power
Pressure
Reduced mass
Relative density
Shear modulus
Shear strain
Shear stress
Sound energy flux
Specific volume
Surface density
Surface tension
Torque, moment of a force
Viscosity (dynamic)
Volume (or bulk) strain
Weight
P
p, P
␮
d
G
␥
␶
P, Pa
␷
␳A , ␳S
␥, ␴
T, (M)
␩, ␮
␪
G, (W, P)
W
Pa, N · m⫺2
kg
N
P ⫽ dW/dt
p ⫽ F/A
␮ ⫽ m1 m2 /(m1 ⫹ m2 )
d ⫽ ␳/␲0
G ⫽ ␶/␥
␥ ⫽ ⌬x/d
␶ ⫽ F/A
P ⫽ dE/dt
␷ ⫽ V/␮ ⫽ 1/␳
␳A ⫽ m/A
␥ ⫽ dW/dA
TⴝrⴛF
␶ x,z ⫽ ␭(d␷x /dz)
␪ ⫽ ⌬V/V0
G⫽m·g
Work
W, w
J
W⫽
N · m2 · kg⫺2
J
m2 · s⫺1
J
J
Pa
Pa
W
m3 · kg⫺1
kg · m⫺2
N · m⫺1, J · m⫺2
N·m
Pa · s
␣a ⫽ 1 ⫺ ␳
␦ ⫽ ␣a ⫺ ␶
␳ ⫽ Pr /P0
␶ ⫽ Ptr /P0
L⫽rⴛp
K ⫽ ⫺V0(dp/dV)
␾ ⫽ 1/␩
冕
冕
⫺ F · ds
F ⴢ ds
short
standard
long
DEAN #37261
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2.16
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TABLE 2.2 Physical and Chemical Symbols and Definitions (Continued)
J. Solid state
Name
Acceptor ionization energy
Bragg angle
Bloch function
Burgers vector
Charge density of electrons
Circular wave vector:
For particles (k)
For phonons (q)
Conductivity tensor
Curie temperature
Debye circular frequency
Debye circular wavenumber
Debye-Waller factor
Density of states
Density of vibrational
modes (spectral)
Diffusion coefficient
Diffusion length
Displacement vector of an
ion
Donor ionization energy
Effective mass
Equilibrium position vector
of an ion
Fermi energy
Gap energy
Grüneisen parameter
Hall coefficient
Lattice plane spacing
Lattice vector
Lorenz coefficient
Symbol
SI unit
Definition
Ea
␪
uk(r)
b
␳
J
rad
m⫺3/2
m
C · m⫺3
k, q
m⫺1
k ⫽ 2␲/␭
␴ik
TC
␻D
qD
B, D
NE
N␻, g
S · m⫺1
K
s⫺1
m⫺1
␴ ⫽ ␳⫺1
J⫺1 · m⫺3
s · m⫺3
NE ⫽ dN(E)/dE
N␻ ⫽ dN(␻)/d␻
D
L
u
m2 · s⫺1
m
m
dN/dt ⫽ ⫺DA dn/dx
L ⫽ (D␶)1/2
u ⴝ R ⴚ R0
Ed
m*
R0
J
kg
m
EF
Eg
␥, ⌫
AH , RH
d
R, R0
L
J
Madelung constant
␣
Mobility
Mobility ratio
Neel temperature
Number density, number
concentration
Order parameters:
Long range
Short range
Order of reflection
Particle position vector:
Electron
Ion position
Peltier coefficient
Reciprocal lattice vector
(circular)
␮
b
TN
n
m3 · C⫺1
m
m
V2 · K⫺2
n␭ ⫽ 2d sin ␪
␺(r) ⫽ uk(r) exp(ik · r)
␳(r) ⫽ ⫺e␺*(r)␺(r)
␥ ⫽ ␣V/␬CV
E ⴝ ␳ ⴢ j ⴙ RH(B ⴛ j)
L ⫽ ␭/␴T
Ecoul ⫽
m2 · V⫺1 · s⫺1
␣ NAz⫹z⫺e2
4␲⑀0R0
␮ ⫽ ␷drift /E
b ⫽ ␮ n /␮ p
K
m⫺3
s
␴
n
r
Rj
⌸
G
m
m
V
m⫺1
G ⴢ R ⴝ 2␲ m
short
standard
long
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TABLE 2.2 Physical and Chemical Symbols and Definitions (Continued)
J. Solid state (continued)
Name
Symbol
SI unit
Definition
Relaxation time
Residual resistivity
Resistivity tensor
Temperature
Thermal conductivity tensor
Thermoelectric force
Thomson coefficient
Translation vectors for the
reciprocal lattice (circular)
Translation vectors for
crystal lattice
Work function
␶
␳R
␳
␪
␭
E
␮
b1; b2; b3
a*; b*; c*
s
m
⍀·m
K
W · m⫺1 · K⫺1
V
V · K⫺1
m⫺1
␶ ⫽ 1/␷F
a1; a2; a3
a; b; c
⌽
m
R ⫽ n1a1 ⫹ n2a2 ⫹ n3a3
J
⌽ ⴝ E⬁ ⴚ EF
Eⴝ␳ⴢj
Jq ⴝ ⴚ ␭ ⴢ grad T
a i ⴢ bk ⫽ 2␲␦ik
K. Space and time
Name
Acceleration
Angular velocity
Area
Breadth
Cartesian space coordinates
Circular frequency, angular
frequency
Diameter
Distance
Frequency
Generalized coordinate
Height
Length
Length of arc
Path length
Period
Plane angle
Position vector
Radius
Relaxation time, time constant
Solid angle
Speed
Spherical polar coordinates
Thickness
Time
Velocity
Volume
Symbol
SI unit
a, (g)
␻
A, As , S
b
x, y, z
␻
m · s⫺2
rad · s⫺1, s⫺1
m2
m
m
rad · s⫺1, s⫺1
d
d
␯, f
q, q i
h
l
s
s
T
␣, ␤, ␥,
␪, ␾
r
r
␶, T
m
m
Hz
(varies)
m
m
m
m
s
rad, l
␻, ⍀
␷, u, w, c
r, ␪, ␾
d, ␦
t
␷, u, w, c
V, (␷)
sr, l
m · s⫺1
m, l, l
m
s
m·s
m3
m
m
s
Definition
a ⫽ d␷/dt
␻ ⫽ d␾/dt
␻ ⫽ 2␲␯
␯ ⫽ 1/T
T ⫽ t/N
␣ ⫽ s/r
r ⫽ xi ⫹ yj ⫹ zk
␶ ⫽ |dt/d ln x|
⍀ ⫽ A/r2
␷ ⫽ |␷ |
␷ ⫽ dr/dt
short
standard
long
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TABLE 2.2 Physical and Chemical Symbols and Definitions (Continued)
L. Spectroscopy
Name
Asymmetry parameter
Symbol
␬
Centrifugal distortion constants:
A reduction
⌬ J ⌬ JK ⌬ K ␦J ␦K
S reduction
DJ DJK DK d1 d2
Degeneracy, statistical
g, d, ␤
weight
Electric dipole moment of a
p, ␮
molecule
Electron spin resonance
(ESR), electron paramagnetic resonance (EPR):
Hyperfine coupling constant:
In liquids
a, A
In solids
T
g factor
g
Electronic term
Te
Harmonic vibration wave␻e; ␻r
number
Inertial defect
⌬
Interatomic distances:
Equilibrium distance
re
Ground state distance
r0
Substitution structure disrs
tance
Zero-point average disrz
tance
Longitudinal relaxation
T1
time
␦
Nuclear magnetic resonance
(NMR), chemical shift, ␦
scale
Coupling constant, direct
DAB
(dipolar)
Magnetogyric ratio
␥
Shielding constant
␴A
Spin-spin coupling conJAB
stant
Principal moments of inerIA ; IB ; IC
tia
Rotational constants:
In frequency
A; B; C
In wavenumber
Ã; B̃; C̃
Rotational term
F
Spin orbit coupling constant
A
Total term
T
Transition dipole moment
of a molecule
SI unit
M, R
Definition
␬⫽
2B ⫺ A ⫺ C
A⫺C
m⫺1
C·m
E p ⫽ ⴚp ⴢ E
Hz
Hz
Ĥhfs /h ⫽ aŜ ⴢ I
Ĥhfs /h ⫽ Ŝ ⴢ T ⴢ I
h␯ ⫽ g␮ B B
Te ⫽ Ee /hc
m⫺1
m⫺1
kg · m2
⌬ ⫽ IC ⫺ IA ⫺ IB
m
m
m
m
s
␦ ⫽ 106(␯ ⫺ ␯0 )/␯0
Hz
Hz
␥ ⫽ 2␲␮ /Ih
BA ⫽ (1 ⫺ ␴A )B
H/h ⫽ JAB IA · IB
kg · m2
IA ⱕ IB ⱕ IC
Hz
m⫺1
m⫺1
m⫺1
m⫺1
A ⫽ h/8␲ 2IA
Ã ⫽ h/8␲ 2IA
F ⫽ Erot /hc
Ts.o. ⫽ A 具 L̃ · S̃ 典
T ⫽ Etot /hc
C·m
M⫽
C · kg⫺1
冕
␺ ⬘p␺ ⬙ d␶
short
standard
long
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TABLE 2.2 Physical and Chemical Symbols and Definitions (Continued)
L. Spectroscopy (continued)
Name
Transition frequency
Transition wavenumber
Transverse relaxation time
Vibrational anharmonicity
constant
Vibrational coordinates:
Internal coordinates
Normal coordinates, dimensionless
Mass adjusted
Vibrational force constants:
Diatomic
Polyatomic
Dimensionless normal
coordinates
Internal coordinates
Symmetry coordinates
Vibrational quantum numbers
Vibrational term
Symbol
␯
␯˜, (␯)
T2
␻e ␹ e; ␹ rs ;
gtt⬘
SI unit
Definition
Hz
m⫺1
s
m⫺1
␯ ⫽ (E⬘ ⫺ E ⬙)/h
␯˜ ⫽ T⬘ ⫺ T ⬙
f, (k)
J · m⫺2
f ⫽ ⭸2V/⭸r2
␾rst . . . ,
krst . . .
f ij
Fij
vr ; lt
m⫺1
(varies)
(varies)
f ij ⫽ ⭸2V/⭸ri ⭸rj
Fij ⫽ ⭸2V/⭸Si⭸Sj
G
m⫺1
G ⫽ Evib /hc
R i , ri , ␪ j ,
etc.
qr
Qr
Quantum number symbol
Angular momentum types
Electron orbital
One electron only
Electron orbital ⫹ spin
Electron spin
One electron only
Internal vibrational:
Spherical top
Other
Nuclear orbital (rotational)
Nuclear spin
Sum of J ⫹ I
Sum of N ⫹ S
Sum of R ⫹ L(⫹j)
Operator
symbol
L̂
l̂
Lˆ ⴙ Sˆ
Ŝ
ŝ
Total
Z axis
L
l
ML
ml
S
s
MS
ms
l(l␨)
l̂
ĵ, ␲ˆ
R̂
Î
F̂
Ĵ
N̂
R
I
F
J
N
z axis
⌳
␭
⍀⫽⌳⫹⌺
␴
⌺
Kl
l(l␨)
KR, kR
MI
MF
MJ
K, k
K, k
M. Thermodynamics
Name
Symbol
SI unit
Definition
Absolute activity
Activity (referenced to
Henry’s law):
␭
␭ B ⫽ exp( ␮ B /RT)
Concentration basis
ac
a c,B ⫽ exp
冋
␮ B ⫺ ␮ B*
RT
册
short
standard
long
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TABLE 2.2 Physical and Chemical Symbols and Definitions (Continued)
M. Thermodynamics (continued)
Name
Symbol
SI unit
Definition
冋
冋
冋
册
册
册
Molality basis
am
a m,B ⫽ exp
␮ B ⫺ ␮ B*
RT
Mole fraction basis
ax
a x,B ⫽ exp
␮ B ⫺ ␮ B*
RT
a
a B ⫽ exp
␥c
␥m
␥x
f
a c,B ⫽ ␥c,B cB /c0
a m,B ⫽ ␥m,B mB /m0
a x,B ⫽ ␥x,B xB
f B ⫽ a B /xB
Activity (referenced to
Raoult’s law)
Activity coefficient (referenced to Henry’s law):
Concentration basis
Molality basis
Mole fraction basis
Activity coefficient (referenced to Raoult’s law)
Affinity of reaction
Celsius temperature
Chemical potential
Compressibility:
Isentropic
Isothermal
Compressibility factor
Cubic expansion coefficient
Enthalpy
Entropy
Equilibrium constant
Equilibrium constant:
Concentration basis
Molality basis
Pressure basis
Fugacity
␮ B ⫺ ␮ B*
RT
A
␪, t
␮
J · mol⫺1
⬚C
J · mol⫺1
A ⫽ ⫺(⭸G/⭸␰)p,T
␪/⬚C ⫽ T/K ⫺ 273.15
␮ B ⫽ (⭸G/⭸n B )T,p,n
␬S
␬T
Z
␣, ␣ V , ␥
H
S
K 0, K
Pa⫺1
Pa⫺1
␬ S ⫽ ⫺(1/V)(⭸V/⭸p)S
␬ T ⫽ ⫺(1/V)(⭸V/⭸p)T
Z ⫽ pVm /RT
␣ ⫽ (1/V)(⭸V/⭸T)p
H ⫽ U ⫹ pV
dS ⱖ dq/T
K⬚ ⫽ exp(⫺⌬ r G⬚/RT)
Kc
Km
Kp
f
K⫺1
J
J · K⫺1
(mol · m⫺3)⌺␯
(mol · m⫺1)⌺␯
Pa⌺␯
Pa
K c ⫽ ⌸ c␯
K m ⫽ ⌸ m␯
K p ⫽ ⌸ p␯
f B ⫽ ␭ B lim (p B /␭ B )T
p:0
Fugacity coefficient
Gibbs energy
Heat
Heat capacity:
At constant pressure
At constant volume
Helmholtz energy
Internal energy
Ionic strength:
Concentration basis
Molality basis
Joule-Thomson coefficient
Linear expansion coefficient
Massieu function
Molar quantity X
Osmotic coefficient:
Molality basis
Mole fraction basis
␾
G
q, Q
J
J
␾ B ⫽ f B /p B
G ⫽ H ⫺ TS
Cp
CV
A
U
J · K⫺1
J · K⫺1
J
C p ⫽ (⭸H/⭸T)p
C V ⫽ (⭸U/⭸T)V
A ⫽ U ⫺ TS
⌬U ⫽ q ⫹ w
Ic, I
Im, I
␮ , ␮ JT
␣1
mol · kg⫺3
mol · kg⫺1
K · Pa⫺1
K⫺1
I c ⫽ 1⁄2⌺m B zB2
I m ⫽ 1⁄2⌺m B z2B
␮ ⫽ (⭸T/⭸p) H
␣ 1 ⫽ (l/l)(⭸l/T)
J
Xm
J · K⫺1
(varies)
J ⫽ ⫺A/T
X m ⫽ X/n
␾m
␾x
␾ m ⫽ ( ␮ A* ⫺ ␮ A )/(RTMA ⌺m B )
␾ x ⫽ ( ␮ A ⫺ ␮ A*)/(RT ln x A )
short
standard
long
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TABLE 2.2 Physical and Chemical Symbols and Definitions (Continued)
M. Thermodynamics (continued)
Name
Symbol
SI unit
Definition
Osmotic pressure (ideal dilute solution)
Partial molar quantity X
Planck function
Pressure coefficient
Ratio of heat capacities
Relative pressure coefficient
Second virial coefficient
Specific quantity X
Standard chemical potential
Standard partial molar enthalpy
Standard partial molar entropy
Standard reaction enthalpy
Standard reaction entropy
Standard reaction Gibbs energy
Surface tension
⌸
Pa
⌸ ⫽ c B RT
XB
Y
␤
␥
␣p
(varies)
J · K⫺1
Pa · K⫺1
X ⫽ (⭸X/⭸n B )T,p,n
Y ⫽ ⫺G/T
␤ ⫽ (⭸P/⭸T)V
␥ ⫽ C p /C V
␣ p ⫽ (1/p)(⭸p/⭸T)V
B
x
␮0
H0
m3 · mol⫺1
(varies)
J · mol⫺1
J · mol⫺1
pVm ⫽ RT(1 ⫹ B/Vm ⫹ · · ·)
x ⫽ X/m
S0
J · mol⫺1 · K⫺1
S 0 ⫽ ⫺(⭸␮ 0/⭸T)
⌬rH0
⌬rS0
⌬rG0
J · mol⫺1
J · mol⫺1 · K⫺1
J · mol⫺1
⌬ r H 0 ⫽ ⌺␯H 0
⌬ r S 0 ⫽ ⌺ ␯S 0
⌬ r G 0 ⫽ ⌺␯␮ 0
␥, ␴
␥ ⫽ (⭸G/⭸A s )T,p
Thermodynamic temperature
Work
T
J · m⫺2,
N · m⫺1
K
w, W
J
K⫺1
H 0 ⫽ ␮ 0 ⫹ TS
Symbols used as subscripts to denote a chemical reaction or process:
ads adsorption
mix mixing of fluids
at
atomization
r
reaction in general
c
combustion reaction
sol solution of solute in solvent
dil dilution of a solution
sub sublimation (solid to gas)
f
formation reaction
trs
transition (two phases)
fus melting, fusion (solid to liquid)
Recommended superscripts:
⫹
activated complex, transition state
⫹
E
excess quantity
id
ideal
⬁
*
⬚
infinite solution
pure substance
standard
N. Transport properties
Name
Coefficient of heat transfer
Diffusion coefficient
Flux (of a quantity X)
Heat flow rate
Heat flux
Mass flow rate
Mass transfer coefficient
Thermal conductance
Thermal conductivity
Symbol
h, (k, K)
D
JX , J
␾
Jq
qm , m
kd
G
␭, k
SI unit
W · m⫺2 · K⫺1
m2 · s⫺1
(varies)
W
W · m⫺2
kg · s⫺1
m · s⫺1
W · K⫺1
W · m⫺1 · K⫺1
Definition
h ⫽ Jq /⌬T
D ⫽ Jn /(dc/dl)
JX ⫽ A⫺1 dX/dt
␾ ⫽ dq/dt
Jq ⫽ ␾/A
qm ⫽ dm/dt
G ⫽ ␾/⌬T
␭ ⫽ Jq /(dT/dl)
short
standard
long
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TABLE 2.2 Physical and Chemical Symbols and Definitions (Continued)
N. Transport properties (continued)
Name
Thermal diffusivity
Thermal resistance
Volume flow rate
Dimensionless quantities:
Alfvén number
Cowling number
Euler number
Fourier number
Fourier number for mass
transfer in binary
mixtures
Froude number
Grashof number
Grashof number for mass
transfer in binary
mixtures
Hartmann number
Knudsen number
Lewis number
Mach number
Magnetic Reynolds
number
Nusselt number
Nusselt number for mass
transfer in binary
mixtures
Péclet number
Péclet number for mass
transfer in binary
mixtures
Prandtl number
Rayleigh number
Reynolds number
Schmidt number
Sherwood number
Stanton number
Stanton number for mass
transfer in binary
mixtures
Strouhal number
Weber number
Symbol
a
R
qv , V
SI unit
m2 · s⫺1
K · W⫺1
m3 · s⫺1
Definition
a ⫽ ␭ /␳c p
R ⫽ 1/G
qv ⫽ dV/dt
Al
Co
Eu
Fo
Fo*
Al ⫽ ␷ (␳␮ )1/2/B
Co ⫽ B 2/␮␳␷ 2
Eu ⫽ ⌬p/␳␷ 2
Fo ⫽ at/l 2
Fo* ⫽ Dt/l 2
Fr
Gr
Gr*
Fr ⫽ ␷ /(lg)1/2
Gr ⫽ l 3 go ⌬T␳2/␩2
Gr* ⫽ l 3g (⭸␳/⭸x)T,p (⌬x ␲ /␩)
Ha
Kn
Le
Ma
Rm, Re m
Ha ⫽ Bl(␬ /␩ )1/2
Kn ⫽ ␭ /l
Le ⫽ a/D
Ma ⫽ ␷ /c
Rm ⫽ ␷␮␬ l
Nu
Nu*
Nu ⫽ hl/k
Nu* ⫽ k d l/D
Pe
Pe*
Pe ⫽ ␷ l/a
Pe* ⫽ ␷ l/D
Pr
Ra
Re
Sc
Sh
St
St*
Pr ⫽ ␩ /␳a
Ra ⫽ l 3 g␣ ⌬T␳/␩ a
Re ⫽ ␳␷ l/␩
Sc ⫽ ␩ /␳D
Sh ⫽ k d l/D
St ⫽ h/␳␷ c p
St* ⫽ k d /␷
Sr
We
Sr ⫽ lf/␷
We ⫽ ␳␷ 2 l/␥
Symbols used in the definitions of dimensionless quantities:
Acceleration of free fall
g
Area
A
Cubic expansion coefficient ␣
Density
␳
Frequency
f
Length
l
Mass
m
Mean free path
␭
Mole fraction
x
Pressure
Speed
Speed of sound
Surface tension
Temperature
Time
Viscosity
Volume
p
␷
c
␥
T
t
␩
V
short
standard
long
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TABLE 2.3 Mathematical Symbols and Abbreviations
Symbol or abbreviation
⫹
⫺
⫾
⫿
⬅
⫻, center dot
⫼
⫽
⬇
⯝
⬎
⬍
⬎⬎
⬍⬍
ⱖ
ⱕ
⬀, ⬃
:
⬁
|a|
an
n
1/n
p a, a
1/2
p a, a
具a典, ā
2.23
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Meaning
Plus
Minus
Plus or minus
Minus or plus
Identically equal to
Multiplied by (ab, a ⫻ b, a · b)
Divided by (a/b, ab⫺1 )
Not equal to
Approximately equal to
Asymptotically equal to
Greater than
Less than
Much greater than
Much less than
Greater than or equal to
Less than or equal to
Proportional to
Tends to, approaches
Infinity
Magnitude of a
nth power of a
nth root of a
Square root of a
Mean value of a
n
兿 a , ⌸a
i⫽1
i
i
log a or log 10 a
log a b
ln b, log e b
e
␲
i
n!
⬔
⬜
||
a⬚
a⬘
a⬙
sin a
cos a
tan a
cot a
sec a
cos a
arcsin a, sin⫺1 a
arccos a, cos⫺1 a
arctan a, tan⫺1 a
sinh a
cosh a
tanh a
Product of a i
Common (or Briggsian) logarithm to the base 10 of a
Logarithm to the base a of b
Natural (Napierian) logarithm (to the base e) of b
Base (2.718) of natural system of logarithms
Pi (3.1416)
Imaginary quantity, square root of minus one
n factorial (n! ⫽ 1 · 2 · 3 · · · n)
Angle
Perpendicular to
Parallel to
a degrees (angle)
a minutes (angle); a prime
a seconds (angle); a double prime
sine of a
cosine of a
tangent of a
cotangent of a
secant of a
cosecant of a
Inverse sine of a (angle whose sine is a)
Inverse sine of a (angle whose cosine is a)
Inverse tangent of a (angle whose tangent is a)
Hyperbolic sine of a
Hyperbolic cosine of a
Hyperbolic tangent of a
short
standard
long
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TABLE 2.3 Mathematical Symbols and Abbreviations (Continued)
Symbol or abbreviation
cotanh a
P(x, y)
P(r, ␪ )
f(x), F(x)
⌬x
dy
dy
or f⬘(x)
dx
2
dy
or f ⬙(x)
dx 2
⭸z
⭸x
⭸ 2z
⭸x ⭸y
Meaning
Hyperbolic cotangent of a
Rectangular coordinate of point P
Polar coordinate of point P
Function of x
Increment of x
Total differential of y
Derivative of y ⫽ f(x) with respect to x
Second derivative of y ⫽ f(x) with respect to x
Partial derivative of z with respect to x
Second partial derivative of z with respect to x and y
冕
冕
Integral of
b
Integral between the limits a and b
a
lim f(x)
limit of f(x) as x tends to a
x:a
n
冘
Summation of a i between the limits 1 and n
i⫽1
TABLE 2.4 SI Prefixes
Submultiple
Prefix
Symbol
Multiple
Prefix
Symbol
10⫺1
10⫺2
10⫺3
10⫺6
10⫺9
10⫺12
10⫺15
10⫺18
10⫺21
10⫺24
deci
centi
milli
micro
nano
pico
femto
atto
zepto
yocto
d
c
m
␮
n
p
f
a
z
y
10
102
103
106
109
1012
1015
1018
1021
1024
deka
hecto
kilo
mega
giga
tera
peta
exa
zetta
yotta
da
h
k
M
G
T
P
E
Z
Y
short
standard
long
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2.25
GENERAL INFORMATION, CONVERSION TABLES, AND MATHEMATICS
top of rh
base of rh
cap height
base of text
TABLE 2.4 SI Prefixes (Continued)
Numerical (multiplying) prefixes
Number
Prefix
Number
Prefix
Number
Prefix
0.5
1
l.5
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
hemi
mono
sesqui
di (bis)*
tri (tris)*
tetra (tetrakis)*
penta
hexa
hepta
octa
nona
deca
undeca
dodeca
trideca
tetradeca
pentadeca
hexadeca
heptadeca
octadeca
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
nonadeca
icosa
henicosa
docosa
tricosa
tetracosa
pentacosa
hexacosa
heptacosa
octacosa
nonacosa
triaconta
hentriaconta
dotriaconta
tritriaconta
tetratriaconta
pentatriaconta
hexatriaconta
heptatriaconta
octatriaconta
39
40
41
42
43
44
45
46
47
48
49
50
60
70
80
90
100
110
120
130
nonatriaconta
tetraconta
hentetraconta
dotetraconta
tritetraconta
tetratetraconta
pentatetraconta
hexatetraconta
heptatetraconta
octatetraconta
nonatetraconta
pentaconta
hexaconta
heptaconta
octaconta
nonaconta
hecta
decahecta
icosahecta
triacontahecta
* In the case of complex entities such as organic ligands (particularly if they are substituted) the multiplying prefixes bis-,
tris-, tetrakis-, pentakis-, . . . are used, i.e., -kis is added starting from tetra-. The modified entity is often placed within parentheses to avoid ambiguity.
TABLE 2.5 Greek Alphabet
Capital
Lower
case
Name
Capital
Lower
case
Name
〈
〉
⌫
⌬
⌭
⌮
⌯
⍜
⌱
⌲
⌳
⌴
␣
␤
␥
␦
⑀
␨
␩
␪
␫
␬
␭
␮
Alpha
Beta
Gamma
Delta
Epsilon
Zeta
Eta
Theta
Iota
Kappa
Lambda
Mu
⌵
⌶
⌷
⌸
⌹
⌺
⌻
⌼
⌽
⌾
⌿
⍀
␯
␰
␱
␲
␳
␴
␶
␷
␾
␹
␺
␻
Nu
Xi
Omicron
Pi
Rho
Sigma
Tau
Upsilon
Phi
Chi
Psi
Omega
short
standard
long
DEAN #37261
LEFT
(McGHP)
2.26
INTERACTIVE
top of rh
base of rh
SECTION 2
cap height
base of text
TABLE 2.6 Abbreviations and Standard Letter Symbols
Abampere
Absolute
Absolute activity
Absorbance (decaidic)
Absorbance (napierian)
Absorptance
Absorption coefficient, linear
decaidic
Absorption coefficient, linear
napierian
Absorption coefficient, molar
decaidic
Absorption coefficient, molar
napierian
Absorption index
Acceleration
Acceleration due to gravity
Acetyl
Acoustic absorption factor
Acoustic dissipation factor
Acoustic reflection factor
Acoustic transmission factor
Activation energy
Activity (referenced to
Raoult’s law)
Activity (referenced to
Henry’s law):
Concentration basis
Molality basis
Mole fraction basis
Activity (radioactive)
Activity coefficient (referenced to Raoult’s law)
Activity coefficient (referenced to Henry’s law):
Concentration basis
Molality basis
Mole fraction basis
Adjusted retention time
Adjusted retention volume
Admittance
Affinity of reaction
Alcohol
Alfvén number
Alkaline
Alpha particle
Alternating current
Amorphous
Amount concentration
Amount of substance
Ampere
Amplification factor
Angle of optical rotation
Angstrom
Angular dispersion
abamp
abs
␭
A
B
␣
a, K
␣
␮, ⑀
␬
k
a
g, gn
Ac
␣a
␦
␳
␶
Ea
a
ac
am
ax
A
f
␥c
␥m
␥x
t⬘R
V⬘R
Y
A
alc
Al
alk
␣
ac
am
c
n
A
␮
␣
Å, A
d␪/d␭
Angular momentum
Angular momentum terms
Angular velocity
Anhydrous
Approximate (circa)
Aqueous solution
Aqueous solution at infinite
dilution
Are, unit of area
Area
Area per molecule
Astronomical unit
Asymmetry parameter
Atmosphere, unit of pressure
Atomic mass
Atomic mass constant
Atomic mass unit
Atomic number
Atomic percent
Atomic weight
Average
Average linear gas velocity
Avogadro constant
Axial angular momentum
Axial spin angular momentum
Bandwidth (10%) of a spectral filter
Band variance
Bar, unit of pressure
Barn, unit of area
Barrel
Base of natural logarithms
Becquerel
Bed volume
Beta particle
Bloch function
Body-centered cubic
Bohr
Bohr magneton
Bohr radius
Boiling point
Boltzmann constant
Bragg angle
Breadth
British thermal unit
Bulk modulus
Bulk strain
Burgers vector
Butyl
Calorie, unit of energy
Calorie, international steam
table
Candela
Capacitance
␲
j, J, l, L, N
␻
anhyd
ca.
aq
aq, ⬁
a
A, S
a, ␴
AU
␬
atm
ma
mu
amu
Z
at.%
at. wt.
av
␮
L, NA
␭, ⌳, ⍀
␴, ⌺
⌬␭0.1
␴2
bar
b
bbl
e
Bq
Vg
␤
uk(r)
bcc
b
␮B
a0
bp
k, kB
␪
b
Btu
K
␪
b
Bu
cal
cal IT
cd
C
short
standard
long
DEAN #37261
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INTERACTIVE
GENERAL INFORMATION, CONVERSION TABLES, AND MATHEMATICS
QV
QW
x, y, z
t,␪
cgs
⌬, ␦
D, d
␳
n
z
CP
␮
␦
ca.
␻
k
q
␲
Cit
h
␴
d
Z
z
Z
Vcol
cf.
n̂
k, K, m, M
␬S
␬T
Z
K
␭c
␭c,n
␭c,p
c
␥
Cmax
CM
Concentration of solute in
stationary phase
Condensed phase (solid or
liquid)
Conductance
Conductivity
Conductivity cell constant
Conductivity tensor
Contact angle
Coordinate, position vector
Coulomb
Counts per minute
Coupling constant, direct dipolar
Critical density
Critical temperature
Cross section
Crystalline
Cubic
Cubic expansion coefficient
Curie
Cycles per second
Curie temperature
Dalton (atomic mass unit)
Day
Debye, unit of electric dipole
Debye circular frequency
Debye circular wave-number
Debye-Waller factor
Decay constant (radioactive)
Decibel
Decompose
Degeneracy, statistical
weight
Degree of dissociation
Degrees Baume
Degrees Celsius
Degrees Fahrenheit
Density (mass)
Density, critical
Density, relative
Density of liquid phase
Density of states
Density of vibrational modes
(spectral)
Detect, determine (d)
Determination
Deuteron
Diamagnetic shielding factor
Diameter
Dielectric polarization
Differential thermal analysis
Diffusion coefficient
Diffusion coefficient, liquid
film
top of rh
base of rh
cap height
base of text
TABLE 2.6 Abbreviations and Standard Letter Symbols (Continued)
Capacity, volume
Capacity, weight
Cartesian space coordinates
Celsius temperature
Centimeter-gram-second
system
Centrifugal distortion constants:
A reduction
S reduction
Charge density of electrons
Charge number of electrochemical reaction
Charge number of an ion
Chemically pure
Chemical potential
Chemical shift
Circa (approximate)
Circular frequency
Circular wave vector:
For particles
For phonons
Circumference divided by
the diameter
Citrate
Coefficient of heat transfer
Collision cross section
Collision diameter
Collision frequency
Collision frequency factor
Collision number
Column volume
Compare (confer)
Complex refractive index
Component of angular momentum
Compressibility:
Isentropic
Isothermal
Compression factor
Compression modulus
Compton wavelength of electron
Compton wavelength of neutron
Compton wavelength of
proton
Concentration (amount of
substance)
Concentration (mass)
Concentration at peak
maximum
Concentration of solute in
mobile phase
2.27
CS
cd
G
␥, ␬
Kcell
␴ik
␪
r
C
cpm, c/m
DAB
dc
tc
␴
cr, cryst
cub
␣, ␣v, ␥
Ci
Hz
Tc
Da
d
D
␻D
qD
D, B
␭
dB
dec
d, g, ␤
␣
⬚Be
⬚C
⬚F
␳, ␥
dc
d
␳L
NE, ␳
N␻
det(d)
detn
d
1⫹␴
d
P
DTA
D
Df
short
standard
long
DEAN #37261
LEFT
(McGHP)
2.28
INTERACTIVE
top of rh
base of rh
SECTION 2
cap height
base of text
TABLE 2.6 Abbreviations and Standard Letter Symbols (Continued)
Diffusion coefficient, mobile
phase
Diffusion coefficient, stationary phase
Diffusion current
Diffusion length
Diffusion rate constant, mass
transfer coefficient
Dilute
Dirac delta function
Direct current
Direct dipolar coupling constant
Disintegration energy
Disintegrations per minute
Displacement vector of an
ion
Dissociation energy
From ground state
From the potential minimum
Distribution ratio
Donor ionization energy
Dropping mercury electrode
Dyne, unit of force
Einstein transition probabilities
Spontaneous emission
Stimulated absorption
Stimulated emission
Electric charge
Electric current
Electric current density
Electric dipole moment of a
molecule
Electric displacement
Electric field gradient
Electric field strength
Electric flux
Electric mobility
Electric polarizability of a
molecule
Electric potential
Electric potential difference
Electric susceptibility
Electrical conductivity
Electrical conductance
Electrical resistance
Electrochemical transfer coefficient
Electrokinetic potential
Electromagnetic unit
Electromotive force
Electron
Electron affinity
DM
DS
id
L
kd
dil
␦
dc
DAB
Q
dpm
u
D, Ed
D0
De
D
Ed
dme
dyn
A, B
Anm
Bmn
Bnm
Q
I
j, J
p, ␮
D
q
E
⌿
u, ␮
␣
V, ␾
U, ⌬V
␹e
␴
G
R
␣
␨
emu
E, emf
e⫺, e
Eea
Electron magnetic moment
Electron paramagnetic resonance
Electron radius
Electron rest mass
Electron spin resonance
Electronvolt
Electrostatic unit
Elementary charge
Elution volume, exclusion
chromatography
Emittance
By blackbody
Energy
Energy density
Energy per electron hole
pair of ion pair in detector
Enthalpy
Entropy
Entropy unit
Equilibrium constant
Concentration basis
Molality basis
Pressure basis
Equilibrium position vector
of an ion
Equivalent weight
Erg, unit of energy
Especially
et alii (and others)
et cetera (and so forth)
Ethyl
Ethylenediamine
Ethylenediamine-N,N,N⬘,N⬘tetraacetic acid
Euler number
Exempli gratia (for example)
Expansion coefficient
Exponential
Extent of reaction
Fano factor
Farad
Faraday constant
Fermi, unit of length
Fermi energy
Film tension
Film thickness
Fine structure constant
Finite change
First radiation constant
Flow rate
Flow rate, column chromatography
Fluid phase (gas or liquid)
Fluidity
␮e
EPR
re
me
ESR
eV
esu
e
Ve
⑀
Mbb
E
w, ␳
⑀
H
S
e.u.
K, K0
Kc
Km
Kp
R0
equiv wt
erg
esp.
et al.
etc.
Et
en
EDTA
Eu
e.g.
␣
exp
␰
F
F
F
f
EF
⌺f
h, t
␣
⌬
c1
q
Fc
fl
␾
short
standard
long
DEAN #37261
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INTERACTIVE
GENERAL INFORMATION, CONVERSION TABLES, AND MATHEMATICS
⌽F
PF
F, J
f
ft
e.g.
F
k
F
Fo
Fr
fp
f, v
f, ␮
Fr
f
␾
gal
⌬␾
␥
␥
Eg
g
R
G
g
G
grad
gr
g
⌶
Gr
grav
G
Gy
␥, ⌫
t1/2
E1/2
AH, RH
H
␻
Ha
Eh
q, Q
C
Cp
Cv
␾
J
ha
h
h
Helmholtz energy
Henry
Hertz
Hexagonal
Horsepower
Hour
Hygroscopic
Hyperfine coupling constant
Hyperfine coupling tensor
ibidem (in the same place)
id est (that is)
Ignition
Impedance
Inch
Indices of a family of crystallographic planes
Indirect spin-spin coupling
constant
Inductance
Inertial defect
Infinitesimal change
Infrared
Inner column volume
Inner electric potential
Inner electrode potential
Inorganic
Inside diameter
Insoluble
Interatomic distances:
Equilibrium distance
Ground-state distance
Substitution structure distance
Zero-point average
distance
Internal energy
Interstitial (outer) volume
In the place cited (loco citato)
In the same place
In the work cited
Ionic conductivity
Ionic strength
Concentration basis
Molality basis
Ionization energy
Irradiance
Joule
Joule-Thomson coefficient
Kelvin
Kilocalorie
Kilogram
Kilogram-force
Kilowatt-hour
Kinematic viscosity
top of rh
base of rh
cap height
base of text
TABLE 2.6 Abbreviations and Standard Letter Symbols (Continued)
Fluorescent efficiency
Fluorescent power
Flux
Focal length
Foot
For example (exempli gratia)
Force
Force constant (vibrational
levels)
Formal concentration
Fourier number
Franklin, unit of electric
charge
Freezing point
Frequency
Friction coefficient
Froude number
Fugacity
Fugacity coefficient
Gallon
Galvani potential difference
Gamma, unit of mass
Gamma radiation
Gap energy (solid state)
Gas (physical state)
Gas constant
Gauss
g factor
Gibbs energy
Grade
Grain, unit of mass
Gram
Grand partition function
Grashof number
Gravimetric
Gravitational constant
Gray
Grüneisen parameter
Half-life
Half-wave potential
Hall coefficient
Hamilton function
Harmonic vibration wavenumber
Hartmann number
Hartree energy
Heat
Heat capacity
At constant pressure
At constant volume
Heat flow rate
Heat flux
Hectare, unit of area
Height
Helion
2.29
A
H
Hz
hex
hp
h
hygr
a, A
T
ibid.
i.e.
ign
Z
in
hkl
JAB
L
⌬
␦
ir
Vi
␾
␾
inorg
i.d.
insol
re
r0
rs
rz
U
Vo
loc. cit.
ibid.
op. cit.
␭, ⌳
I
Ic
Im
Ei
E
J
␮, ␮ JT
K
kcal
kg
kgf
kWh
v, ␾
short
standard
long
DEAN #37261
LEFT
(McGHP)
2.30
INTERACTIVE
top of rh
base of rh
SECTION 2
cap height
base of text
TABLE 2.6 Abbreviations and Standard Letter Symbols (Continued)
Kinetic energy
Knudsen number
Kovats retention indices
Lagrange function
Lambda, unit of volume
Landé g-factor
Larmor circular frequency
Larmor frequency
Lattice plane spacing
Lattice vector
Lattice vectors
Length
Length of arc
Lewis number
Light year
Limit (mathematics)
Linear expansion coefficient
Linear reciprocal dispersion
Linear strain
Liquid
Liquid crystal
Liter
loco citato (in the place
cited)
Logarithm, common
Logarithm, base e
Longitudinal relaxation time
Lorenz coefficient
Loss angle
Lumen
Luminous intensity
Lux
Mach number
Madelung constant
Magnetic dipole moment of
a molecule
Magnetic field strength
Magnetic flux
Magnetic flux density
Magnetic moment of protons
in water
Magnetic quantum number
Magnetic Reynolds number
Magnetic susceptibility
Magnetic vector potential
Magnetizability
Magnetization
Magnetogyric ratio
Mass
Mass absorption coefficient
Mass concentration
Mass density
Mass flow rate
Mass fraction
Massieu function
K, T, Ek
Kn
RI
L
␭
g, ge
␻L
vL
d
R, R0
a, b, c
l, L
s
Le
l.y.
lim
␣1
D⫺1, d␭/dx
e, ⑀
l, lq
lc
L, 1
loc. cit.
log
ln
T1
L
␦
lm
I
lx
Ma
␣
m, ␮
H
⌽
B
␮p/␮B
Mj
Rm
␬, ␹
A
␰
M
␥
m
␮/␳, ␮m
␥, ␳
␳
qm
w
J
Mass number
Mass of atom
Mass transfer coefficient
Matrix volume
Maximum
Maxwell, unit of magnetic
flux
Mean ionic activity
Mean ionic activity coefficient
Mean ionic mobility
Melting point
Metallic
Metastable
Metastable peaks
Meter
Methyl
Micrometer
Micron
Mile
Miller indices
Milliequivalent
Millimeters of mercury, unit
of pressure
Millimole
Minimum
Minute
Mixture
Mobility
Mobility ratio
Modulus of elasticity
Molal
Molality
Molar
Molar (decadic) absorption
coefficient
Molar ionic conductivity
Molar magnetic susceptibility
Molar mass
Molar quantity X
Molar refraction
Molar volume
Mole
Mole fraction, condensed
phase
Gaseous mixtures
Mole percent
Molecular weight
Moment of inertia
Momentum
Monoclinic
Monomeric form
Muon, negative
Muon, positive
A
m, ma
kd
Vg
max
Mx
a⫾
␥⫾
W⫾
mp
met
m
m*
m
Me
␮m
␮
mi
h, l, k
meq
mmHg
mM
min
m, min
mixt
␮
b
E
m
b
M, M
⑀
␭, ⌳
␹m
M
Xm
R, Rm
Vm
mol
x
y
mol %
mol wt
I, J
p
mn
mon
␮⫺
␮⫹
short
standard
long
DEAN #37261
(McGHP)
RIGHT
INTERACTIVE
GENERAL INFORMATION, CONVERSION TABLES, AND MATHEMATICS
M, L
B
e
␬
TN
VN
ve
n
␮N
N
mn
N
N
␴
NMR
␮N
I
A
C
n
N
NA
Nu
␥
Oe
⍀
op. cit.
f /number
L ⫽ 0, 1, 2, 3, . . .
S, P, D, F, . . .
⌳ ⫽ 0, 1, 2, . . .
⌺, ⌸, ⌬, . . .
l ⫽ 0, 1, 2, 3, . . .
s, p, d, f, . . .
n
n
n
s
␴
org
o-rh
␾
␾m
␾x
⌸
oz
Outer diameter
Outer electric potential
Overall order of reaction
Overpotential
Oxalate
Oxidant
Packing uniformity factor
Page(s)
Parsec, unit of length
Partial molar quantity
Partial order of reaction
Particle diameter
Particle position vector:
Electron
Ion position
Partition coefficient
Partition function
Partition ratio
Parts per billion, volume
Parts per billion, weight
Parts per million, volume
Parts per million, weight
Pascal
Path length (absorbing)
Peak asymmetry factor
Peak resolution
Péclet number
Peltier coefficient
Percent
Period of time
Permeability
Permeability of vacuum
Permittivity
Permittivity of vacuum
pH, expressed in activity
Expressed in molarity
Phenyl
Phosphorescent efficiency
Phosphorescent power
Photochemical yield
Photoluminescence power
Photon
Pion
Planck constant
Planck constant/2␲
Planck function
Plane angle
Plate height
Plate number, effective
Poise
Polymeric form
Porosity, column
Positron
Potential energy
Pound
top of rh
base of rh
cap height
base of text
TABLE 2.6 Abbreviations and Standard Letter Symbols (Continued)
Mutual inductance
Napierian absorbance
Napierian base
Napierian molar absorption
coefficient
Neel temperature
Net retention volume
Neutrino
Neutron
Neutron magnetic moment
Neutron number
Neutron rest mass
Newton
Normal concentration
Normal stress
Nuclear magnetic resonance
Nuclear magneton
Nuclear spin angular momentum
Nucleon number
Number concentration
Number density
Number of entities
Numerical aperture
Nusselt number
Obstruction factor
Oersted, unit of magnetic
field
Ohm
opere citato (in the work
cited)
Optical speed
Orbital angular momentum:
Quantum number
Series symbol
Orbital angular momentum
(molecules):
Quantum number
Symbol
Orbital angular momenta of
individual electrons
Order of Bragg reflection
Order of reaction
Order of reflection
Order parameters (solid
state), long range
Short range
Organic
Orthorhombic
Osmotic coefficient
Molality basis
Mole fraction basis
Osmotic pressure (ideal dilute solution)
Ounce
2.31
o.d.
␺
n
␩
Ox
ox
␭
p. (pp.)
pc
X
nB
dp
r
Rj
K
q, Q, z, Z, ⍀
k⬘
ng/mL
ng/g
␮g/mL
␮/g
Pa
l
AF
Rs
Pe
⌸
%
T
␮
␮0
⑀
⑀0
paH
pH
Ph, ␾
⌽P
PP
␾
P
␥
␲
h
ប
Y
␣, ␤, ␥, ␪, ␾
H
Neff
P
pol
⑀
␤⫹
V, ⌽, EP
lb
short
standard
long
DEAN #37261
LEFT
(McGHP)
2.32
INTERACTIVE
top of rh
base of rh
SECTION 2
cap height
base of text
TABLE 2.6 Abbreviations and Standard Letter Symbols (Continued)
Pounds per square inch
Powder
Power
Poynting vector
Prandtl number
Pressure (partial)
Pressure (total)
Pressure coefficient
Pressure, column inlet
Pressure, column outlet
Pressure, critical
Pressure drop
Pressure-gradient correction
Principal moments of inertia
Principal quantum number
Probability
Probability density
Product sign
Propyl
Proton
Proton magnetic resonance
Proton magnetrogyric ratio
Proton number
Proton rest mass
Pyridine
Quadrupole interaction energy tensor
Quadrupole moment of a
molecule
Quantity of electricity, electric charge
Quantum of energy
Quantum yield
Rad, unit of radiation dose
Radian
Radiant energy
Radiant energy density
Radiant energy flux
Radiant exitance
Radiant flux received
Radiant intensity
Radiant intensity at time t
after termination of excitation
Radiant power
Radiant power incident on
sample
Radiofrequency
Radius
Rate of concentration change
Rate constant
Rate of reaction
Ratio of heat capacities
Reactance
Reciprocal lattice
psi
pwd
p
S
Pr
p
p, P
␤
pi
po
pc
⌬P
j
IA; IB; IC
n
P
P
⌸
Pr
p
pmr
␥p
Z
mp
py
␹
Q, ⍜
Q
hv
␾
rad
rad
Q, W
␳, w
dQ/dt
M
E
I
I(t)
⌽
P0
rf
r
r
k
v
␥
X
a*, b*, c*
Reciprocal lattice vector (circular)
Vectors for
Reciprocal radius of ionic atmosphere
Reciprocal temperature parameter, 1/kT
Reciprocal thickness of double layer
Reduced column length
Reduced mass
Reduced plate height
Reduced velocity
Reductant
Reference
Reflectance
Reflection plane
Refractive index
Relative permeability
Relative permittivity (dielectric constant)
Relative pressure coefficient
Relative retention ratio
Relaxation time
Rem, unit of dose equivalent
Residual resistivity (solid
state)
Resistivity tensor
Retardation factor
Retarded van der Waals constant
Retention time
Retention volume
Revolutions per minute
Reynolds number
Rhombic
Rhombohedral
Roentgen
Root-mean-square
Rotational constants:
In frequency
In wavenumber
Rotational term (spectroscopy)
Rotation-reflection
Rydberg, unit of energy
Rydberg constant
Saturated
Saturated calomel electrode
Schmidt number
Second
Second radiation constant
Second virial coefficient
Sedimentation coefficient
Selectivity coefficient
G
a*; b*, c*
␬
␤
␬
␭
␮
h
v
red
ref
␳
␴
n
␮r
⑀r
␣p
␣
␶
rem
␳R
␳
Rf
B, ␤
tR
VR
rpm
Re
rh
rh-hed
R
rms
A, B, C
Ã; B̃; C̃
F
Sn
Ry
R, R⬁
satd
SCE
Sc
s
c2
B
s
k
short
standard
long
DEAN #37261
(McGHP)
RIGHT
INTERACTIVE
GENERAL INFORMATION, CONVERSION TABLES, AND MATHEMATICS
L
␣
G
␥
␶
Sh
␴
␴
S
Sv
S/N
sl
c, s
␻, ⍀
⍀F(P,DF)
⍀A
s
sol
soln, sln
solv
P, Pa
d
ads
sp gr
V0g
s
v, ␷
⌬␭em
⌬␭ex
⌬␭m
Q␭ , dQ/d␭
␳␯ , w␯
␳␯ , w␯
␳␯˜ , w␯˜
d␾/d␭
g
u, w
c
c0
r, ␪, ␾
s, S
T1
Spin orbit coupling constant
Spin-spin coupling constant
Spin-spin (or transverse) relaxation time
Spin wavefunctions
Square
Standard
Standard enthalpy of activation
Standard enthalpy of formation
Standard entropy
Standard entropy of activation
Standard Gibbs energy of activation
Standard Gibbs energy of
formation
Standard heat capacity
Standard hydrogen electrode
Standard partial molar enthalpy
Standard partial molar entropy
Standard potential of electrochemical cell reaction
Standard reaction enthalpy
Standard reaction entropy
Standard reaction Gibbs energy
Standard temperature and
pressure
Stanton number
Statistical weight
Statistical weight of atomic
states
Stefan-Boltzmann constant
Steradian
Stoichiometric number
Stokes
Summation sign
Surface charge density
Surface concentration
Surface coverage
Surface density
Surface electric potential
Surface pressure
Surface tension
Susceptance
Svedberg, unit of time
Symmetrical
Symmetry coordinate
Symmetry number
Tartrate
Temperature
top of rh
base of rh
cap height
base of text
TABLE 2.6 Abbreviations and Standard Letter Symbols (Continued)
Self-inductance
Separation factor
Shear modulus
Shear strain
Shear stress
Sherwood number
Shielding constant (NMR)
Short-range order parameter
Siemens
Sievert
Signal-to-noise ratio
Slightly
Solid
Solid angle
Solid angle over which luminescence is measured (F,
fluorescence; P, phosphorescence; DF, delayed fluorescence)
Solid angle over which radiation is absorbed in cell
Solubility
Soluble
Solution
Solvent
Sound energy flux
Spacing between crystal diffracting planes
Species adsorbed on a substance
Specific gravity
Specific retention volume
Specific surface area
Specific volume
Spectral bandwidth of emission monochromator
Spectral bandwidth of excitation monochromator
Spectral bandwidth of monochromator
Spectral radiant energy
Spectral radiant energy density:
In terms of frequency
In terms of wavelength
In terms of wavenumber
Spectral radiant energy flux
Spectroscopic splitting factor
Speed
Speed of light:
In a medium
In vacuum
Spherical polar coordinates
Spin angular momentum
Spin-lattice relaxation time
2.33
A
JAB
T2
␣, ␤
sq
std
H⫹⫹
⌬Hf 0
S0
⌬S⫹⫹
⌬G⫹⫹
⌬Gf 0
Cp
SHE
H0
S0
E0
⌬rH 0
⌬rS 0
⌬rG 0
STP
St
W, ␤, ␻
g
␴
sr
v
St
⌺
␴
⌫
␪
␳A, ␳S
␹
␲
␥, ␴
B
Sv
sym
S
s, ␴
Tart
␪, ⍜
short
standard
long
DEAN #37261
LEFT
(McGHP)
2.34
INTERACTIVE
top of rh
base of rh
SECTION 2
cap height
base of text
TABLE 2.6 Abbreviations and Standard Letter Symbols (Continued)
Temperature, thermodynamic
Temperature at boiling point
Term value spectroscopy
Tesla
Tetragonal
Thermal conductance
Thermal conductivity
Thermal diffusivity
Thermal resistance
Thermoelectric force
Thickness of diffusion layer
Thickness of layer
Thickness (effective) of stationary phase
Thickness of surface layer
Thickness of various layers
Thomson coefficient
Thomson cross section
Time
Time interval, characteristic
Tonne
Torr (mm of mercury)
Torque
Total bed volume
Total term (spectroscopy)
Transconductance
Transfer coefficient
Transit time of nonretained
solute
Transition
Transition dipole moment of
a molecule
Transition frequency
Transition wavenumber
Translation (circular)
Translation vectors for
crystal lattice
Transmission factor
Transmittance
Transport number
Transverse relaxation time
Triclinic
Trigonal
Triton (tritium nucleus)
Ultrahigh frequency
Ultraviolet
Unified atomic mass unit
United States Pharmacopoeia
Vacuum
van der Waals constant
Vapor pressure
Velocity
Versus
T
Tb
T
T
tetr
G
␭, k
a
R
E
␦
t
df
␶
␦
␮, ␶
␴e
t
T, ␶
t, ton
Torr
T
Vtot
T
gm
␣
tM, t0
tr
M, R
v
ṽ
b1; b2; b3
a 1; a 2; a 3
a; b; c
␶
T, ␶
t
T2
tric
trig
t
uhf
uv
u
USP
vac
␭
p, vp
u, w
vs
Vibrational anharmonicity
constant
Vibrational coordinates:
Internal coordinates
Normal coordinates, dimensionless
Mass adjusted
Vibrational force constants:
Diatomic
Polyatomic, dimensionless
normal coordinates
Internal coordinates
Symmetry coordinates
Vibrational quantum number
Vibrational term
Viscosity
Vitreous substance
Volt
Volt-ampere-reactive
Volta potential difference
Volume
Volume flow rate
Volume fraction
Volume in space phase
Volume liquid phase in column
Volume mobile phase in column
Volume of activation
Volume percent
Volume per volume
Volume strain
Watt
Wavefunction
Wavelength
Wavenumber (in a medium)
Wavenumber in vacuum
Weber
Weber number
Weight
Weight of liquid phase
Weight percent
Weight per volume
Wien displacement constant
Work
Work function
x unit
Yard
Young’s modulus
Zeeman splitting constant
Zone width at baseline
Zone width at one-half peak
height
␹
Ri, rI, ␪j, etc.
qr
Qr
f
␾rst
. . .
; krst
. . .
f ij
Fij
v
G
␩, ␮
vit
V
var
⌬␺
V, v
qv
␾
⍀
VL
VM
⌬⫹⫹V
vol %
v/v
␪
W
␾, ␺, ⌿
␭
␴
ṽ
Wb
We
W
wL
wt %
w/v
b
w, W
⌽
X
yd
E
␮B/hc
Wb
W1/2
short
standard
long
DEAN #37261
(McGHP)
RIGHT
INTERACTIVE
2.35
GENERAL INFORMATION, CONVERSION TABLES, AND MATHEMATICS
top of rh
base of rh
cap height
base of text
TABLE 2.7 Conversion Factors
The data were compiled by L. P. Buseth for the 13th edition; some entries have been added or modified in view
of recent data and SI units.
Relations which are exact are indicated by an asterisk (*). Factors in parentheses are also exact. Other factors
are within ⫾5 in the last significant figure.
To convert
Abampere
Abcoulomb
Abfarad
Abhenry
Abmho
Abvolt
Acre
Acre (U.S. survey)
Acre-foot
Acre-inch
Ampere per square centimeter
Ampere-hour
Ampere-turn
Ampere-turn per centimeter
Ångström
Apostilb
Are
Assay ton
Astronomical unit
Atmosphere
Atomic mass unit
Avogadro number
Into
ampere*
coulomb*
statcoulomb
farad*
henry*
siemens*
volt
hectare or square hectometer
square chain (Gunter’s)*
square kilometer*
square meter*
square mile*
square rod*
square yard*
square meter
cubic foot*
cubic meter
gallon (U.S.)
cubic foot*
cubic meter
ampere per square inch*
coulomb*
faraday
gilbert
ampere-turn per inch
meter*
nanometer*
candela per square meter
lambert*
acre
square meter*
gram
meter
light-year
bar*
foot of water (at 4⬚C)
inch of mercury (at 0⬚C)
kilogram per square centimeter
millimeter of mercury*
millimeter of water (4⬚C)
newton per square meter*
pascal*
pound per square inch
ton per square inch
torr*
gram
molecules per mole
Multiply by
10
10
2.998 ⫻ 1010
109
10⫺9
109
10⫺8
0.404 685 64
10
0.004 046 873
4046.873
(1/640)
160
4840
4046.873
4.3560 ⫻ 104
1233.482
3.259 ⫻ 105
3630
102.7902
6.4516
3600
0.037 31
1.256 637
2.540
10⫺10
0.1
0.318 309 9; (1/␲)
10⫺4
0.024 710 54
100
29.1667
1.496 00 ⫻ 10⫺11
1.581 284 ⫻ 10⫺5
1.013 25.0
33.898 54
29.921 26
1.033 227
760
1.033 227 ⫻ 104
1.013 250 ⫻ 105
101 325.0
14.695 95
0.007 348
760
1.6605 ⫻ 10⫺24
6.022 137 ⫻ 1023
short
standard
long
DEAN #37261
LEFT
(McGHP)
2.36
INTERACTIVE
top of rh
base of rh
SECTION 2
cap height
base of text
TABLE 2.7 Conversion Factors (Continued)
To convert
Bar
Barn
Barrel (British)
Barrel (petroleum)
Barrel (U.S. dry)
Barrel (U.S. liquid)
Barye
Becquerel
Biot
Board foot
Bohr
Bohr magneton
Bolt (U.S. cloth)
Boltzmann constant
British thermal unit (Btu)
Btu per foot3
Btu (International table)/ft3
Btu (thermochemical)/ft3
Btu (International table)/hour
Btu (thermochemical)/hour
Btu (International table)/pound
Btu (thermochemical)/pound
Btu (thermochemical)/(ft2 · h)
Btu (thermochemical)/minute
Btu (thermochemical)/pound
Btu per square foot
Bucket (British, dry)
Into
atmosphere
dyne per square centimeter*
kilogram per square centimeter
millimeter of mercury
millimeter of water (4⬚C)
newton per square meter
pascal*
pound per square inch
square meter*
gallon (British)*
liter
gallon (British)
gallon (U.S.)*
liter
bushel (U.S.)
cubic foot
liter
quart (U.S. dry)
gallon (U.S.)
liter
dyne per square centimeter*
curie*
ampere*
cubic foot
cubic meter
meter
joule per tesla
foot*
meter
joule per degree
calorie
cubic foot-atmosphere
erg
foot-pound
horsepower-hour (British)
horsepower-hour (metric)
joule (International table)
joule (thermochemical)
kilogram-calorie
kilogram-meter
kilowatt-hour
liter-atmosphere
kilocalorie per cubic meter
joule per meter3
joule per meter3
watt
watt
joule per kilogram*
joule per kilogram
watt per meter2
watt
joule per kilogram
joule per square meter
gallon (British)*
Multiply by
0.986 923
106
1.019 716
750.062
1.019 716 ⫻ 104
105
105
14.503 77
10⫺28
36
163.659
34.9723
42
158.987
3.281 22
4.083 33
115.6271
104.9990
31.5 (variable)
119.2405
1
2.7 ⫻ 10⫺11
10
(1/12)
2.359 737 ⫻ 10⫺3
5.291 77 ⫻ 10⫺11
9.274 02 ⫻ 10⫺24
120
36.576
1.3806 ⫻ 10⫺23
251.996
0.367 717
1.0550 ⫻ 1010
778.169
3.930 15 ⫻ 10⫺4
3.984 66 ⫻ 10⫺4
1055.056
1054.350
0.2520
107.5
2.930 71 ⫻ 10⫺4
10.4126
8.899 15
3.725 895 ⫻ 104
3.723 402 ⫻ 104
0.293 071 1
0.292 875 1
2.326 ⫻ 103
2.324 444 ⫻ 103
3.154 591
17.572 50
2.324 444 ⫻ 103
1.135 65 ⫻ 104
4
short
standard
long
DEAN #37261
(McGHP)
RIGHT
INTERACTIVE
2.37
GENERAL INFORMATION, CONVERSION TABLES, AND MATHEMATICS
cap height
base of text
TABLE 2.7 Conversion Factors (Continued)
To convert
Bushel (British)
Bushel (U.S.)
Cable length (international)
Cable length (U.S. or British)
Caliber
Calorie
Calorie (15⬚C)
Calorie (international)
Calorie per minute
Candela
Candela per square centimeter
Carat (metric)
Celsius temperature
Centigrade heat unit or chu
Centimeter
Centimeter of mercury (0⬚C)
Centimeter of water (4⬚C)
Centimeter per second
top of rh
base of rh
Into
bushel (U.S.)
cubic foot
gallon (British)*
gallon (U.S.)
liter
barrel (U.S., dry)
bushel (British)
cubic foot
cubic meter
gallon (British)
gallon (U.S.)
liter
peck (U.S.)*
pint (U.S., dry)*
foot
meter*
mile (nautical)*
foot*
meter
mile (nautical)
mile (statute)
inch*
millimeter*
Btu
foot-pound
foot-poundal
horsepower-hour (British)
joule*
kilowatt-hour
liter-atmosphere
joule
joule
foot-pound per second
horsepower (British)
watt*
Hefner unit
lumen per steradian*
candela per square foot*
candela per square meter*
lambert
gram*
Fahrenheit temperature
kelvin
Btu*
calorie
joule
foot
inch
mil
pascal
pascal
foot per minute
kilometer per hour*
Multiply by
1.032 057
1.284 35
8
9.607 60
36.3687
0.304 765
0.968 939
1.244 456
0.035 239 07
7.751 51
9.309 18
35.239 07
4
64
607.611 55
185.2
0.1
720
219.456
0.118 407
0.136 364
0.01
0.254
0.003 968 320
3.088 03
99.3543
1.559 61 ⫻ 10⫺6
4.184
1.163 ⫻ 10⫺6
0.041 320 5
4.1858
4.1868
0.051 467 1
9.357 65 ⫻ 10⫺5
0.069 78
1.11
1
929.0304
104
3.141 593; (␲)
0.2
(9/5)⬚C ⫹ 32
⬚C ⫺ 273.15
1.8
453.592
1899.10
0.032 808 4
0.393 700 8
393.700 8
1333.22
98.063 8
1.986 50
0.036
short
standard
long
DEAN #37261
LEFT
(McGHP)
2.38
INTERACTIVE
top of rh
base of rh
SECTION 2
cap height
base of text
TABLE 2.7 Conversion Factors (Continued)
To convert
Centimeter per second
(continued)
Centimeter per second squared
Centimeter-dyne
Centimeter-gram
Centipoise
Chain (Ramsden’s)
Chain (Gunter’s)
Circular inch
Circular millimeter
Circumference
Cord
Coulomb
Coulomb per square centimeter
Cubic centimeter
Cubic centimeter-atmosphere
Cubic centimeter per gram
Cubic centimeter per second
Cubic decimeter (dm3 )
Cubic foot
Into
knot
mile per hour
foot per second squared
meter per second squared*
erg*
joule*
meter-kilogram
pound-foot
erg*
joule*
kilogram per (meter-second)*
pascal-second*
pound per (foot-second)
foot*
meter*
foot*
meter*
circular mil*
square centimeter
square inch
square millimeter
degree*
gon (grade)
radian
cord foot*
cubic foot*
ampere-second*
coulomb per square inch*
cubic foot
cubic inch
dram (U.S., fluid)
gallon (British)
gallon (U.S.)
liter*
minim (U.S.)
ounce (British, fluid)
ounce (U.S., fluid)
pint (British)
pint (U.S., dry)
pint (U.S., liquid)
joule*
watt-hour
cubic foot per pound
cubic foot per minute
liter per hour*
liter*
acre-foot
board foot*
cord*
cord foot*
cubic inch*
cubic meter*
cubic yard
Multiply by
0.019 438 4
0.022 369 4
0.032 808 4
0.01
1
10⫺7
1.020 ⫻ 10⫺8
7.376 ⫻ 10⫺8
980.665
9.806 65 ⫻ 10⫺5
0.001
0.001
0.006 72
100
30.48
66
20.1168
106
5.067 075
(␲/4)
(␲/4)
360
400
(2␲)
8
128
1
6.4516
3.531 47 ⫻ 10⫺5
0.061 023 744
0.270 512 2
2.199 69 ⫻ 10⫺4
2.641 72 ⫻ 10⫺4
0.001
16.230 73
0.035 195 1
0.033 814 02
0.001 759 75
0.001 816 17
0.002 113 376
0.101 325
2.814 58 ⫻ 10⫺5
0.016 018 5
0.002 118 88
3.6
1
2.295 68 ⫻ 10⫺5
12
(1/128)
(1/16)
1728
0.028 316 846 592
(1/27)
short
standard
long
DEAN #37261
(McGHP)
RIGHT
INTERACTIVE
2.39
GENERAL INFORMATION, CONVERSION TABLES, AND MATHEMATICS
cap height
base of text
TABLE 2.7 Conversion Factors (Continued)
To convert
Cubic foot (continued)
Cubic foot per hour
Cubic foot per pound
Cubic foot-atmosphere
Cubic inch
Cubic inch per minute
Cubic kilometer
Cubic meter per kilogram
Cubic yard
Cubic yard per minute
Cubit
Cup (U.S.)
Cup (metric)
Curie
Cycle per second
Dalton
Day (mean solar)
Debye
Decibel
Degree (plane angle)
Degree (angle) per foot
Degree (angle) per second
Degree Celsius
Degree Fahrenheit
Degree Rankine
Denier
Dipole length (e cm)
top of rh
base of rh
Into
gallon (British)
gallon (U.S.)
liter
liter per minute
cubic meter per kilogram
Btu
calorie
joule
kilogram-meter
liter-atmosphere
watt-hour
cubic foot
milliliter*
cubic centimeter per second
cubic mile
cubic foot per pound
bushel (British)
bushel (U.S.)
cubic foot*
cubic meter
liter
cubic foot per second*
gallon (British) per second
gallon (U.S.) per second
liter per second
inch*
milliliter; centimeter3
cubic centimeter*
becquerel*
hertz*
kilogram
unified atomic mass*
hour*
minute*
second*
coulomb-meter
neper
circumference
gon (grade)
minute (angle)*
quadrant
radian
revolution
second (angle)*
radian per meter
radian per second
degree Fahrenheit*
degree Rankine*
kelvin*
degree Celsius
kelvin
tex
coulomb-meter
Multiply by
6.228 835
7.480 519
28.316 847
0.471 947
0.062 428 0
2.719 48
685.298
2869.205
292.577
28.3168
0.797 001
(1/1728)
16.387 064
0.273 118
0.239 913
16.0185
21.0223
21.6962
27
0.764 554 86
764.555
0.45
2.802 98
3.366 23
12.742 58
18
236.6
200
3.7 ⫻ 1010
1
1.660 54 ⫻ 10⫺27
1
24
1440
86 400
3.335 64 ⫻ 10⫺30
0.115 129 255
(1/366)
1.111 11
60
(1/90)
(␲/180)
(1/360)
3600
0.057 261 5
0.017 453 3
1.8
1.8
1
(5/9)
(5/9)
(1/9)
1.602 18 ⫻ 10⫺21
short
standard
long
DEAN #37261
LEFT
(McGHP)
2.40
INTERACTIVE
top of rh
base of rh
SECTION 2
cap height
base of text
TABLE 2.7 Conversion Factors (Continued)
To convert
Drachm (British)
Drachm (British, fluid)
Dram (apothecaries or troy)
Dram (weight)
Dram (U.S., fluid)
Dyne
Dyne per centimeter
Dyne per square centimeter
Dyne-centimeter
Dyne-second/centimeter2
Electron charge
Electron charge-centimeter
(e cm)
Electron charge-centimeter2
Electron mass
Electronvolt
Ell
Em, pica
Into
Multiply by
dram (apothecaries or troy)*
cubic centimeter
dram (U.S., fluid)
minim (British)
ounce (British, fluid)
dram (weight)
grain*
gram*
ounce (troy)*
pennyweight*
pound (troy)*
scruple*
grain*
gram
ounce (weight)
pound (weight)
cubic centimeter
gallon (U.S.)
gill (U.S.)
milliliter
minim (U.S.)*
ounce (U.S., fluid)
pint (U.S., fluid)
kilogram (force)
newton*
pound (force)
newton per meter*
bar*
kilogram per square centimeter
millimeter of mercury (0⬚C)
millimeter of water (4⬚C)
newton per square meter*
pascal*
pound per square inch (psi)
erg*
foot-pound (force)
foot-poundal
joule*
kilogram-meter (force)
newton-meter*
poise*
pascal-second*
coulomb
coulomb-meter
1
3.551 633
0.960 760
60
(1/8)
2.194 285 7
60
3.887 934 6
(1/8)
2.5
(1/96)
3
27.343 75
1.771 845 2
(1/16)
(1/256)
3.696 691 2
(1/1024)
(1/32)
3.696 691 2
60
(1/8)
(1/128)
1.019 716 ⫻ 10⫺6
10⫺5
2.248 09 ⫻ 10⫺6
0.001
10⫺6
1.019 716 ⫻ 10⫺6
7.500 617 ⫻ 10⫺4
0.010 197 16
0.1
0.1
1.450 38 ⫻ 10⫺5
1
7.375 62 ⫻ 10⫺8
2.373 04 ⫻ 10⫺6
10⫺7
1.019 716 ⫻ 10⫺8
10⫺7
1
0.1
1.602 18 ⫻ 10⫺19
1.602 18 ⫻ 10⫺21
coulomb-meter squared
atomic mass unit
gram
erg
joule
kilojoule per mole
inch*
inch
millimeter
1.602 18 ⫻ 10⫺23
0.000 548 6
9.1096 ⫻ 10⫺28
1.602 18 ⫻ 10⫺12
1.602 18 ⫻ 10⫺19
96.4853
45
0.167
4.217 52
short
standard
long
DEAN #37261
(McGHP)
RIGHT
INTERACTIVE
2.41
GENERAL INFORMATION, CONVERSION TABLES, AND MATHEMATICS
cap height
base of text
TABLE 2.7 Conversion Factors (Continued)
To convert
EMU1 of capacitance
EMU of current
EMU of electric potential
EMU of inductance
EMU of quantity (charge)
EMU of resistance
EMU of work
ESU2 of capacitance
ESU of current
ESU of electric potential
ESU of inductance
ESU of quantity (charge)
ESU of resistance
ESU of work
Erg
Erg per second
Erg per (cm2 ⫻ second)
Erg per gauss
Fahrenheit scale
Fahrenheit temperature (⬚F)
Faraday (based on carbon-12)
Faraday (chemical)
Faraday (physical)
Fathom
Fermi
Foot
Foot of water (4⬚C)
Foot per minute
Foot-candle
Foot-lambert
1
2
top of rh
base of rh
Into
farad*
ampere*
volt*
henry*
coulomb
ohm
joule
farad
ampere
volt
henry
coulomb
ohm
joule
dyne-centimeter*
joule*
watt-hour
Btu
watt*
watt per square meter*
ampere-centimeter squared*
joule per tesla*
centigrade scale
Celsius temperature (⬚C)
coulomb
coulomb
coulomb
foot*
meter
meter*
centimeter*
inch*
mile (nautical)
mile (statute)
yard
atmosphere
bar
gram per square centimeter
inch of mercury (0⬚C)
pascal
centimeter per second*
knot
mile per hour
lumen per square foot*
lumen per square meter
lux
candela per square centimeter
candela per square foot
lambert
meter-lambert
EMU, the electromagnetic system of electrical units based on dynamics.
ESU, the electrostatic system of electrical units based on static data.
Multiply by
109
10
10⫺8
10⫺9
10
10⫺9
10⫺7
1.112 650 ⫻ 10⫺12
3.335 641⫻ 10⫺10
299.792 5
8.987 552 ⫻ 1011
3.335 556 ⫻ 10⫺11
8.987 552 ⫻ 1011
10⫺7
1
10⫺7
2.777 78 ⫻ 10⫺11
5.69 ⫻ 10⫺6
10⫺7
0.001
10
0.001
(5/9)
(⬚F ⫺ 32)(5/9)
96 487.0
96 495.7
96 521.9
6
1.828 8
10⫺15
30.48
12
1.645 788 ⫻ 10⫺4
1.893 939 ⫻ 10⫺4
(1/3)
0.029 499 8
0.029 499 8
30.48
0.882 671
2989.067
0.508
0.009 874 73
0.011 363 6
1
10.7639
10.76391
3.426 26 ⫻ 10⫺4
(1/␲)
0.001 076 39
10.7639
short
standard
long
DEAN #37261
LEFT
(McGHP)
2.42
INTERACTIVE
top of rh
base of rh
SECTION 2
cap height
base of text
TABLE 2.7 Conversion Factors (Continued)
To convert
Foot-pound
Foot-pound per minute
Foot-poundal
Franklin
Franklin per cm3
Franklin per cm2
Furlong
Gallon (British, imperial)
Gallon (U.S.)
Gallon (U.S.) per minute
Gamma
Gas constant
Gauss
Gilbert
Into
Btu
calorie
foot-poundal
horsepower (British)
joule
kilogram-meter
liter-atmosphere
newton-meter
watt-hour
horsepower (British)
horsepower (metric)
watt
Btu
calorie
foot-pound
joule
kilogram-meter
liter-atmosphere
watt-hour
coulomb
coulomb per cubic meter
coulomb per square meter
chain (Gunter’s)*
foot*
meter*
mile
bushel (British)
cubic decimeter, liter*
cubic foot
gallon (U.S., fluid)
gill (British)*
liter
ounce (British)*
quart (British)*
barrel (petroleum)
cubic decimeter, liter
cubic foot
gallon (British)
liter
ounce (U.S., fluid)*
quart (U.S., fluid)*
cubic foot per hour
cubic meter per hour
liter per minute
microgram*
calorie per mole-degree
joule per mole-degree
liter-atmosphere per
mole-degree
tesla*
weber per square meter*
ampere-turn
Multiply by
0.001 285 07
0.323 832
32.1740
5.050 51 ⫻ 10⫺7
1.355 818
0.138 255
0.013 380 9
1.355 818
3.766 161 ⫻ 10⫺4
3.030 30 ⫻ 10⫺5
3.072 33 ⫻ 10⫺5
0.022 597 0
3.994 11 ⫻ 10⫺5
0.010 064 99
0.031 081 0
0.042 140 11
0.004 297 10
4.158 91 ⫻ 10⫺4
1.170 56 ⫻ 10⫺5
3.335 64 ⫻ 10⫺10
3.335 64 ⫻ 10⫺4
3.335 64 ⫻ 10⫺6
10
600
201.168
(1/8)
(1/8)
4.546 90
0.160 544
1.200 95
32
4.546 09
160
4
(1/42)
3.785 41
0.133 680 56
0.832 674
3.785 41
128
4
8.020 83
0.227 125
3.785 412
1
1.987
8.3143
0.082 057
10⫺4
10⫺4
0.795 775
short
standard
long
DEAN #37261
(McGHP)
RIGHT
INTERACTIVE
2.43
GENERAL INFORMATION, CONVERSION TABLES, AND MATHEMATICS
cap height
base of text
TABLE 2.7 Conversion Factors (Continued)
To convert
Gill (British)
Gill (U.S.)
Gon (grade)
Grade
Grain
Gram
Gram per (centimeter-second)
Gram per cubic centimeter
Gram per square meter
Gram per ton (long)
Gram (force)
Gram per square centimeter
Gram-centimeter
Gram-square centimeter
Gray
Hartree
Hectare
Hefner unit
Hemisphere
top of rh
base of rh
Into
cubic centimeter, mL
cubic inch
gallon (British)
gill (U.S.)
ounce (British, fluid)*
pint (British)
cubic centimeter, mL
gallon (U.S.)
liter
ounce (U.S., fluid)*
quart (U.S.)
circumference
minute (angle)*
radian
radian
carat (metric)*
milligram*
ounce (weight)
ounce (troy)
pennyweight
pound
scruple
carat (metric)*
dram
grain
ounce (weight)
ounce (troy)
pennyweight
pound
ton (metric)*
poise*
kilogram per liter*
pound per cubic foot
pound per gallon (U.S.)
ounce per square foot
gram per ton (metric)
gram per ton (short)
dyne*
newton*
pascal*
joule*
pound-square foot
joule per kilogram*
electron volt
hertz
joule
acre
are*
meter squared
candela
sphere*
spherical right angle*
steradian
Multiply by
142.065
8.669 36
(1/32)
1.200 95
5
(1/4)
118.2941
(1/32)
0.118 294 1
4
(1/8)
(1/400)
54
(2␲/400)
(2␲/400)
0.323 994 55
64.798 91
0.002 285 714 3
(1/480)
(1/24)
(1/7000)
(1/20)
5
0.564 383 39
15.432 358
0.035 273 962
0.032 150 747
0.643 014 93
0.002 204 622 6
10⫺6
1
1
62.4280
8.345 40
0.327 706
0.984 207
0.892 857
980.665
0.009 806 65
98.0665
9.806 65 ⫻ 10⫺5
2.373 04 ⫻ 10⫺6
1
27.211 40
6.579 683 90 ⫻ 1015
4.359 75 ⫻ 10⫺18
2.471 054
100
104
0.9
0.5
4
(2␲)
short
standard
long
DEAN #37261
LEFT
(McGHP)
2.44
INTERACTIVE
top of rh
base of rh
SECTION 2
cap height
base of text
TABLE 2.7 Conversion Factors (Continued)
To convert
Hertz
Hogshead
Horsepower (British)
Horsepower (electric)
Horsepower-hour (British)
Hour (mean solar)
Hundredweight (long)
Hundredweight (short)
Inch
Inch of mercury (0⬚C)
Inch of water (4⬚C)
Inch per minute
Joule
Into
cycle per second*
gallon (U.S.)*
Btu per hour
foot pound per hour*
horsepower (metric)
joule per second
kilocalorie per hour
kilogram-meter per second
watt
watt*
Btu
foot-pound*
joule
kilocalorie
kilogram-meter
watt-hour
day
minute*
second*
week
kilogram*
pound*
ton (long)
ton (metric)
ton (short)*
hundredweight (long)
centimeter*
foot
mil*
atmosphere
inch of water (4⬚C)
millibar
millimeter of water (4⬚C)
pascal
pound per square inch, psi
inch of mercury (0⬚C)
millibar
millimeter of mercury (0⬚C)
pascal
pound per square inch, psi
foot per hour*
meter per hour*
millimeter per second
Btu
calorie*
centigrade heat unit, chu
centimeter-dyne*
cubic foot-atmosphere
cubic foot-(pound per in2)
erg*
foot-pound
foot-poundal
horsepower-hour (British)
liter-atmosphere
Multiply by
1
63
2544.43
1.98 ⫻ 106
1.013 87
745.700
641.186
76.0402
745.70
746
2544.43
1.98 ⫻ 106
2.684 52 ⫻ 106
641.186
2.737 45 ⫻ 105
745.7
(1/24)
60
3600
(1/168)
50.802 345 44
112
(1/20)
0.050 802 345
0.056
0.892 857
2.54
(1/12)
1000
0.033 421 05
13.5951
33.863 88
345.316
3386.388
0.491 1541
0.073 5559
2.490 89
1.868 32
249.089
0.036 1273
5
1.524
0.423 333
9.478 170 ⫻ 10⫺4
0.2390
5.265 65
107
0.000 348 529
0.005 121 959
107
0.737 562
23.7304
3.725 06 ⫻ 10⫺7
0.009 869 233
short
standard
long
DEAN #37261
(McGHP)
RIGHT
INTERACTIVE
2.45
GENERAL INFORMATION, CONVERSION TABLES, AND MATHEMATICS
cap height
base of text
TABLE 2.7 Conversion Factors (Continued)
To convert
Joule (continued)
Joule per centimeter
Joule per gram
Joule per second
Kilogram (force)
Kilometer
Kilowatt
Kilowatt-hour
Knot
Lambda
Lambert
Langley
League (nautical)
League (statute)
Light-year
Link
Liter
Liter per minute
Liter-atmosphere
top of rh
base of rh
Into
newton-meter*
watt-second*
kilogram (force)
newton*
pound (force)
Btu per pound
kilocalorie per kilogram
watt-hour per pound
watt*
dyne*
newton*
pound (force)
poundal
astronomical unit
mile (nautical)
mile (statute)
Btu per minute
foot-pound per second
horsepower (British)
horsepower (metric)
joule per second*
kilocalorie per hour
Btu
horsepower-hour (British)
joule*
kilocalorie
foot per minute
kilometer per hour*
mile (nautical) per hour*
mile (statute) per hour
decimeter cubed*
microliter*
candela per square meter
candela per square inch
foot-lambert
joule per square meter*
mile (nautical)*
mile (statute)*
astronomical unit
meter
chain*
cubic decimeter (dm3)*
cubic foot
gallon (British)
gallon (U.S.)
quart (British)
quart (U.S.)
cubic foot per hour
gallon (British) per hour
gallon (U.S.) per hour
Btu
calorie
cubic foot-atmosphere
cubic foot-pound per in2
Multiply by
1
1
10.197 16
100
22.4809
0.429 923
0.238 846
0.125 998
1
9.806 65 ⫻ 105
9.806 65
2.204 62
70.9316
6.684 59 ⫻ 10⫺9
0.539 956 80
0.621 371 192
56.8690
737.562
1.341 02
1.359 62
1000
859.845
3412.14
1.341 02
3.6 ⫻ 106
859.845
101.2686
1.852
1
1.150 78
10⫺6
1
(1/␲) ⫻ 104; 3183.099
2.053 61
929.030
4.184 ⫻ 104
3
3
6.323 97 ⫻ 104
9.460 73 ⫻ 1015
0.01
1
0.035 314 67
0.219 969
0.264 172 1
0.879 877
1.056 688
2.118 88
13.198
15.8503
0.096 037 6
24.2011
0.035 314 7
0.518 983
short
standard
long
DEAN #37261
LEFT
(McGHP)
2.46
INTERACTIVE
top of rh
base of rh
SECTION 2
cap height
base of text
TABLE 2.7 Conversion Factors (Continued)
To convert
Liter-atmosphere (continued)
Lumen per square centimeter
Lumen per square meter
Lux
Maxwell
Meter
Meter per second
Meter-candle
Meter-lambert
Mho (ohm⫺1)
Micron
Mil
Mile (nautical)
Mile (statute)
Mile per gallon (British)
Mile per gallon (U.S.)
Mile per hour
Milliliter
Millimeter of mercury (0⬚C)
Into
horsepower (British)
horsepower (metric)
joule*
kilogram-meter
watt-hour
lux*
phot*
lumen per square foot
lumen per square meter*
weber*
ångström*
fathom
foot
inch
mile (nautical)
mile (statute)
foot per minute
kilometer per hour*
knot
mile per hour
lux*
candela per square meter
foot-lambert
lambert*
siemen*
meter
inch*
micrometer*
foot
kilometer*
mile (statute)
chain (Gunter’s)*
chain (Ramsden’s)*
foot*
furlong*
kilometer*
light-year
link (Gunter’s)*
link (Ramsden’s)*
mile (nautical)
rod*
kilometer per liter
kilometer per liter
foot per minute
kilometer per hour*
knot
cubic centimeter*
atmosphere
dyne per square centimeter
millimeter of water (4⬚C)
pascal
pound per square inch (psi)
torr*
Multiply by
3.774 42 ⫻ 10⫺5
3.826 77 ⫻ 10⫺5
101.325
10.332 27
0.028 145 8
104
1
0.092 903 0
1
10⫺8
1010
0.546 807
3.280 839 895
39.370 078 740
5.399 568 ⫻ 10⫺4
6.213 712 ⫻ 10⫺4
196.850
3.6
1.943 844
2.236 936
1
(1/␲)
0.092 903 0
10⫺4
1
10⫺6
0.001
25.4
6076.115 49
1.852
1.150 78
80
52.8
5280
8
1.609 344
1.701 11 ⫻ 10⫺11
8000
5280
0.868 976
320
0.354 006
0.425 144
88
1.609 344
0.868 976
1
(1/760)
1333.224
13.5951
133.322
0.019 336 8
1
short
standard
long
DEAN #37261
(McGHP)
RIGHT
INTERACTIVE
2.47
GENERAL INFORMATION, CONVERSION TABLES, AND MATHEMATICS
cap height
base of text
TABLE 2.7 Conversion Factors (Continued)
To convert
Millimeter of water (4⬚C)
Minim (British)
Minim (U.S.)
Minute (plane angle)
Minute
Month (mean of 4-year period)
Nail (British)
Nanometer
Neper
Nuclear magneton
Neutron mass
Newton
Newton per square meter
Newton-meter
Nit
Noggin (British)
Nox
Oersted
Ohm (mean international)
Ohm (U.S. international)
Ohm per foot
Ounce (avoirdupois)
Ounce (troy)
Ounce (British, fluid)
top of rh
base of rh
Into
atmosphere
millibar*
millimeter of mercury (0⬚C)
pascal*
pound per square inch
milliliter
minim (U.S.)
milliliter
circumference
degree (angle)
gon
radian
hour
second
day
hour
week
inch*
ångström*
decibel
joule per tesla
atomic mass unit
gram
dyne*
kilogram (force)
pound (force)
poundal
See pascal
foot-pound
joule*
kilogram-meter
watt-second*
candela per square meter*
gill (British)*
lux*
ampere per meter (in practice)
ohm
ohm
ohm per meter
dram*
grain*
gram*
ounce (troy)
pound
grain*
gram*
ounce (avoirdupois)
pennyweight*
pound (avoirdupois)
scruple*
cubic centimeter
gallon (British)
milliliter
Multiply by
0.009 678 41
0.098 066 5
0.073 555 9
9.806 65
0.001 422 33
0.059 193 9
0.960 760
0.061 611 5
4.629 63 ⫻ 10⫺5
(1/60)
(1/54)
(␲/10,800)
(1/60)
60
30.4375
730.5
4.348 21
2.25
10
8.685 890
5.050 79 ⫻ 10⫺27
1.008 66
1.6749 ⫻ 10⫺24
105
0.101 971 6
0.224 809
7.233 01
0.737 562
1
0.101 971 6
1
1
1
0.001
(1000/4␲); 79.577 47
1.000 49
1.000 495
3.280 84
16
437.5
28.349 5
0.911 458 33
(1/16)
480
31.1035
1.097 142 9
20
0.068 571 429
24
28.413 06
(1/160)
28.413 06
short
standard
long
DEAN #37261
LEFT
(McGHP)
2.48
INTERACTIVE
top of rh
base of rh
SECTION 2
cap height
base of text
TABLE 2.7 Conversion Factors (Continued)
To convert
Ounce (British, fluid)
(continued)
Ounce (U.S., fluid)
Ounce (avoirdupois) per cubic
foot
Ounce (avoirdupois)/gallon
(U.S.)
Ounce (avoirdupois) per ton
(long)
Ounce (avoirdupois) per ton
(short)
Parsec
Part per million
Pascal
Pascal-second
Peck (British)
Peck (U.S.)
Pennyweight
Phot
Pica (printer’s)
Pint (British)
Pint (U.S., dry)
Pint (U.S., fluid)
Into
Multiply by
minim (British)
ounce (U.S., fluid)
pint (British)
quart (British)
cubic centimeter
gallon (U.S.)
milliliter
pint (U.S., fluid)
quart (U.S., fluid)
kilogram per cubic meter
480
0.960 760
(1/20)
(1/40)
29.573 530
(1/128)
29.573 530
(1/16)
(1/32)
1.001 154
gram per liter
7.489 15
gram per ton (metric)
27.9018
milligram per kilogram
gram per ton (metric)*
27.9018
31.25
milligram per kilogram*
light-year
milligram per kilogram*
milliliter per cubic meter*
atmosphere
bar*
dyne per square centimeter*
inch of mercury
millimeter of mercury
millimeter of water
newton per square meter*
pound per square inch
poundal per square foot
poise*
gallon (British)*
bushel (U.S.)*
grain*
gram*
ounce (troy)
pound
lux*
inch
point*
gallon (British)
liter
pint (U.S., fluid)
quart (British)
bushel (U.S.)
liter
peck (U.S.)
pint (British)
quart (U.S., dry)
gallon (U.S.)
liter
31.25
3.261 636
1
1
9.869 233 ⫻ 10⫺6
10⫺5
10
2.953 00 ⫻ 10⫺4
7.500 62 ⫻ 10⫺3
0.101 972
1
1.450 377 ⫻ 10⫺4
0.671 969
10
2
0.25
24
1.555 173 84
(1/20)
0.003 428 571 4
104
0.167
12
(1/8)
0.568 261
1.200 95
0.5
(1/64)
0.550 610 5
(1/16)
0.968 939
0.5
(1/8)
0.473 176 5
short
standard
long
DEAN #37261
(McGHP)
RIGHT
INTERACTIVE
2.49
GENERAL INFORMATION, CONVERSION TABLES, AND MATHEMATICS
cap height
base of text
TABLE 2.7 Conversion Factors (Continued)
To convert
Pint (U.S., fluid)
(continued)
Planck’s constant
Point (printer’s, Didot)
Point (printer’s, U.S.)
Poise
Polarizability volume (4␲⑀0 cm3)
Pole (British)
Pottle (British)
Pound
Pound (troy)
Pound per cubic foot
Pound per cubic inch
Pound
Pound
Pound
Pound
Pound
Pound
Pound
Pound
per foot
per (foot-second)
per gallon (U.S.)
per hour
per inch
per minute
per square foot
(force)
Pound per square inch
Pound-second per square inch
Poundal
Poundal per square foot
Poundal-foot
Poundal-second per square foot
Proof (U.S.)
Proton mass
Puncheon (British)
top of rh
base of rh
Into
pint (British)
quart (U.S., fluid)*
joule-second
millimeter
millimeter*
dyne-second per square centimeter*
pascal-second*
coulomb squared-(meter squared per joule)
foot*
gallon (British)*
gram*
ounce (weight)*
ton (long)
ton (short)
grain
gram*
ounce (troy)*
pennyweight
pound (weight)
scruple*
kilogram per cubic meter
gram per cubic centimeter
pound per cubic foot*
kilogram per meter
pascal-second
gram per liter
kilogram per day
kilogram per meter
kilogram per hour
kilogram per square meter
kilogram (force)
newton
poundal
atmosphere
bar
inch of mercury (0⬚C)
millimeter of mercury (0⬚C)
millimeter of water (4⬚C)
pascal
pound per square foot
pascal-second
gram (force)
newton
pound (force)
pascal
newton-meter
pascal-second
percent alcohol by volume*
atomic mass unit
gram
gallon (British)
Multiply by
0.832 674
0.5
6.626 08 ⫻ 10⫺34
0.376 065 03
0.351 459 8
1
0.1
1.112 65 ⫻ 10⫺16
16.5
0.5
453.592 37
16
4.464 285 7 ⫻ 10⫺4
(1/2000)
5760
373.241 721 6
12
240
0.822 857 14
288
16.018 46
27.679 905
1728
1.488 16
1.488 16
119.8264
10.886 22
17.857 97
27.215 54
4.882 43
0.453 592
4.448 222
32.1740
0.068 046 0
0.068 948 0
2.036 02
51.7149
703.070
6894.757
144
6894.76
14.0981
0.138 255
0.031 081 0
1.488 164
0.042 140 1
1.488 164
0.5
1.007 28
1.6726 ⫻ 10⫺24
70
short
standard
long
DEAN #37261
LEFT
(McGHP)
2.50
INTERACTIVE
top of rh
base of rh
SECTION 2
cap height
base of text
TABLE 2.7 Conversion Factors (Continued)
To convert
Quad
Quadrant
Quadrupole area (e cm2)
Quart (British)
Quart (U.S., dry)
Quart (U.S., fluid)
Quartern (British, fluid)
Quintal (metric)
Rad (absorbed dose)
Radian
Radian per centimeter
Radian per second
Radian per second squared
Rankin (degree)
Ream
Register ton
Rem (dose equivalent)
Revolution
Revolution per minute
Revolution per minute squared
Revolution per second squared
Reyn
Into
Btu
joule
circumference*
degree (angle)*
gon (grade)*
minute (angle)*
radian
coulomb meter squared
gallon (British)*
liter
ounce (British, fluid)*
pint (British)*
quart (U.S., fluid)
bushel (U.S.)
cubic foot
liter
peck (U.S.)
pint (U.S., dry)*
gallon (U.S.)*
liter
ounce (U.S., fluid)*
pint (U.S., fluid)
quart (British)
gill (British)*
kilogram*
gray*
joule per kilogram*
circumference
degree (angle)
minute (angle)
quadrant
revolution
degree per millimeter
degree per inch
revolution per minute
revolution per minute squared
kelvin
quire*
sheet
cubic foot*
cubic meter
sievert*
degree (angle)
gon*
quadrant*
radian
degree (angle) per second*
radian per second
radian per second squared
radian per second squared
revolution per minute squared
pascal-second
pound-second per square inch
Multiply by
1015
1.055 ⫻ 1018
0.25
90
100
5400
(␲/2)
1.602 18 ⫻ 10⫺23
0.25
1.136 523
40
2
1.200 95
(1/32)
0.038 889 25
1.101 221
(1/8)
2
0.25
0.946 529
32
2
0.832 674
0.5
100
0.01
0.01
(1/2␲)
57.295 780
3437.75
(2/␲)
(1/2␲)
5.729 58
145.531
9.549 30
572.958
(5/9)
20
480 or 500
100
2.831 685
0.01
360
400
4
(2␲)
6
0.104 720
0.001 745 33
6.283 185
3600
6894.76
1
short
standard
long
DEAN #37261
(McGHP)
RIGHT
INTERACTIVE
2.51
GENERAL INFORMATION, CONVERSION TABLES, AND MATHEMATICS
cap height
base of text
TABLE 2.7 Conversion Factors (Continued)
To convert
Rhe
Right angle
Rod (British, volume)
Rod (surveyer’s measure)
Roentgen
Rood (British)
Rydberg
Scruple
Second (plane angle)
Section
Siemens
Slug
Speed of light
Sphere
Square centimeter
Square chain (Gunter’s)
Square chain (Ramsden’s)
Square degree (angle)
Square foot
Square inch
Square kilometer
Square link (Gunter’s)
Square link (Ramsden’s)
Square meter
top of rh
base of rh
Into
per pascal-second*
degree*
radian
cubic foot*
chain (Gunter’s)*
foot*
link (Gunter’s)*
meter*
coulomb per kilogram
acre*
square meter
joule
dram (troy)
grain*
gram*
ounce (weight)
ounce (troy)
pennyweight
pound
degree
minute
radian
square mile*
mho (ohm⫺1)*
geepound*
kilogram
pound
centimeter per second
steradian
circular mil
circular millimeter
square inch
acre*
square foot*
square meter
square foot*
steradian
acre
square centimeter
square meter
square rod
circular mil
circular millimeter
square centimeter
acre
hectare*
square mile
square foot*
square foot*
are*
square foot
square mile
Multiply by
10
90
(␲/2)
1000
0.25
16.5
25
5.0292
2.58 ⫻ 10⫺4
0.25
1011.714 1
2.179 87 ⫻ 10⫺18
(1/3)
20
1.295 978 2
0.045 714 286
(1/24)
(10/12)
(1/350)
2.777 78 ⫻ 10⫺4
(1/60)
(␲/6.48 ⫻ 105)
1
1
1
14.593 90
32.1740
2.997 924 58 ⫻ 1010
(4␲)
1.973 53 ⫻ 105
127.3240
0.155 000 31
0.1
4356
404.686
104
3.046 17 ⫻ 10⫺4
2.295 68 ⫻ 10⫺5
929.0304
0.092 903 04
0.003 673 09
1.273 240 ⫻ 106
821.4432
6.4516
247.1054
100
0.386 102 16
0.4356
1
0.01
10.763 91
3.861 01 ⫻ 10⫺7
short
standard
long
DEAN #37261
LEFT
(McGHP)
2.52
INTERACTIVE
top of rh
base of rh
SECTION 2
cap height
base of text
TABLE 2.7 Conversion Factors (Continued)
To convert
Square meter (continued)
Square mile
Square rod
Square yard
Statampere
Statcoulomb
Statfarad
Stathenry
Statmho
Statohm
Statvolt
Statweber
Steradian
Stere
Stilb
Stokes (kinematic viscosity)
Stone (British)
Svedberg
Tablespoon (metric)
Teaspoon (metric)
Tesla
Tex
Therm
Ton (assay)
Ton (long)
Ton (metric)
Ton (short)
Ton (force, long)
Ton (force, metric)
Into
square rod
square yard
acre*
square kilometer
township
acre
square foot
square meter
square foot*
square inch*
square meter*
square rod
ampere
coulomb
farad
henry
siemens
ohm
volt
weber
sphere
spherical right angle
square degree
cubic meter*
candela/cm2
square meter per second*
pound*
second*
cubic centimeter*; milliter
cubic centimeter*; milliliter
weber per square meter*
denier*
gram per kilometer*
Btu*
joule*
gram
hundredweight (long)*
hundredweight (short)*
kilogram
pound*
ton (metric)
ton (short)
hundredweight (long)
hundredweight (short)
kilogram*
pound
ton (long)
ton (short)*
kilogram
pound*
newton
newton
Multiply by
0.039 536 9
1.195 990
640
2.589 988 110
(1/36)
(1/160)
272.25
25.292 853
9
1296
0.836 127 36
0.033 057 85
3.335 641 ⫻ 10⫺10
3.335 641 ⫻ 10⫺10
1.112 650 ⫻ 10⫺12
8.987 552 ⫻ 1011
1.112 650 ⫻ 10⫺12
8.987 552 ⫻ 1011
299.7925
299.7925
(1/4␲)
(2/␲)
3282.81
1
1
10⫺4
14
10⫺13
14.79
4.929
1
9
1
105
1.054 804 ⫻ 108
29.166 67
20
22.4
1016.046 908 8
2240
1.016 046 9
1.12
19.684 131
22.046 226
1000
2204.6226
0.984 206 53
1.102 311 3
907.184 74
2000
1186.553
9806.65
short
standard
long
DEAN #37261
(McGHP)
RIGHT
INTERACTIVE
2.53
GENERAL INFORMATION, CONVERSION TABLES, AND MATHEMATICS
cap height
base of text
TABLE 2.7 Conversion Factors (Continued)
To convert
Ton (force, short)
Ton (force, long)/ft2
Ton (force, metric)/m2
Ton (force, short)/ft2
Tonne (metric)
Torr
Township (U.S.)
Unified atomic mass unit
Unit pole
Volt (mean international)
Volt (U.S. international)
Volt-second
Watt
Watt per square inch
Watt-hour
Watt-second
Weber
Week
Wey (British, capacity)
Wey (British, mass)
X unit
Yard
Year (mean of 4-years)
Year (sidereal)
top of rh
base of rh
Into
newton
bar
pascal
bar
pascal
bar
pascal
kilogram*
atmosphere
millibar
millimeter of mercury* (0⬚C)
pascal
square kilometer
square mile*
kilogram
weber
volt
volt
weber*
Btu per hour
calorie per minute
erg per second*
foot-pound per minute
horsepower (British)
horsepower (metric)
joule per second*
kilogram-meter per second
watt per square meter
Btu
calorie
foot-pound
horsepower-hour (British)
horsepower-hour (metric)
joule*
liter-atmosphere
joule*
maxwell*
day*
hour*
bushel (British)
pound
meter
fathom*
meter
day
week
day (mean solar)
Multiply by
8896.44
1.072 518
1.072 518 ⫻ 105
0.098 066 5
9806.65
0.957 605
9.576 05 ⫻ 104
1000
(1/760)
1.333 224
1
133.322; (101 325/760)
93.2396
36
1.660 54 ⫻ 10⫺27
1.256 637 ⫻ 10⫺7
1.000 34
1.000 330
1
3.412 14
14.3308
107
44.2537
0.001 341 02
0.001 359 62
1
0.101 972
1550.003
3.412 14
859.845
2655.22
0.001 341 02
0.001 359 62
3600
35.5292
1
108
7
168
40 (variable)
252 (variable)
1.002 02 ⫻ 10⫺13
0.5
0.9144
365.25
52.178 87
365.256 36
short
standard
long
DEAN #37261
LEFT
(McGHP)
2.54
INTERACTIVE
top of rh
base of rh
SECTION 2
cap height
base of text
TABLE 2.8 Temperature Conversion Table
The column of figures in bold and which is headed “Reading in ⬚F. or ⬚C. to be converted” refers to the
temperature either in degrees Fahrenheit or Celsius which it is desired to convert into the other scale. If converting
from Fahrenheit degrees to Celsius degrees, the equivalent temperature will be found in the column headed
“⬚C.”; while if converting from degrees Celsius to degrees Fahrenheit, the equivalent temperature will be found
in the column headed “⬚F.” This arrangement is very similar to that of Sauveur and Boylston, copyrighted 1920,
and is published with their permission.
⬚F.
Reading in
⬚F. or ⬚C.
to be
converted
Reading in
⬚F. or ⬚C.
to be
converted
⬚C.
⬚C.
........
........
........
........
........
⫺458
⫺456
⫺454
⫺452
⫺450
⫺272.22
⫺271.11
⫺270.00
⫺268.89
⫺267.78
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
⫺378
⫺376
⫺374
⫺372
⫺370
⫺227.78
⫺226.67
⫺225.56
⫺224.44
⫺223.33
........
........
........
........
........
⫺448
⫺446
⫺444
⫺442
⫺440
⫺266.67
⫺265.56
⫺264.44
⫺263.33
⫺262.22
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
⫺368
⫺366
⫺364
⫺362
⫺360
⫺222.22
⫺221.11
⫺220.00
⫺218.89
⫺217.78
........
........
........
........
........
⫺438
⫺436
⫺434
⫺432
⫺430
⫺261.11
⫺260.00
⫺258.89
⫺257.78
⫺256.67
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
⫺358
⫺356
⫺354
⫺352
⫺350
⫺216.67
⫺215.56
⫺214.44
⫺213.33
⫺212.22
........
........
........
........
........
⫺428
⫺426
⫺424
⫺422
⫺420
⫺255.56
⫺254.44
⫺253.33
⫺252.22
⫺251.11
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
⫺348
⫺346
⫺344
⫺342
⫺340
⫺211.11
⫺210.00
⫺208.89
⫺207.78
⫺206.67
........
........
........
........
........
⫺418
⫺416
⫺414
⫺412
⫺410
⫺250.00
⫺248.89
⫺247.78
⫺246.67
⫺245.56
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
⫺338
⫺336
⫺334
⫺332
⫺330
⫺205.56
⫺204.44
⫺203.33
⫺202.22
⫺201.11
........
........
........
........
........
⫺408
⫺406
⫺404
⫺402
⫺400
⫺244.44
⫺243.33
⫺242.22
⫺241.11
⫺240.00
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
⫺328
⫺326
⫺324
⫺322
⫺320
⫺200.00
⫺198.89
⫺197.78
⫺196.67
⫺195.56
........
........
........
........
........
⫺398
⫺396
⫺394
⫺392
⫺390
⫺238.89
⫺237.78
⫺236.67
⫺235.56
⫺234.44
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
⫺318
⫺316
⫺314
⫺312
⫺310
⫺194.44
⫺193.33
⫺192.22
⫺191.11
⫺190.00
........
........
........
........
........
⫺388
⫺386
⫺384
⫺382
⫺380
⫺233.33
⫺232.22
⫺231.11
⫺230.00
⫺228.89
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
⫺308
⫺306
⫺304
⫺302
⫺300
⫺188.89
⫺187.78
⫺186.67
⫺185.56
⫺184.44
⬚F.
short
standard
long
DEAN #37261
(McGHP)
RIGHT
INTERACTIVE
2.55
GENERAL INFORMATION, CONVERSION TABLES, AND MATHEMATICS
cap height
base of text
TABLE 2.8 Temperature Conversion Table (Continued)
⬚F.
Reading in
⬚F. or ⬚C.
to be
converted
⬚C.
top of rh
base of rh
⬚F.
Reading in
⬚F. or ⬚C.
to be
converted
⬚C.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.....
.....
.....
.....
.....
⫺298
⫺296
⫺294
⫺292
⫺290
⫺183.33
⫺182.22
⫺181.11
⫺180.00
⫺178.89
⫺342.4
⫺338.8
⫺335.2
⫺331.6
⫺328.0
⫺208
⫺206
⫺204
⫺202
⫺200
⫺133.33
⫺132.22
⫺131.11
⫺130.00
⫺128.89
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.....
.....
.....
.....
.....
⫺288
⫺286
⫺284
⫺282
⫺280
⫺177.78
⫺176.67
⫺175.56
⫺174.44
⫺173.33
⫺324.4
⫺320.8
⫺317.2
⫺313.6
⫺310.0
⫺198
⫺196
⫺194
⫺192
⫺190
⫺127.78
⫺126.67
⫺125.56
⫺124.44
⫺123.33
........
........
........
⫺457.6
⫺454.0
⫺278
⫺276
⫺274
⫺272
⫺270
⫺172.22
⫺171.11
⫺170.00
⫺168.89
⫺167.78
⫺306.4
⫺302.8
⫺299.2
⫺295.6
⫺292.0
⫺188
⫺186
⫺184
⫺182
⫺180
⫺122.22
⫺121.11
⫺120.00
⫺118.89
⫺117.78
⫺450.4
⫺446.8
⫺443.2
⫺439.6
⫺436.0
⫺268
⫺266
⫺264
⫺262
⫺260
⫺166.67
⫺165.56
⫺164.44
⫺163.33
⫺162.22
⫺288.4
⫺284.8
⫺281.2
⫺277.6
⫺274.0
⫺178
⫺176
⫺174
⫺172
⫺170
⫺116.67
⫺115.56
⫺114.44
⫺113.33
⫺112.22
⫺432.4
⫺428.8
⫺425.2
⫺421.6
⫺418.0
⫺258
⫺256
⫺254
⫺252
⫺250
⫺161.11
⫺160.00
⫺158.89
⫺157.78
⫺156.67
⫺270.4
⫺266.8
⫺263.2
⫺259.6
⫺256.0
⫺168
⫺166
⫺164
⫺162
⫺160
⫺111.11
⫺110.00
⫺108.89
⫺107.78
⫺106.67
⫺414.4
⫺410.8
⫺407.2
⫺403.6
⫺400.0
⫺248
⫺246
⫺244
⫺242
⫺240
⫺155.56
⫺154.44
⫺153.33
⫺152.22
⫺151.11
⫺252.4
⫺248.8
⫺245.2
⫺241.6
⫺238.0
⫺158
⫺156
⫺154
⫺152
⫺150
⫺105.56
⫺104.44
⫺103.33
⫺102.22
⫺101.11
⫺396.4
⫺392.8
⫺389.2
⫺385.6
⫺382.0
⫺238
⫺236
⫺234
⫺232
⫺230
⫺150.00
⫺148.89
⫺147.78
⫺146.67
⫺145.56
⫺234.4
⫺230.8
⫺227.2
⫺223.6
⫺220.0
⫺148
⫺146
⫺144
⫺142
⫺140
⫺100.00
⫺98.89
⫺97.78
⫺96.67
⫺95.56
⫺378.4
⫺374.8
⫺371.2
⫺367.6
⫺364.0
⫺228
⫺226
⫺224
⫺222
⫺220
⫺144.44
⫺143.33
⫺142.22
⫺141.11
⫺140.00
⫺216.4
⫺212.8
⫺209.2
⫺205.6
⫺202.0
⫺138
⫺136
⫺134
⫺132
⫺130
⫺94.44
⫺93.33
⫺92.22
⫺91.11
⫺90.00
⫺360.4
⫺356.8
⫺353.2
⫺349.6
⫺346.0
⫺218
⫺216
⫺214
⫺212
⫺210
⫺138.89
⫺137.78
⫺136.67
⫺135.56
⫺134.44
⫺198.4
⫺194.8
⫺191.2
⫺187.6
⫺184.0
⫺128
⫺126
⫺124
⫺122
⫺120
⫺88.89
⫺87.78
⫺86.67
⫺85.56
⫺84.44
short
standard
long
DEAN #37261
LEFT
(McGHP)
2.56
INTERACTIVE
top of rh
base of rh
SECTION 2
cap height
base of text
TABLE 2.8 Temperature Conversion Table (Continued)
⬚F.
Reading in
⬚F. or ⬚C.
to be
converted
⬚C.
⫺180.4
⫺176.8
⫺173.2
⫺169.6
⫺166.0
⫺118
⫺116
⫺114
⫺112
⫺110
⫺83.33
⫺82.22
⫺81.11
⫺80.00
⫺78.89
⫺18.4
⫺14.8
⫺11.2
⫺7.6
⫺4.0
⫺28
⫺26
⫺24
⫺22
⫺20
⫺33.33
⫺32.22
⫺31.11
⫺30.00
⫺28.89
⫺162.4
⫺158.8
⫺155.2
⫺151.6
⫺148.0
⫺108
⫺106
⫺104
⫺102
⫺100
⫺77.78
⫺76.67
⫺75.56
⫺74.44
⫺73.33
⫺0.4
⫹3.2
⫹6.8
⫹10.4
⫹14.0
⫺18
⫺16
⫺14
⫺12
⫺10
⫺27.78
⫺26.67
⫺25.56
⫺24.44
⫺23.33
⫺144.4
⫺140.8
⫺137.2
⫺133.6
⫺130.0
⫺98
⫺96
⫺94
⫺92
⫺90
⫺72.22
⫺71.11
⫺70.00
⫺68.89
⫺67.78
⫹17.6
⫹19.4
⫹21.2
⫹23.0
⫹24.8
⫺8
⫺7
⫺6
⫺5
⫺4
⫺22.22
⫺21.67
⫺21.11
⫺20.56
⫺20.00
⫺126.4
⫺122.8
⫺119.2
⫺115.6
⫺112.0
⫺88
⫺86
⫺84
⫺82
⫺80
⫺66.67
⫺65.56
⫺64.44
⫺63.33
⫺62.22
⫹26.6
⫹28.4
⫹30.2
⫹32.0
⫹33.8
⫺3
⫺2
⫺1
⫾0
⫹1
⫺19.44
⫺18.89
⫺18.33
⫺17.78
⫺17.22
⫺108.4
⫺104.8
⫺101.2
⫺97.6
⫺94.0
⫺78
⫺76
⫺74
⫺72
⫺70
⫺61.11
⫺60.00
⫺58.89
⫺57.78
⫺56.67
⫹35.6
⫹37.4
⫹39.2
⫹41.0
⫹42.8
⫹2
⫹3
⫹4
⫹5
⫹6
⫺16.67
⫺16.11
⫺15.56
⫺15.00
⫺14.44
⫺90.4
⫺86.8
⫺83.2
⫺79.6
⫺76.0
⫺68
⫺66
⫺64
⫺62
⫺60
⫺55.56
⫺54.44
⫺53.33
⫺52.22
⫺51.11
⫹44.6
⫹46.4
⫹48.2
⫹50.0
⫹51.8
⫹7
⫹8
⫹9
⫹10
⫹11
⫺13.89
⫺13.33
⫺12.78
⫺12.22
⫺11.67
⫺72.4
⫺68.8
⫺65.2
⫺61.6
⫺58.0
⫺58
⫺56
⫺54
⫺52
⫺50
⫺50.00
⫺48.89
⫺47.78
⫺46.67
⫺45.56
⫹53.6
⫹55.4
⫹57.2
⫹59.0
⫹60.8
⫹12
⫹13
⫹14
⫹15
⫹16
⫺11.11
⫺10.56
⫺10.00
⫺9.44
⫺8.89
⫺54.4
⫺50.8
⫺47.2
⫺43.6
⫺40.0
⫺48
⫺46
⫺44
⫺42
⫺40
⫺44.44
⫺43.33
⫺42.22
⫺41.11
⫺40.00
⫹62.6
⫹64.4
⫹66.2
⫹68.0
⫹69.8
⫹17
⫹18
⫹19
⫹20
⫹21
⫺8.33
⫺7.78
⫺7.22
⫺6.67
⫺6.11
⫺36.4
⫺32.8
⫺29.2
⫺25.6
⫺22.0
⫺38
⫺36
⫺34
⫺32
⫺30
⫺38.89
⫺37.78
⫺36.67
⫺35.56
⫺34.44
⫹71.6
⫹73.4
⫹75.2
⫹77.0
⫹78.8
⫹22
⫹23
⫹24
⫹25
⫹26
⫺5.56
⫺5.00
⫺4.44
⫺3.89
⫺3.33
⬚F.
Reading in
⬚F. or ⬚C.
to be
converted
⬚C.
short
standard
long
DEAN #37261
(McGHP)
RIGHT
INTERACTIVE
2.57
GENERAL INFORMATION, CONVERSION TABLES, AND MATHEMATICS
cap height
base of text
TABLE 2.8 Temperature Conversion Table (Continued)
⬚F.
Reading in
⬚F. or ⬚C.
to be
converted
⬚C.
top of rh
base of rh
⬚F.
Reading in
⬚F. or ⬚C.
to be
converted
⬚C.
⫹80.6
⫹82.4
⫹84.2
⫹86.0
⫹87.8
⫹27
⫹28
⫹29
⫹30
⫹31
⫺2.78
⫺2.22
⫺1.67
⫺1.11
⫺0.56
⫹161.6
⫹163.4
⫹165.2
⫹167.0
⫹168.8
⫹72
⫹73
⫹74
⫹75
⫹76
⫹22.22
⫹22.78
⫹23.33
⫹23.89
⫹24.44
⫹89.6
⫹91.4
⫹93.2
⫹95.0
⫹96.8
⫹32
⫹33
⫹34
⫹35
⫹36
⫾0.00
⫹0.56
⫹1.11
⫹1.67
⫹2.22
⫹170.6
⫹172.4
⫹174.2
⫹176.0
⫹177.8
⫹77
⫹78
⫹79
⫹80
⫹81
⫹25.00
⫹25.56
⫹26.11
⫹26.67
⫹27.22
⫹98.6
⫹100.4
⫹102.2
⫹104.0
⫹105.8
⫹37
⫹38
⫹39
⫹40
⫹41
⫹2.78
⫹3.33
⫹3.89
⫹4.44
⫹5.00
⫹179.6
⫹181.4
⫹183.2
⫹185.0
⫹186.8
⫹82
⫹83
⫹84
⫹85
⫹86
⫹27.78
⫹28.33
⫹28.89
⫹29.44
⫹30.00
⫹107.6
⫹109.4
⫹111.2
⫹113.0
⫹114.8
⫹42
⫹43
⫹44
⫹45
⫹46
⫹5.56
⫹6.11
⫹6.67
⫹7.22
⫹7.78
⫹188.6
⫹190.4
⫹192.2
⫹194.0
⫹195.8
⫹87
⫹88
⫹89
⫹90
⫹91
⫹30.56
⫹31.11
⫹31.67
⫹32.22
⫹32.78
⫹116.6
⫹118.4
⫹120.2
⫹122.0
⫹123.8
⫹47
⫹48
⫹49
⫹50
⫹51
⫹8.33
⫹8.89
⫹9.44
⫹10.00
⫹10.56
⫹197.6
⫹199.4
⫹201.2
⫹203.0
⫹204.8
⫹92
⫹93
⫹94
⫹95
⫹96
⫹33.33
⫹33.89
⫹34.44
⫹35.00
⫹35.56
⫹125.6
⫹127.4
⫹129.2
⫹131.0
⫹132.8
⫹52
⫹53
⫹54
⫹55
⫹56
⫹11.11
⫹11.67
⫹12.22
⫹12.78
⫹13.33
⫹206.6
⫹208.4
⫹210.2
⫹212.0
⫹213.8
⫹97
⫹98
⫹99
⫹100
⫹101
⫹36.11
⫹36.67
⫹37.22
⫹37.78
⫹38.33
⫹134.6
⫹136.4
⫹138.2
⫹140.0
⫹141.8
⫹57
⫹58
⫹59
⫹60
⫹61
⫹13.89
⫹14.44
⫹15.00
⫹15.56
⫹16.11
⫹215.6
⫹217.4
⫹219.2
⫹221.0
⫹222.8
⫹102
⫹103
⫹104
⫹105
⫹106
⫹38.89
⫹39.44
⫹40.00
⫹40.56
⫹41.11
⫹143.6
⫹145.4
⫹147.2
⫹149.0
⫹150.8
⫹62
⫹63
⫹64
⫹65
⫹66
⫹16.67
⫹17.22
⫹17.78
⫹18.33
⫹18.89
⫹224.6
⫹226.4
⫹228.2
⫹230.0
⫹231.8
⫹107
⫹108
⫹109
⫹110
⫹111
⫹41.67
⫹42.22
⫹42.78
⫹43.33
⫹43.89
⫹152.6
⫹154.4
⫹156.2
⫹158.0
⫹159.8
⫹67
⫹68
⫹69
⫹70
⫹71
⫹19.44
⫹20.00
⫹20.56
⫹21.11
⫹21.67
⫹233.6
⫹235.4
⫹237.2
⫹239.0
⫹240.8
⫹112
⫹113
⫹114
⫹115
⫹116
⫹44.44
⫹45.00
⫹45.56
⫹46.11
⫹46.67
short
standard
long
DEAN #37261
LEFT
(McGHP)
2.58
INTERACTIVE
top of rh
base of rh
SECTION 2
cap height
base of text
TABLE 2.8 Temperature Conversion Table (Continued)
⬚F.
Reading in
⬚F. or ⬚C.
to be
converted
⬚F.
Reading in
⬚F. or ⬚C.
to be
converted
⬚C.
⬚C.
⫹242.6
⫹244.4
⫹246.2
⫹248.0
⫹249.8
⫹117
⫹118
⫹119
⫹120
⫹121
⫹47.22
⫹47.78
⫹48.33
⫹48.89
⫹49.44
⫹323.6
⫹325.4
⫹327.2
⫹329.0
⫹330.8
⫹162
⫹163
⫹164
⫹165
⫹166
⫹72.22
⫹72.78
⫹73.33
⫹73.89
⫹74.44
⫹251.6
⫹253.4
⫹255.2
⫹257.0
⫹258.8
⫹122
⫹123
⫹124
⫹125
⫹126
⫹50.00
⫹50.56
⫹51.11
⫹51.67
⫹52.22
⫹332.6
⫹334.4
⫹336.2
⫹338.0
⫹339.8
⫹167
⫹168
⫹169
⫹170
⫹171
⫹75.00
⫹75.56
⫹76.11
⫹76.67
⫹77.22
⫹260.6
⫹262.4
⫹264.2
⫹266.0
⫹267.8
⫹127
⫹128
⫹129
⫹130
⫹131
⫹52.78
⫹53.33
⫹53.89
⫹54.44
⫹55.00
⫹341.6
⫹343.4
⫹345.2
⫹347.0
⫹348.8
⫹172
⫹173
⫹174
⫹175
⫹176
⫹77.78
⫹78.33
⫹78.89
⫹79.44
⫹80.00
⫹269.6
⫹271.4
⫹273.2
⫹275.0
⫹276.8
⫹132
⫹133
⫹134
⫹135
⫹136
⫹55.56
⫹56.11
⫹56.67
⫹57.22
⫹57.78
⫹350.6
⫹352.4
⫹354.2
⫹356.0
⫹357.8
⫹177
⫹178
⫹179
⫹180
⫹181
⫹80.56
⫹81.11
⫹81.67
⫹82.22
⫹82.78
⫹278.6
⫹280.4
⫹282.2
⫹284.0
⫹285.8
⫹137
⫹138
⫹139
⫹140
⫹141
⫹58.33
⫹58.89
⫹59.44
⫹60.00
⫹60.56
⫹359.6
⫹361.4
⫹363.2
⫹365.0
⫹366.8
⫹182
⫹183
⫹184
⫹185
⫹186
⫹83.33
⫹83.89
⫹84.44
⫹85.00
⫹85.56
⫹287.6
⫹289.4
⫹291.2
⫹293.0
⫹294.8
⫹142
⫹143
⫹144
⫹145
⫹146
⫹61.11
⫹61.67
⫹62.22
⫹62.78
⫹63.33
⫹368.6
⫹370.4
⫹372.2
⫹374.0
⫹375.8
⫹187
⫹188
⫹189
⫹190
⫹191
⫹86.11
⫹86.67
⫹87.22
⫹87.78
⫹88.33
⫹296.6
⫹298.4
⫹300.2
⫹302.0
⫹303.8
⫹147
⫹148
⫹149
⫹150
⫹151
⫹63.89
⫹64.44
⫹65.00
⫹65.56
⫹66.11
⫹377.6
⫹379.4
⫹381.2
⫹383.0
⫹384.8
⫹192
⫹193
⫹194
⫹195
⫹196
⫹88.89
⫹89.44
⫹90.00
⫹90.56
⫹91.11
⫹305.6
⫹307.4
⫹309.2
⫹311.0
⫹312.8
⫹152
⫹153
⫹154
⫹155
⫹156
⫹66.67
⫹67.22
⫹67.78
⫹68.33
⫹68.89
⫹386.6
⫹388.4
⫹390.2
⫹392.0
⫹393.8
⫹197
⫹198
⫹199
⫹200
⫹201
⫹91.67
⫹92.22
⫹92.78
⫹93.33
⫹93.89
⫹314.6
⫹316.4
⫹318.2
⫹320.0
⫹321.8
⫹157
⫹158
⫹159
⫹160
⫹161
⫹69.44
⫹70.00
⫹70.56
⫹71.11
⫹71.67
⫹395.6
⫹397.4
⫹399.2
⫹401.0
⫹402.8
⫹202
⫹203
⫹204
⫹205
⫹206
⫹94.44
⫹95.00
⫹95.56
⫹96.11
⫹96.67
short
standard
long
DEAN #37261
(McGHP)
RIGHT
INTERACTIVE
2.59
GENERAL INFORMATION, CONVERSION TABLES, AND MATHEMATICS
cap height
base of text
TABLE 2.8 Temperature Conversion Table (Continued)
⬚F.
Reading in
⬚F. or ⬚C.
to be
converted
⬚C.
⫹404.6
⫹406.4
⫹408.2
⫹410.0
⫹411.8
⫹207
⫹208
⫹209
⫹210
⫹211
⫹413.6
⫹415.4
⫹417.2
⫹419.0
⫹420.8
top of rh
base of rh
⬚F.
Reading in
⬚F. or ⬚C.
to be
converted
⬚C.
⫹97.22
⫹97.78
⫹98.33
⫹98.89
⫹99.44
⫹543.2
⫹546.8
⫹550.4
⫹554.0
⫹557.6
⫹284
⫹286
⫹288
⫹290
⫹292
⫹140.00
⫹141.11
⫹142.22
⫹143.33
⫹144.44
⫹212
⫹213
⫹214
⫹215
⫹216
⫹100.00
⫹100.56
⫹101.11
⫹101.67
⫹102.22
⫹561.2
⫹564.8
⫹568.4
⫹572.0
⫹575.6
⫹294
⫹296
⫹298
⫹300
⫹302
⫹145.56
⫹146.67
⫹147.78
⫹148.89
⫹150.00
⫹422.6
⫹424.4
⫹426.2
⫹428.0
⫹431.6
⫹217
⫹218
⫹219
⫹220
⫹222
⫹102.78
⫹103.33
⫹103.89
⫹104.44
⫹105.56
⫹579.2
⫹582.8
⫹586.4
⫹590.0
⫹593.6
⫹304
⫹306
⫹308
⫹310
⫹312
⫹151.11
⫹152.22
⫹153.33
⫹154.44
⫹155.56
⫹435.2
⫹438.8
⫹442.4
⫹446.0
⫹449.6
⫹224
⫹226
⫹228
⫹230
⫹232
⫹106.67
⫹107.78
⫹108.89
⫹110.00
⫹111.11
⫹597.2
⫹600.8
⫹604.4
⫹608.0
⫹611.6
⫹314
⫹316
⫹318
⫹320
⫹322
⫹156.67
⫹157.78
⫹158.89
⫹160.00
⫹161.11
⫹453.2
⫹456.8
⫹460.4
⫹464.0
⫹467.6
⫹234
⫹236
⫹238
⫹240
⫹242
⫹112.22
⫹113.33
⫹114.44
⫹115.56
⫹116.67
⫹615.2
⫹618.8
⫹622.4
⫹626.0
⫹629.6
⫹324
⫹326
⫹328
⫹330
⫹332
⫹162.22
⫹163.33
⫹164.44
⫹165.56
⫹166.67
⫹471.2
⫹474.8
⫹478.4
⫹482.0
⫹485.6
⫹244
⫹246
⫹248
⫹250
⫹252
⫹117.78
⫹118.89
⫹120.00
⫹121.11
⫹122.22
⫹633.2
⫹636.8
⫹640.4
⫹644.0
⫹647.6
⫹334
⫹336
⫹338
⫹340
⫹342
⫹167.78
⫹168.89
⫹170.00
⫹171.11
⫹172.22
⫹489.2
⫹492.8
⫹496.4
⫹500.0
⫹503.6
⫹254
⫹256
⫹258
⫹260
⫹262
⫹123.33
⫹124.44
⫹125.56
⫹126.67
⫹127.78
⫹651.2
⫹654.8
⫹658.4
⫹662.0
⫹665.6
⫹344
⫹346
⫹348
⫹350
⫹352
⫹173.33
⫹174.44
⫹175.56
⫹176.67
⫹177.78
⫹507.2
⫹510.8
⫹514.4
⫹518.0
⫹521.6
⫹264
⫹266
⫹268
⫹270
⫹272
⫹128.89
⫹130.00
⫹131.11
⫹132.22
⫹133.33
⫹669.2
⫹672.8
⫹676.4
⫹680.0
⫹683.6
⫹354
⫹356
⫹358
⫹360
⫹362
⫹178.89
⫹180.00
⫹181.11
⫹182.22
⫹183.33
⫹525.2
⫹528.8
⫹532.4
⫹536.0
⫹539.6
⫹274
⫹276
⫹278
⫹280
⫹282
⫹134.44
⫹135.56
⫹136.67
⫹137.78
⫹138.89
⫹687.2
⫹690.8
⫹694.4
⫹698.0
⫹701.6
⫹364
⫹366
⫹368
⫹370
⫹372
⫹184.44
⫹185.56
⫹186.67
⫹187.78
⫹188.89
short
standard
long
DEAN #37261
LEFT
(McGHP)
2.60
INTERACTIVE
top of rh
base of rh
SECTION 2
cap height
base of text
TABLE 2.8 Temperature Conversion Table (Continued)
⬚F.
Reading in
⬚F. or ⬚C.
to be
converted
⬚F.
Reading in
⬚F. or ⬚C.
to be
converted
⬚C.
⬚C.
⫹705.2
⫹708.8
⫹712.4
⫹716.0
⫹719.6
⫹374
⫹376
⫹378
⫹380
⫹382
⫹190.00
⫹191.11
⫹192.22
⫹193.33
⫹194.44
⫹867.2
⫹870.8
⫹874.4
⫹878.0
⫹881.6
⫹464
⫹466
⫹468
⫹470
⫹472
⫹240.00
⫹241.11
⫹242.22
⫹243.33
⫹244.44
⫹723.2
⫹726.8
⫹730.4
⫹734.0
⫹737.6
⫹384
⫹386
⫹388
⫹390
⫹392
⫹195.56
⫹196.67
⫹197.78
⫹198.89
⫹200.00
⫹885.2
⫹888.8
⫹892.4
⫹896.0
⫹899.6
⫹474
⫹476
⫹478
⫹480
⫹482
⫹245.56
⫹246.67
⫹247.78
⫹248.89
⫹250.00
⫹741.2
⫹744.8
⫹748.4
⫹752.0
⫹755.6
⫹394
⫹396
⫹398
⫹400
⫹402
⫹201.11
⫹202.22
⫹203.33
⫹204.44
⫹205.56
⫹903.2
⫹906.8
⫹910.4
⫹914.0
⫹917.6
⫹484
⫹486
⫹488
⫹490
⫹492
⫹251.11
⫹252.22
⫹253.33
⫹254.44
⫹255.56
⫹759.2
⫹762.8
⫹766.4
⫹770.0
⫹773.6
⫹404
⫹406
⫹408
⫹410
⫹412
⫹206.67
⫹207.78
⫹208.89
⫹210.00
⫹211.11
⫹921.2
⫹924.8
⫹928.4
⫹932.0
⫹935.6
⫹494
⫹496
⫹498
⫹500
⫹502
⫹256.67
⫹257.78
⫹258.89
⫹260.00
⫹261.11
⫹777.2
⫹780.8
⫹784.4
⫹788.0
⫹791.6
⫹414
⫹416
⫹418
⫹420
⫹422
⫹212.22
⫹213.33
⫹214.44
⫹215.56
⫹216.67
⫹939.2
⫹942.8
⫹946.4
⫹950.0
⫹953.6
⫹504
⫹506
⫹508
⫹510
⫹512
⫹262.22
⫹263.33
⫹264.44
⫹265.56
⫹266.67
⫹795.2
⫹798.8
⫹802.4
⫹806.0
⫹809.6
⫹424
⫹426
⫹428
⫹430
⫹432
⫹217.78
⫹218.89
⫹220.00
⫹221.11
⫹222.22
⫹957.2
⫹960.8
⫹964.4
⫹968.0
⫹971.6
⫹514
⫹516
⫹518
⫹520
⫹522
⫹267.78
⫹268.89
⫹270.00
⫹271.11
⫹272.22
⫹813.2
⫹816.8
⫹820.4
⫹824.0
⫹827.6
⫹434
⫹436
⫹438
⫹440
⫹442
⫹223.33
⫹224.44
⫹225.56
⫹226.67
⫹227.78
⫹975.2
⫹978.8
⫹982.4
⫹986.0
⫹989.6
⫹524
⫹526
⫹528
⫹530
⫹532
⫹273.33
⫹274.44
⫹275.56
⫹276.67
⫹277.78
⫹831.2
⫹834.8
⫹838.4
⫹842.0
⫹845.6
⫹444
⫹446
⫹448
⫹450
⫹452
⫹228.89
⫹230.00
⫹231.11
⫹232.22
⫹233.33
⫹993.2
⫹996.8
⫹1000.4
⫹1004.0
⫹1007.6
⫹534
⫹536
⫹538
⫹540
⫹542
⫹278.89
⫹280.00
⫹281.11
⫹282.22
⫹283.33
⫹849.2
⫹852.8
⫹856.4
⫹860.0
⫹863.6
⫹454
⫹456
⫹458
⫹460
⫹462
⫹234.44
⫹235.56
⫹236.67
⫹237.78
⫹238.89
⫹1011.2
⫹1014.8
⫹1018.4
⫹1022.0
⫹1025.6
⫹544
⫹546
⫹548
⫹550
⫹552
⫹284.44
⫹285.56
⫹286.67
⫹287.78
⫹288.89
short
standard
long
DEAN #37261
(McGHP)
RIGHT
INTERACTIVE
2.61
GENERAL INFORMATION, CONVERSION TABLES, AND MATHEMATICS
cap height
base of text
TABLE 2.8 Temperature Conversion Table (Continued)
⬚F.
Reading in
⬚F. or ⬚C.
to be
converted
⬚C.
⫹1029.2
⫹1032.8
⫹1036.4
⫹1040.0
⫹1043.6
⫹554
⫹556
⫹558
⫹560
⫹562
⫹1047.2
⫹1050.8
⫹1054.4
⫹1058.0
⫹1061.6
top of rh
base of rh
⬚F.
Reading in
⬚F. or ⬚C.
to be
converted
⬚C.
⫹290.00
⫹291.11
⫹292.22
⫹293.33
⫹294.44
⫹1191.2
⫹1194.8
⫹1198.4
⫹1202.0
⫹1205.6
⫹644
⫹646
⫹648
⫹650
⫹652
⫹340.00
⫹341.11
⫹342.22
⫹343.33
⫹344.44
⫹564
⫹566
⫹568
⫹570
⫹572
⫹295.56
⫹296.67
⫹297.78
⫹298.89
⫹300.00
⫹1209.2
⫹1212.8
⫹1216.4
⫹1220.0
⫹1223.6
⫹654
⫹656
⫹658
⫹660
⫹662
⫹345.56
⫹346.67
⫹347.78
⫹348.89
⫹350.00
⫹1065.2
⫹1068.8
⫹1072.4
⫹1076.0
⫹1079.6
⫹574
⫹576
⫹578
⫹580
⫹582
⫹301.11
⫹302.22
⫹303.33
⫹304.44
⫹305.56
⫹1227.2
⫹1230.8
⫹1234.4
⫹1238.0
⫹1241.6
⫹664
⫹666
⫹668
⫹670
⫹672
⫹351.11
⫹352.22
⫹353.33
⫹354.44
⫹355.56
⫹1083.2
⫹1086.8
⫹1090.4
⫹1094.0
⫹1097.6
⫹584
⫹586
⫹588
⫹590
⫹592
⫹306.67
⫹307.78
⫹308.89
⫹310.00
⫹311.11
⫹1245.2
⫹1248.8
⫹1252.4
⫹1256.0
⫹1259.6
⫹674
⫹676
⫹678
⫹680
⫹682
⫹356.67
⫹357.78
⫹358.89
⫹360.00
⫹361.11
⫹1101.2
⫹1104.8
⫹1108.4
⫹1112.0
⫹1115.6
⫹594
⫹596
⫹598
⫹600
⫹602
⫹312.22
⫹313.33
⫹314.44
⫹315.56
⫹316.67
⫹1263.2
⫹1266.8
⫹1270.4
⫹1274.0
⫹1277.6
⫹684
⫹686
⫹688
⫹690
⫹692
⫹362.22
⫹363.33
⫹364.44
⫹365.56
⫹366.67
⫹1119.2
⫹1122.8
⫹1126.4
⫹1130.0
⫹1133.6
⫹604
⫹606
⫹608
⫹610
⫹612
⫹317.78
⫹318.89
⫹320.00
⫹321.11
⫹322.22
⫹1281.2
⫹1284.8
⫹1288.4
⫹1292.0
⫹1295.6
⫹694
⫹696
⫹698
⫹700
⫹702
⫹367.78
⫹368.89
⫹370.00
⫹371.11
⫹372.22
⫹1137.2
⫹1140.8
⫹1144.4
⫹1148.0
⫹1151.6
⫹614
⫹616
⫹618
⫹620
⫹622
⫹323.33
⫹324.44
⫹325.56
⫹326.67
⫹327.78
⫹1299.2
⫹1302.8
⫹1306.4
⫹1310.0
⫹1313.6
⫹704
⫹706
⫹708
⫹710
⫹712
⫹373.33
⫹374.44
⫹375.56
⫹376.67
⫹377.78
⫹1155.2
⫹1158.8
⫹1162.4
⫹1166.0
⫹1169.6
⫹624
⫹626
⫹628
⫹630
⫹632
⫹328.89
⫹330.00
⫹331.11
⫹332.22
⫹333.33
⫹1317.2
⫹1320.8
⫹1324.4
⫹1328.0
⫹1331.6
⫹714
⫹716
⫹718
⫹720
⫹722
⫹378.89
⫹380.00
⫹381.11
⫹382.22
⫹383.33
⫹1173.2
⫹1176.8
⫹1180.4
⫹1184.0
⫹1187.6
⫹634
⫹636
⫹638
⫹640
⫹642
⫹334.44
⫹335.56
⫹336.67
⫹337.78
⫹338.89
⫹1335.2
⫹1338.8
⫹1342.4
⫹1346.0
⫹1349.6
⫹724
⫹726
⫹728
⫹730
⫹732
⫹384.44
⫹385.56
⫹386.67
⫹387.78
⫹388.89
short
standard
long
DEAN #37261
LEFT
(McGHP)
2.62
INTERACTIVE
top of rh
base of rh
SECTION 2
cap height
base of text
TABLE 2.8 Temperature Conversion Table (Continued)
⬚F.
Reading in
⬚F. or ⬚C.
to be
converted
⬚F.
Reading in
⬚F. or ⬚C.
to be
converted
⬚C.
⬚C.
⫹1353.2
⫹1356.8
⫹1360.4
⫹1364.0
⫹1367.6
⫹734
⫹736
⫹738
⫹740
⫹742
⫹390.00
⫹391.11
⫹392.22
⫹393.33
⫹394.44
⫹1515.2
⫹1518.8
⫹1522.4
⫹1526.0
⫹1529.6
⫹824
⫹826
⫹828
⫹830
⫹832
⫹440.00
⫹441.11
⫹442.22
⫹443.33
⫹444.44
⫹1371.2
⫹1374.8
⫹1378.4
⫹1382.0
⫹1385.6
⫹744
⫹746
⫹748
⫹750
⫹752
⫹395.56
⫹396.67
⫹397.78
⫹398.89
⫹400.00
⫹1533.2
⫹1536.8
⫹1540.4
⫹1544.0
⫹1547.6
⫹834
⫹836
⫹838
⫹840
⫹842
⫹445.56
⫹446.67
⫹447.78
⫹448.89
⫹450.00
⫹1389.2
⫹1392.8
⫹1396.4
⫹1400.0
⫹1403.6
⫹754
⫹756
⫹758
⫹760
⫹762
⫹401.11
⫹402.22
⫹403.33
⫹404.44
⫹405.56
⫹1551.2
⫹1554.8
⫹1558.4
⫹1562.0
⫹1565.6
⫹844
⫹846
⫹848
⫹850
⫹852
⫹451.11
⫹452.22
⫹453.33
⫹454.44
⫹455.56
⫹1407.2
⫹1410.8
⫹1414.4
⫹1418.0
⫹1421.6
⫹764
⫹766
⫹768
⫹770
⫹772
⫹406.67
⫹407.78
⫹408.89
⫹410.00
⫹411.11
⫹1569.2
⫹1572.8
⫹1576.4
⫹1580.0
⫹1583.6
⫹854
⫹856
⫹858
⫹860
⫹862
⫹456.67
⫹457.78
⫹458.89
⫹460.00
⫹461.11
⫹1425.2
⫹1428.8
⫹1432.4
⫹1436.0
⫹1439.6
⫹774
⫹776
⫹778
⫹780
⫹782
⫹412.22
⫹413.33
⫹414.44
⫹415.56
⫹416.67
⫹1587.2
⫹1590.8
⫹1594.4
⫹1598.0
⫹1601.6
⫹864
⫹866
⫹868
⫹870
⫹872
⫹462.22
⫹463.33
⫹464.44
⫹465.56
⫹466.67
⫹1443.2
⫹1446.8
⫹1450.4
⫹1454.0
⫹1457.6
⫹784
⫹786
⫹788
⫹790
⫹792
⫹417.78
⫹418.89
⫹420.00
⫹421.11
⫹422.22
⫹1605.2
⫹1608.8
⫹1612.4
⫹1616.0
⫹1619.6
⫹874
⫹876
⫹878
⫹880
⫹882
⫹467.78
⫹468.89
⫹470.00
⫹471.11
⫹472.22
⫹1461.2
⫹1464.8
⫹1468.4
⫹1472.0
⫹1475.6
⫹794
⫹796
⫹798
⫹800
⫹802
⫹423.33
⫹424.44
⫹425.56
⫹426.67
⫹427.78
⫹1623.2
⫹1626.8
⫹1630.4
⫹1634.0
⫹1637.6
⫹884
⫹886
⫹888
⫹890
⫹892
⫹473.33
⫹474.44
⫹475.56
⫹476.67
⫹477.78
⫹1479.2
⫹1482.8
⫹1486.4
⫹1490.0
⫹1493.6
⫹804
⫹806
⫹808
⫹810
⫹812
⫹428.89
⫹430.00
⫹431.11
⫹432.22
⫹433.33
⫹1641.2
⫹1644.8
⫹1648.4
⫹1652.0
⫹1655.6
⫹894
⫹896
⫹898
⫹900
⫹902
⫹478.89
⫹480.00
⫹481.11
⫹482.22
⫹483.33
⫹1497.2
⫹1500.8
⫹1504.4
⫹1508.0
⫹1511.6
⫹814
⫹816
⫹818
⫹820
⫹822
⫹434.44
⫹435.56
⫹436.67
⫹437.78
⫹438.89
⫹1659.2
⫹1662.8
⫹1666.4
⫹1670.0
⫹1673.6
⫹904
⫹906
⫹908
⫹910
⫹912
⫹484.44
⫹485.56
⫹486.67
⫹487.78
⫹488.89
short
standard
long
DEAN #37261
(McGHP)
RIGHT
INTERACTIVE
2.63
GENERAL INFORMATION, CONVERSION TABLES, AND MATHEMATICS
cap height
base of text
TABLE 2.8 Temperature Conversion Table (Continued)
⬚F.
Reading in
⬚F. or ⬚C.
to be
converted
⬚C.
⫹1677.2
⫹1680.8
⫹1684.4
⫹1688.0
⫹1691.6
⫹914
⫹916
⫹918
⫹920
⫹922
⫹1695.2
⫹1698.8
⫹1702.4
⫹1706.0
⫹1709.6
top of rh
base of rh
⬚F.
Reading in
⬚F. or ⬚C.
to be
converted
⬚C.
⫹490.00
⫹491.11
⫹492.22
⫹493.33
⫹494.44
⫹1868.0
⫹1886.0
⫹1904.0
⫹1922.0
⫹1940.0
⫹1020
⫹1030
⫹1040
⫹1050
⫹1060
⫹548.89
⫹554.44
⫹560.00
⫹565.56
⫹571.11
⫹924
⫹926
⫹928
⫹930
⫹932
⫹495.56
⫹496.67
⫹497.78
⫹498.89
⫹500.00
⫹1958.0
⫹1976.0
⫹1994.0
⫹2012.0
⫹2030.0
⫹1070
⫹1080
⫹1090
⫹1100
⫹1110
⫹576.67
⫹582.22
⫹587.78
⫹593.33
⫹598.89
⫹1713.2
⫹1716.8
⫹1720.4
⫹1724.0
⫹1727.6
⫹934
⫹936
⫹938
⫹940
⫹942
⫹501.11
⫹502.22
⫹503.33
⫹504.44
⫹505.56
⫹2048.0
⫹2066.0
⫹2084.0
⫹2102.0
⫹2120.0
⫹1120
⫹1130
⫹1140
⫹1150
⫹1160
⫹604.44
⫹610.00
⫹615.56
⫹621.11
⫹626.67
⫹1731.2
⫹1734.8
⫹1738.4
⫹1742.0
⫹1745.6
⫹944
⫹946
⫹948
⫹950
⫹952
⫹506.67
⫹507.78
⫹508.89
⫹510.00
⫹511.11
⫹2138.0
⫹2156.0
⫹2174.0
⫹2192.0
⫹2210.0
⫹1170
⫹1180
⫹1190
⫹1200
⫹1210
⫹632.22
⫹637.78
⫹643.33
⫹648.89
⫹654.44
⫹1749.2
⫹1752.8
⫹1756.4
⫹1760.0
⫹1763.6
⫹954
⫹956
⫹958
⫹960
⫹962
⫹512.22
⫹513.33
⫹514.44
⫹515.56
⫹516.67
⫹2228.0
⫹2246.0
⫹2264.0
⫹2282.0
⫹2300.0
⫹1220
⫹1230
⫹1240
⫹1250
⫹1260
⫹660.00
⫹665.56
⫹671.11
⫹676.67
⫹682.22
⫹1767.2
⫹1770.8
⫹1774.4
⫹1778.0
⫹1781.6
⫹964
⫹966
⫹968
⫹970
⫹972
⫹517.78
⫹518.89
⫹520.00
⫹521.11
⫹522.22
⫹2318.0
⫹2336.0
⫹2354.0
⫹2372.0
⫹2390.0
⫹1270
⫹1280
⫹1290
⫹1300
⫹1310
⫹687.78
⫹693.33
⫹698.89
⫹704.44
⫹710.00
⫹1785.2
⫹1788.8
⫹1792.4
⫹1796.0
⫹1799.6
⫹974
⫹976
⫹978
⫹980
⫹982
⫹523.33
⫹524.44
⫹525.56
⫹526.67
⫹527.78
⫹2408.0
⫹2426.0
⫹2444.0
⫹2462.0
⫹2480.0
⫹1320
⫹1330
⫹1340
⫹1350
⫹1360
⫹715.56
⫹721.11
⫹726.67
⫹732.22
⫹737.78
⫹1803.2
⫹1806.8
⫹1810.4
⫹1814.0
⫹1817.6
⫹984
⫹986
⫹988
⫹990
⫹992
⫹528.89
⫹530.00
⫹531.11
⫹532.22
⫹533.33
⫹2498.0
⫹2516.0
⫹2534.0
⫹2552.0
⫹2570.0
⫹1370
⫹1380
⫹1390
⫹1400
⫹1410
⫹743.33
⫹748.89
⫹754.44
⫹760.00
⫹765.56
⫹1821.2
⫹1824.8
⫹1828.4
⫹1832.0
⫹1850.0
⫹994
⫹996
⫹998
⫹1000
⫹1010
⫹534.44
⫹535.56
⫹536.67
⫹537.78
⫹543.33
⫹2588.0
⫹2606.0
⫹2624.0
⫹2642.0
⫹2660.0
⫹1420
⫹1430
⫹1440
⫹1450
⫹1460
⫹771.11
⫹776.67
⫹782.22
⫹787.78
⫹793.33
short
standard
long
DEAN #37261
LEFT
(McGHP)
2.64
INTERACTIVE
top of rh
base of rh
SECTION 2
cap height
base of text
TABLE 2.8 Temperature Conversion Table (Continued)
⬚F.
Reading in
⬚F. or ⬚C.
to be
converted
⬚F.
Reading in
⬚F. or ⬚C.
to be
converted
⬚C.
⬚C.
⫹2678.0
⫹2696.0
⫹2714.0
⫹2732.0
⫹2750.0
⫹1470
⫹1480
⫹1490
⫹1500
⫹1510
⫹798.89
⫹804.44
⫹810.00
⫹815.56
⫹821.11
⫹3488.0
⫹3506.0
⫹3524.0
⫹3542.0
⫹3560.0
⫹1920
⫹1930
⫹1940
⫹1950
⫹1960
⫹1048.9
⫹1054.4
⫹1060.0
⫹1065.6
⫹1071.1
⫹2768.0
⫹2786.0
⫹2804.0
⫹2822.0
⫹2840.0
⫹1520
⫹1530
⫹1540
⫹1550
⫹1560
⫹826.67
⫹832.22
⫹837.78
⫹843.33
⫹848.89
⫹3578.0
⫹3596.0
⫹3614.0
⫹3632.0
⫹3650.0
⫹1970
⫹1980
⫹1990
⫹2000
⫹2010
⫹1076.7
⫹1082.2
⫹1087.8
⫹1093.3
⫹1098.9
⫹2858.0
⫹2876.0
⫹2894.0
⫹2912.0
⫹2930.0
⫹1570
⫹1580
⫹1590
⫹1600
⫹1610
⫹854.44
⫹860.00
⫹865.56
⫹871.11
⫹876.67
⫹3668.0
⫹3686.0
⫹3704.0
⫹3722.0
⫹3740.0
⫹2020
⫹2030
⫹2040
⫹2050
⫹2060
⫹1104.4
⫹1110.0
⫹1115.6
⫹1121.1
⫹1126.7
⫹2948.0
⫹2966.0
⫹2984.0
⫹3002.0
⫹3020.0
⫹1620
⫹1630
⫹1640
⫹1650
⫹1660
⫹882.22
⫹887.78
⫹893.33
⫹898.89
⫹904.44
⫹3758.0
⫹3776.0
⫹3794.0
⫹3812.0
⫹3830.0
⫹2070
⫹2080
⫹2090
⫹2100
⫹2110
⫹1132.2
⫹1137.8
⫹1143.3
⫹1148.9
⫹1154.4
⫹3038.0
⫹3056.0
⫹3074.0
⫹3092.0
⫹3110.0
⫹1670
⫹1680
⫹1690
⫹1700
⫹1710
⫹910.00
⫹915.56
⫹921.11
⫹926.67
⫹932.22
⫹3848.0
⫹3866.0
⫹3884.0
⫹3902.0
⫹3920.0
⫹2120
⫹2130
⫹2140
⫹2150
⫹2160
⫹1160.0
⫹1165.6
⫹1171.1
⫹1176.7
⫹1182.2
⫹3128.0
⫹3146.0
⫹3164.0
⫹3182.0
⫹3200.0
⫹1720
⫹1730
⫹1740
⫹1750
⫹1760
⫹937.78
⫹943.33
⫹948.89
⫹954.44
⫹960.00
⫹3938.0
⫹3956.0
⫹3974.0
⫹3992.0
⫹4010.0
⫹2170
⫹2180
⫹2190
⫹2200
⫹2210
⫹1187.8
⫹1193.3
⫹1198.9
⫹1204.4
⫹1210.0
⫹3218.0
⫹3236.0
⫹3254.0
⫹3272.0
⫹3290.0
⫹1770
⫹1780
⫹1790
⫹1800
⫹1810
⫹965.56
⫹971.11
⫹976.67
⫹982.22
⫹987.78
⫹4028.0
⫹4046.0
⫹4064.0
⫹4082.0
⫹4100.0
⫹2220
⫹2230
⫹2240
⫹2250
⫹2260
⫹1215.6
⫹1221.1
⫹1226.7
⫹1232.2
⫹1237.8
⫹3308.0
⫹3326.0
⫹3344.0
⫹3362.0
⫹3380.0
⫹1820
⫹1830
⫹1840
⫹1850
⫹1860
⫹993.33
⫹998.89
⫹1004.4
⫹1010.0
⫹1015.6
⫹4118.0
⫹4136.0
⫹4154.0
⫹4172.0
⫹4190.0
⫹2270
⫹2280
⫹2290
⫹2300
⫹2310
⫹1243.3
⫹1248.9
⫹1254.4
⫹1260.0
⫹1265.6
⫹3398.0
⫹3416.0
⫹3434.0
⫹3452.0
⫹3470.0
⫹1870
⫹1880
⫹1890
⫹1900
⫹1910
⫹1021.1
⫹1026.7
⫹1032.2
⫹1037.8
⫹1043.3
⫹4208.0
⫹4226.0
⫹4244.0
⫹4262.0
⫹4280.0
⫹2320
⫹2330
⫹2340
⫹2350
⫹2360
⫹1271.1
⫹1276.7
⫹1282.2
⫹1287.8
⫹1293.3
short
standard
long
DEAN #37261
(McGHP)
RIGHT
INTERACTIVE
2.65
GENERAL INFORMATION, CONVERSION TABLES, AND MATHEMATICS
cap height
base of text
TABLE 2.8 Temperature Conversion Table (Continued)
⬚F.
Reading in
⬚F. or ⬚C.
to be
converted
⬚C.
⫹4298.0
⫹4316.0
⫹4334.0
⫹4352.0
⫹4370.0
⫹2370
⫹2380
⫹2390
⫹2400
⫹2410
⫹4388.0
⫹4406.0
⫹4424.0
⫹4442.0
⫹4460.0
top of rh
base of rh
⬚F.
Reading in
⬚F. or ⬚C.
to be
converted
⬚C.
⫹1298.9
⫹1304.4
⫹1310.0
⫹1315.6
⫹1321.1
⫹4964.0
⫹4982.0
⫹5000.0
⫹5018.0
⫹5036.0
⫹2740
⫹2750
⫹2760
⫹2770
⫹2780
⫹1504.4
⫹1510.0
⫹1515.6
⫹1521.1
⫹1526.7
⫹2420
⫹2430
⫹2440
⫹2450
⫹2460
⫹1326.7
⫹1332.2
⫹1337.8
⫹1343.3
⫹1348.9
⫹5054.0
⫹5072.0
⫹5090.0
⫹5108.0
⫹5126.0
⫹2790
⫹2800
⫹2810
⫹2820
⫹2830
⫹1532.2
⫹1537.8
⫹1543.3
⫹1548.9
⫹1554.4
⫹4478.0
⫹4496.0
⫹4514.0
⫹4532.0
⫹4550.0
⫹4568.0
⫹4586.0
⫹2470
⫹2480
⫹2490
⫹2500
⫹2510
⫹2520
⫹2530
⫹1354.4
⫹1360.0
⫹1365.6
⫹1371.1
⫹1376.7
⫹1382.2
⫹1387.8
⫹5144.0
⫹5162.0
⫹5180.0
⫹5198.0
⫹5216.0
⫹2840
⫹2850
⫹2860
⫹2870
⫹2880
⫹1560.0
⫹1565.6
⫹1571.1
⫹1576.7
⫹1582.2
⫹4604.0
⫹4622.0
⫹4640.0
⫹4658.0
⫹4676.0
⫹2540
⫹2550
⫹2560
⫹2570
⫹2580
⫹1393.3
⫹1398.9
⫹1404.4
⫹1410.0
⫹1415.6
⫹5234.0
⫹5252.0
⫹5270.0
⫹5288.0
⫹5306.0
⫹2890
⫹2900
⫹2910
⫹2920
⫹2930
⫹1587.8
⫹1593.3
⫹1598.9
⫹1604.4
⫹1610.0
⫹4694.0
⫹4712.0
⫹4730.0
⫹4748.0
⫹4766.0
⫹2590
⫹2600
⫹2610
⫹2620
⫹2630
⫹1421.1
⫹1426.7
⫹1432.2
⫹1437.8
⫹1443.3
⫹5324.0
⫹5342.0
⫹5360.0
⫹5378.0
⫹5396.0
⫹2940
⫹2950
⫹2960
⫹2970
⫹2980
⫹1615.6
⫹1621.1
⫹1626.7
⫹1632.2
⫹1637.8
⫹4784.0
⫹4802.0
⫹4820.0
⫹4838.0
⫹4856.0
⫹2640
⫹2650
⫹2660
⫹2670
⫹2680
⫹1448.9
⫹1454.4
⫹1460.0
⫹1465.6
⫹1471.1
⫹5414.0
⫹5432.0
⫹5450.0
⫹5468.0
⫹5486.0
⫹2990
⫹3000
⫹3010
⫹3020
⫹3030
⫹1643.3
⫹1648.9
⫹1654.4
⫹1660.0
⫹1665.6
⫹4874.0
⫹4892.0
⫹4910.0
⫹4928.0
⫹4946.0
⫹2690
⫹2700
⫹2710
⫹2720
⫹2730
⫹1476.7
⫹1482.2
⫹1487.8
⫹1493.3
⫹1498.9
⫹5504.0
⫹5522.0
⫹5540.0
⫹5558.0
⫹5576.0
⫹5594.0
⫹5612.0
⫹3040
⫹3050
⫹3060
⫹3070
⫹3080
⫹3090
⫹3100
⫹1671.1
⫹1676.7
⫹1682.2
⫹1687.8
⫹1693.3
⫹1698.9
⫹1704.4
short
standard
long
DEAN #37261
LEFT
(McGHP)
2.66
2.1.1
INTERACTIVE
top of rh
base of rh
SECTION 2
cap height
base of text
Conversion of Thermometer Scales
The following abbreviations are used: ⬚F, degrees Fahrenheit; ⬚C, degrees Celsius; ⬚K, degrees
Kelvin; ⬚Ré, degrees Reaumur; ⬚R, degrees Rankine; ⬚Z, degrees on any scale; (fp)“Z”, the freezing
point of water on the Z scale; and (bp)“Z”, the boiling point of water on the Z scale. Reference:
Dodds, Chemical and Metallurgical Engineering 38:476 (1931).
⬚F ⫺ 32
⬚C
⬚Ré K ⫺ 273 ⬚R ⫺ 492
⬚Z ⫺ (fp)“Z”
⫽
⫽
⫽
⫽
⫽
180
100
80
100
180
(bp)“Z” ⫺ (fp)“Z”
Examples
(1) To find the Fahrenheit temperature corresponding to ⫺20⬚C:
⬚F ⫺ 32
⬚C
⫽
180
100
⬚F ⫺ 32 ⫽
⬚F ⫺ 32 ⫺20
⫽
180
100
or
(⫺20)(180)
⫽ ⫺36
100
⬚F ⫽ ⫺4
(2) To find the Reaumur temperature corresponding to 20⬚F:
⬚F ⫺ 32 ⬚Ré 20 ⫺ 32 ⬚Ré
⫽
⫽
⫽
180
80
180
80
i.e.,
20⬚F ⫽ ⫺5.33⬚Ré
(3) To find the correct temperature on a thermometer reading 80⬚C and that shows a reading of
⫺0.30⬚C in a melting ice/water mixture and 99.0⬚C in steam at 760 mm pressure of mercury:
⬚C
Z ⫺ (fp)“Z”
80 ⫺ (⫺0.30)
⫽
⫽
100 (bp)“Z” ⫺ (fp)“Z” 99.0 ⫺ (⫺0.30)
i.e.,
2.1.2
2.1.2.1
⬚C ⫽ 80.87
(corrected)
Density and Specific Gravity
Hydrometers. Various hydrometers and the relation between the various scales.
Alcoholometer. This hydrometer is used in determining the density of aqueous ethyl alcohol
solutions; the reading in degrees is numerically the same as the percentage of alcohol by volume.
The scale known as Tralle gives the percentage by volume. Wine and Must hydrometer relations
are given below.
Ammoniameter. This hydrometer, employed in finding the density of aqueous ammonia solutions, has a scale graduated in equal divisions from 0⬚ to 40⬚. To convert the reading to specific
gravity multiply by 3 and subtract the resulting number from 1000.
Balling Hydrometer. See under Saccharometers.
Barkometer or Barktrometer. This hydrometer, which is used in determining the density of
tanning liquors, has a scale from 0⬚ to 80⬚ Bk; the number to the right of the decimal point of a
specific gravity reading is the corresponding Bk degree; thus, a specific gravity of 1.015 is
15⬚ Bk.
short
standard
long
DEAN #37261
(McGHP)
RIGHT
INTERACTIVE
GENERAL INFORMATION, CONVERSION TABLES, AND MATHEMATICS
2.67
Baumé Hydrometers. For liquids heavier than water: This hydrometer was originally based on
the density of a 10% sodium chloride solution, which was given the value of 10⬚, and the density
of pure water, which was given the value of 0⬚; the interval between these two values was divided
into 10 equal parts. Other reference points have been taken with the result that so much confusion
exists that there are about 36 different scales in use, many of which are incorrect. In general a
Baumé hydrometer should have inscribed on it the temperature at which it was calibrated and
also the temperature of the water used in relating the density to a specific gravity. The following
expression gives the relation between the specific gravity and several of the Baumé scales:
Specific gravity ⫽
m ⫽ 145 at 60⬚/60⬚F (15.56⬚C)
⫽ 144
top of rh
base of rh
cap height
base of text
m
m ⫺ Baumé
for the American Scale
for the old scale used in Holland
⫽ 146.3 at 15⬚C
for the Gerlach Scale
⫽ 144.3 at 15⬚C
for the Rational Scale generally used in Germany
For liquids lighter than water: Originally the density of a solution of 1 gram of sodium chloride
in 9 grams of water at 12.5⬚C was given a value of 10⬚Bé. The scale between these points was
divided into ten equal parts and these divisions were repeated throughout the scale giving a
relation which could be expressed by the formula: Specific gravity ⫽ 145.88/(135.88 ⫹ Bé),
which is approximately equal to 146/(136 ⫹ Bé). Other scales have since come into more general
use such as that of the Bureau of Standards in which the specific gravity at 60⬚/60⬚F ⫽
140/(130 ⫹ Bé) and that of the American Petroleum Institute (A.P.I. Scale) in which the specific
gravity at 60⬚/60⬚F ⫽ 141.5/(131.5 ⫹ API⬚).
See also special table for conversion to density and Twaddell scale.
Beck’s Hydrometer. This hydrometer is graduated to show a reading of 0⬚ in pure water and a
reading of 30⬚ in a solution with a specific gravity of 0.850, with equal scale divisions above and
below these two points.
Brix Hydrometer. See under Saccharometers.
Cartier’s Hydrometer. This hydrometer shows a reading of 22⬚ when immersed in a solution
having a density of 22⬚ Baumé but the scale divisions are smaller than on the Baumé hydrometer
in the ratio of 16 Cartier to 15 Baumé.
Fatty Oil Hydrometer. The graduations on this hydrometer are in specific gravity within the
range 0.908 to 0.938. The letters on the scale correspond to the specific gravity of the various
common oils as follows: R, rape; O, olive; A, almond; S, sesame; HL, hoof oil; HP, hemp; C,
cotton seed; L, linseed. See also Oleometer below.
Lactometers. These hydrometers are used in determining the density of milk. The various scales
in common use are the following:
New York Board of Health has a scale graduated into 120 equal parts, 0⬚ being equal to the
specific gravity of water and 100⬚ being equal to a specific gravity of 1.029.
Quevenne lactometer is graduated from 15⬚ to 40⬚ corresponding to specific gravities from
1.015 to 1.040.
Soxhlet lactometer has a scale from 25⬚ to 35⬚ corresponding to specific gravities from 1.025
to 1.035 respectively.
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DEAN #37261
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(McGHP)
2.68
INTERACTIVE
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SECTION 2
Oleometer. A hydrometer for determining the density of vegetable and sperm oils with a scale
from 50⬚ to 0⬚ corresponding to specific gravities from 0.870 to 0.970. See also Fatty Oil Hydrometer above.
cap height
base of text
Saccharometers. These hydrometers are used in determining the density of sugar solutions.
Solutions of the same concentration but of different carbohydrates have very nearly the same
specific gravity and in general a concentration of 10 grams of carbohydrate per 100 mL of solution
shows a specific gravity of 1.0386. Thus, the wt. of sugar in 1000 mL soln. is (a) for conc.
⬍12g/100 mL: (wt. of 1000 mL soln. ⫺ 1000) ⫼ 0.386; (b) for conc. ⬎12g/100 mL: (wt of
1000 mL soln. ⫺ 1000) ⫼ 0.385.
Brix hydrometer is graduated so that the number of degrees is identical with the percentage
by weight of cane sugar and is used at the temperature indicated on the hydrometer.
Balling’s saccharometer is used in Europe and is practically identical with the Brix hydrometer.
Bates brewers’ saccharometer which is used in determining the density of malt worts is
graduated so that the divisions express pounds per barrel (32 gallons). The relation between
degrees Bates (⫽b) and degrees Balling (⫽B) is shown by the following formula: B ⫽
260b/(360 ⫹ b).
See also below under Wine and Must Hydrometer.
Salinometer. This hydrometer, which is used in the pickling and meat packing plants, is graduated to show percentage of saturation of a sodium chloride solution. An aqueous solution is
completely saturated when it contains 26.4% pure sodium chloride. The range from 0% to 26.4%
is divided into 100 parts, each division therefore representing 1% of saturation. In another type
of salinometer, the degrees correspond to percentages of sodium chloride expressed in grams of
sodium chloride per 100 mL of water.
Sprayometer (Parrot and Stewart). This hydrometer which is used in determining the density
of lime sulfur solutions has two scales; one scale is graduated from 0⬚ to 38⬚ Baumé and the
other scale is from 1.000 to 1.350 specific gravity.
Tralle Hydrometer. See Alcoholometer above.
Twaddell Hydrometer. This hydrometer, which is used only for liquids heavier than water, has
a scale such that when the reading is multiplied by 5 and added to 1000 the resulting number is
the specific gravity with reference to water as 1000. To convert specific gravity at 60⬚/60⬚F to
Twaddell degrees, take the decimal portion of the specific gravity value and multiply it by 200;
thus a specific gravity of 1.032 ⫽ 0.032 ⫻ 200 ⫽ 6.4⬚ Tw. See also special table for conversion
to density and Baumé scale.
Wine and Must Hydrometer. This instrument has three scales. One scale shows readings of 0⬚
to 15⬚ Brix for sugar (see Brix Hydrometer above); another scale from 0⬚ to 15⬚ Tralle is used
for sweet wines to indicate the percentage of alcohol by volume; and a third scale from 0⬚ to 20⬚
Tralle is used for tart wines to indicate the percentage of alcohol by volume.
2.1.2.2 Conversion of Specific Gravity at 25⬚/25⬚C to Density at any Temperature from 0⬚ to
40⬚C.* Liquids change volume with change in temperature, but the amount of this change, ␤
(coefficient of cubical expansion), varies widely with different liquids, and to some extent for the
same liquid at different temperatures.
The table below, which is calculated from the relationship:
F␤ t ⫽
density of water at 25⬚C (⫽0.99705)
1 ⫺ ␤(25 ⫺ t)
* Cf. Dreisbach, Ind. Eng. Chem., Anal. Ed. 12:160 (1940).
(2.1)
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standard
long
DEAN #37261
(McGHP)
RIGHT
INTERACTIVE
2.69
GENERAL INFORMATION, CONVERSION TABLES, AND MATHEMATICS
may be used to find d t, the density (weight of 1 mL) of a liquid at any temperature (t) between 0⬚
and 40⬚C if the specific gravity at 25⬚/25⬚C (S) and the coefficient of cubical expansion (␤) are
known. Substitutions are made in the equations:
d t ⫽ SF␤ t
S⫽
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base of rh
cap height
base of text
(2.2)
dt
F␤ t
(2.3)
Factors (F␤ t)
Density t⬚C ⫽ sp. gr. 25⬚/25⬚ ⫻ F␤ t
⬚C.
*b ⫻ 103
1.3
1.2
1.1
1.0
0.9
0.8
0.7
0.6
0.5
0.
0
5
10
15
20
25
30
35
40
1.0306
1.0279
1.0253
1.0227
1.0200
1.0174
1.0148
1.0122
1.0097
1.0071
1.0237
1.0216
1.0195
1.0174
1.0153
1.0133
1.0113
1.0092
1.0072
1.0051
1.0169
1.0154
1.0138
1.0123
1.0107
1.0092
1.0077
1.0061
1.0046
1.0031
1.0102
1.0092
1.0082
1.0072
1.0060
1.0051
1.0041
1.0031
1.0021
1.0011
1.0036
1.0031
1.0026
1.0021
1.0016
1.0011
1.0006
1.0001
0.99958
0.99908
0.99705
0.99705
0.99705
0.99705
0.99705
0.99705
0.99705
0.99705
0.99705
0.99705
0.99065
0.9911
0.9916
0.9921
0.99262
0.9931
0.9935
0.9941
0.9944
0.9951
0.9843
0.9853
0.9963
0.9872
0.9882
0.98918
0.99015
0.9911
0.9921
0.9931
0.9780
0.9794
0.9809
0.98234
0.9838
0.9851
0.98672
0.9882
0.9897
0.9911
* ␤ ⫽ coefficient of cubical expansion.
Examples. All examples are based upon an assumed coefficient of cubical expansion, ␤, of
1.3 ⫻ 10⫺3.
Example 1. To find the density of a liquid at 20⬚C, d 20, which has a specific gravity (S) of
1.250025
25 :
From the table above F␤ t at 20⬚C ⫽ 1.0036.
d 20 ⫽ d t ⫽ SF␤ t ⫽ 1.2500 ⫻ 1.0036 ⫽ 1.2545
Example 2. To find the density at 20⬚C (d 20) of a liquid which has a specific gravity of
1.2500174 :
Since the density of water at 4⬚C is equal to 1, specific gravity at 17⬚/4⬚ ⫽ d17 ⫽ 1.2500.
Substitution in Equation 3 with F␤ t at 17⬚C, by interpolation from the table, equal to 1.00756,
gives
Sp. gr. 25⬚/25⬚ ⫽ S ⫽ 1.2500 ⫼ 1.00756
Substitution of this value for S in Equation 2 with F␤ t at 20⬚C, from the table, equal to 1.0036,
gives
d 20 ⫽ d t ⫽ (1.2500 ⫼ 1.00756) ⫻ 1.0036 ⫽ 1.2451
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DEAN #37261
2.70
LEFT
(McGHP)
INTERACTIVE
SECTION 2
Example 3. To find the specific gravity at 20⬚/4⬚C of a liquid which has a specific gravity of
1.2500254 :
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Since the density of water at 4⬚C, is equal to 1, specific gravity 25⬚/4⬚ ⫽ d 25 ⫽ 1.2500; and,
specific gravity 20⬚/4⬚ ⫽ d 20.
Substitution in Equation 3, with d t ⫽ 1.2500; and, with F␤ t at 25⬚C, from the table, equal to
0.99705, gives
Sp. gr. 25⬚/25⬚ ⫽ S ⫽ 1.2500 ⫼ 0.99705
Substitution of this value for S in Equation 2, with F␤ t at 20⬚C, from the table, equal to 1.0036,
gives
Sp. gr. 20⬚/4⬚ ⫽ d 20 ⫽ (1.2500 ⫼ 0.99705) ⫻ 1.0036 ⫽ 1.2582
Example 4. To find the density at 25⬚C of a liquid which has a specific gravity of 1.250015
15 :
Since the density of water at 15⬚C ⫽ 0.99910,
d15 ⫽ sp. gr. 15⬚/15⬚ ⫻ 0.99910 ⫽ 1.2500 ⫻ 0.99910
Substitution in Equation 3, with F␤ t at 15⬚C, from the table, equal to 1.0102, gives
Sp. gr. 25⬚/25⬚ ⫽ S ⫽ (1.2500 ⫻ 0.99910) ⫼ 1.0102
Substitution of this value for S in Equation 2, with F␤ t at 25⬚, from the table, equal to 0.99705,
gives
d 26 ⫽ d t ⫽ (1.2500 ⫻ 0.99910 ⫼ 1.0102) ⫻ 0.99705 ⫽ 1.2326
2.1.3
Barometry and Barometric Corrections
In principle, the mercurial barometer balances a column of pure mercury against the weight of the
atmosphere. The height of the column above the level of the mercury in the reservoir can be measured
and serves as a direct index of atmospheric pressure. The space above the mercury in a barometer
tube should be a Torricellian vacuum, perfect except for the practically negligible vapor pressure of
mercury. The perfection of the vacuum is indicated by the sharpness of the click noted when the
barometer tube is inclined. A barometer should be in a vertical position, suspended rather than
fastened to a wall, and in a good light but not exposed to direct sunlight or too near a source of
heat. The standard conditions for barometric measurements are 0⬚C and gravity as at 45⬚ latitude
and sea level. There are numerous sources of error, but corrections for most of these are readily
applied. Some of the corrections are very small, and their application may be questionable in view
of the probably larger errors. The degree of consistency to be expected in careful measurements is
about 0.13 mm with a 6.4-mm tube, increasing to 0.04 mm with a tube 12.7 mm in diameter.
In reading a barometer of the Fortin type (the usual laboratory instrument for precision measurements), the procedure should be as follows: (1) Observe and record the temperature as indicated
by the thermometer attached to the barometer. The temperature correction is very important and may
be affected by heat from the observer’s body. (2) Set the mercury in the reservoir at zero level, so
that the point of the pin above the mercury just touches the surface, making a barely noticeable
dimple therein. Tap the tube at the top and verify the zero setting. (3) Bring the vernier down until
the view at the light background is cut off at the highest point of the meniscus. Record the reading.
The corrections to be made on the reading are as follows: (1) Temperature, to correct for the
difference in thermal expansion of the mercury and the brass (or glass) to which the scale is attached.
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DEAN #37261
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INTERACTIVE
GENERAL INFORMATION, CONVERSION TABLES, AND MATHEMATICS
2.71
This correction converts the reading into the value of 0⬚C. The brass scale table is applicable to the
Fortin barometer. See Tables 2.10 latitude-gravity correction, and 2.11 altitude-gravity correction,
to compensate for differences in gravity, which would affect the height of the mercury column by
variation in mass. If local gravity is unknown, an approximate correction may be made from the
tables. Local values of gravity are often subject to irregularities which lead to errors even when the
corrections here provided are made. It is, therefore, advisable to determine the local value of gravity,
from which the correction can be effected in the following manner:
Bt ⫽ Br ⫹
冉
g1 ⫺ g0
g0
冊
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base of rh
cap height
base of text
⫻ Br
in which Bt and Br are the true and the observed heights of the barometer, respectively. g0 is standard
gravity (980 665 cm · s⫺2), and g1 is the local gravity. It may be noted that for most localities, g1 is
smaller than g0 , which makes the correction negative. These corrections compensate the reading to
gravity at 45⬚ latitude and sea level. (3) Correction for capillary depression of the level of the
meniscus. This varies with the tube diameter and actual height of the meniscus in a particular case.
Some barometers are calibrated to allow for an average value of the latter and approximating the
correction. See table. (4) Correction for vapor pressure of mercury. This correction is usually negligible, being only 0.001 mm at 20⬚C and 0.006 mm at 40⬚C. This correction is added. See table of
vapor pressure of mercury.
The corrections above do not apply to aneroid barometers. These instruments should be calibrated
at regular intervals by checking them against a corrected mercurial barometer.
For records on weather maps, meteorologists customarily correct barometer readings to sea level,
and some barometers may be calibrated accordingly. Such instruments are not suitable for laboratory
use where true pressure under standard conditions is required. Scale corrections should be specified
in the maker’s instructions with the instrument, and are also indicated by the lack of correspondence
between a gauge mark usually placed exactly 76.2 cm from the zero point and the 76.2-cm scale
graduation.
short
standard
long
DEAN #37261
LEFT
(McGHP)
2.72
INTERACTIVE
top of rh
base of rh
SECTION 2
cap height
base of text
TABLE 2.9 Barometer Temperature Correction— Metric Units
The values in the table below are to be subtracted from the observed readings to correct for the difference
in the expansion of the mercury and the glass scale at different temperatures.
A. Glass scale
Observed barometer height in millimeters
740
750
760
700
730
Temp.
⬚C.
770
800
mm.
mm.
mm.
mm.
mm.
mm.
mm.
0
1
2
3
4
0.00
0.12
0.24
0.36
0.49
0.00
0.13
0.25
0.38
0.51
0.00
0.13
0.26
0.38
0.51
0.00
0.13
0.26
0.39
0.52
0.00
0.13
0.26
0.40
0.53
0.00
0.13
0.27
0.40
0.53
0.00
0.14
0.27
0.42
0.55
5
6
7
8
9
0.61
0.73
0.85
0.97
1.09
0.63
0.76
0.89
1.01
1.14
0.64
0.77
0.90
1.03
1.15
0.65
0.78
0.91
1.04
1.17
0.66
0.79
0.92
1.05
1.18
0.67
0.80
0.93
1.07
1.20
0.69
0.83
0.97
1.11
1.25
10
11
12
13
14
1.21
1.33
1.45
1.58
1.70
1.26
1.39
1.52
1.64
1.77
1.28
1.41
1.54
1.67
1.79
1.30
1.43
1.56
1.69
1.82
1.32
1.45
1.58
1.71
1.84
1.33
1.47
1.60
1.73
1.87
1.39
1.52
1.66
1.80
1.94
15
16
17
18
19
1.82
1.94
2.06
2.18
2.30
1.90
2.02
2.15
2.27
2.40
1.92
2.05
2.18
2.30
2.43
1.95
2.08
2.21
2.33
2.46
1.97
2.10
2.23
2.37
2.50
2.00
2.13
2.26
2.40
2.53
2.08
2.21
2.35
2.49
2.63
20
21
22
23
24
2.42
2.54
2.66
2.78
2.90
2.52
2.65
2.78
2.90
3.03
2.56
2.69
2.81
2.94
3.07
2.59
2.72
2.85
2.98
3.11
2.63
2.76
2.89
3.02
3.15
2.66
2.79
2.93
3.06
3.19
2.77
2.90
3.04
3.18
3.32
25
26
27
28
29
3.02
3.14
3.26
3.38
3.50
3.15
3.28
3.40
3.53
3.65
3.20
3.32
3.45
3.58
3.70
3.24
3.37
3.50
3.63
3.75
3.28
3.41
3.54
3.67
3.80
3.32
3.46
3.59
3.72
3.85
3.45
3.59
3.73
3.87
4.00
30
31
32
33
34
35
3.62
3.74
3.86
3.98
4.10
4.22
3.78
3.90
4.03
4.15
4.28
4.40
3.83
3.96
4.08
4.21
4.34
4.47
3.88
4.01
4.14
4.27
4.40
4.53
3.93
4.06
4.20
4.33
4.46
4.59
3.99
4.12
4.25
4.38
4.51
4.65
4.14
4.28
4.42
4.55
4.69
4.83
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standard
long
DEAN #37261
(McGHP)
RIGHT
INTERACTIVE
GENERAL INFORMATION, CONVERSION TABLES, AND MATHEMATICS
2.73
top of rh
base of rh
cap height
base of text
TABLE 2.9 Barometer Temperature Correction— Metric Units (Continued)
The values in the table below are to be subtracted from the observed readings to correct for the difference
in the expansion of the mercury and the glass scale at different temperatures.
B. Brass scale
Observed barometer height in millimeters
660
670
680
640
650
Temp.
⬚C.
690
700
mm.
mm.
mm.
mm.
mm.
mm.
mm.
0
1
2
3
4
0.00
0.10
0.21
0.31
0.42
0.00
0.11
0.21
0.32
0.42
0.00
0.11
0.22
0.32
0.43
0.00
0.11
0.22
0.33
0.44
0.00
0.11
0.22
0.33
0.44
0.00
0.11
0.23
0.34
0.45
0.00
0.11
0.23
0.34
0.46
5
6
7
8
9
0.52
0.63
0.73
0.84
0.94
0.53
0.64
0.74
0.85
0.95
0.54
0.65
0.75
0.86
0.97
0.55
0.66
0.76
0.87
0.98
0.55
0.66
0.78
0.89
1.00
0.56
0.67
0.79
0.90
1.01
0.57
0.68
0.80
0.91
1.03
10
11
12
13
14
1.04
1.15
1.25
1.35
1.46
1.06
1.16
1.27
1.38
1.48
1.07
1.18
1.29
1.40
1.50
1.09
1.20
1.31
1.42
1.53
1.11
1.22
1.33
1.44
1.55
1.12
1.24
1.35
1.46
1.57
1.14
1.25
1.37
1.48
1.59
15
16
17
18
19
1.56
1.67
1.77
1.87
1.98
1.59
1.69
1.80
1.90
2.01
1.61
1.72
1.82
1.93
2.04
1.64
1.74
1.85
1.96
2.07
1.66
1.77
1.88
1.99
2.10
1.68
1.80
1.91
2.02
2.13
1.71
1.82
1.94
2.05
2.16
20
21
22
23
24
2.08
2.18
2.29
2.39
2.49
2.11
2.22
2.32
2.43
2.53
2.15
2.25
2.36
2.47
2.57
2.18
2.29
2.40
2.50
2.61
2.21
2.32
2.43
2.54
2.65
2.24
2.35
2.47
2.58
2.69
2.28
2.39
2.50
2.62
2.73
25
26
27
28
29
2.60
2.70
2.81
2.91
3.01
2.64
2.74
2.85
2.95
3.06
2.68
2.79
2.89
3.00
3.11
2.72
2.83
2.94
3.05
3.15
2.76
2.87
2.98
3.09
3.20
2.80
2.91
3.02
3.14
3.25
2.84
2.96
3.07
3.18
3.29
30
31
32
33
34
35
3.12
3.22
3.32
3.42
3.53
3.63
3.16
3.27
3.37
3.48
3.58
3.69
3.21
3.32
3.43
3.53
3.64
3.74
3.26
3.37
3.48
3.59
3.69
3.80
3.31
3.42
3.53
3.64
3.75
3.86
3.36
3.47
3.58
3.69
3.80
3.91
3.41
3.52
3.63
3.75
3.86
3.97
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standard
long
DEAN #37261
LEFT
(McGHP)
2.74
INTERACTIVE
top of rh
base of rh
SECTION 2
cap height
base of text
TABLE 2.9 Barometer Temperature Correction— Metric Units (Continued)
B. Brass scale (continued)
710
720
Observed barometer height in millimeters
730
740
750
760
770
780
mm.
mm.
mm.
mm.
mm.
mm.
mm.
mm.
Temp.
⬚C.
0.00
0.12
0.23
0.35
0.46
0.00
0.12
0.23
0.35
0.47
0.00
0.12
0.24
0.36
0.48
0.00
0.12
0.24
0.36
0.48
0.00
0.12
0.24
0.37
0.49
0.00
0.12
0.25
0.37
0.50
0.00
0.13
0.25
0.38
0.50
0.00
0.13
0.25
0.38
0.51
0
1
2
3
4
0.58
0.69
0.81
0.93
1.04
0.59
0.70
0.82
0.94
1.06
0.59
0.71
0.83
0.95
1.07
0.60
0.72
0.84
0.96
1.08
0.61
0.73
0.86
0.98
1.10
0.62
0.74
0.87
0.99
1.11
0.63
0.75
0.88
1.00
1.13
0.64
0.76
0.89
1.02
1.14
5
6
7
8
9
1.16
1.27
1.39
1.50
1.62
1.17
1.29
1.41
1.52
1.64
1.19
1.31
1.43
1.54
1.66
1.21
1.33
1.45
1.57
1.69
1.22
1.34
1.47
1.59
1.71
1.24
1.36
1.48
1.61
1.73
1.25
1.38
1.50
1.63
1.75
1.27
1.40
1.52
1.65
1.78
10
11
12
13
14
1.73
1.85
1.96
2.08
2.19
1.76
1.87
1.99
2.11
2.22
1.78
1.90
2.02
2.14
2.25
1.81
1.93
2.05
2.17
2.29
1.83
1.95
2.07
2.20
2.32
1.85
1.98
2.10
2.22
2.35
1.88
2.00
2.13
2.25
2.38
1.90
2.03
2.16
2.28
2.41
15
16
17
18
19
2.31
2.42
2.54
2.65
2.77
2.34
2.46
2.57
2.69
2.81
2.37
2.49
2.61
2.73
2.85
2.41
2.53
2.65
2.77
2.88
2.44
2.56
2.68
2.80
2.92
2.47
2.59
2.72
2.84
2.96
2.50
2.63
2.75
2.88
3.00
2.54
2.66
2.79
2.91
3.04
20
21
22
23
24
2.88
3.00
3.11
3.23
3.34
2.92
3.04
3.16
3.27
3.39
2.96
3.08
3.20
3.32
3.44
3.00
3.12
3.24
3.36
3.48
3.05
3.17
3.29
3.41
3.53
3.09
3.21
3.33
3.45
3.58
3.13
3.25
3.38
3.50
3.62
3.17
3.29
3.42
3.54
3.67
25
26
27
28
29
3.46
3.57
3.68
3.80
3.91
4.03
3.50
3.62
3.74
3.85
3.97
4.09
3.55
3.67
3.79
3.91
4.02
4.14
3.60
3.72
3.84
3.96
4.08
4.20
3.65
3.77
3.89
4.01
4.13
4.26
3.70
3.82
3.94
4.07
4.19
4.31
3.75
3.87
4.00
4.12
4.24
4.37
3.80
3.92
4.05
4.17
4.30
4.43
30
31
32
33
34
35
short
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long
DEAN #37261
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2.75
GENERAL INFORMATION, CONVERSION TABLES, AND MATHEMATICS
top of rh
base of rh
cap height
base of text
TABLE 2.9 Barometer Temperature Correction— Metric Units (Continued)
C. Correction of a barometer for capillarity (Smithsonian Tables)
Diameter
of tube,
millimeters
0.4
4
5
6
7
8
9
10
11
12
13
0.83
0.47
0.27
0.18
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
0.6
Height of meniscus in millimeters
0.8
1.0
1.2
1.4
1.6
1.8
. . . .
. . . .
1.43
0.97
0.65
0.46
0.33
0.24
0.18
0.13
. . . .
. . . .
. . . .
1.13
0.77
0.52
0.37
0.27
0.19
0.14
Correction to be added in millimeters
1.22
0.65
0.41
0.28
0.20
0.15
. . . .
. . . .
. . . .
. . . .
1.54
0.86
0.56
0.40
0.29
0.21
0.15
0.10
0.07
0.04
1.98
1.19
0.78
0.53
0.38
0.28
0.20
0.14
0.10
0.07
2.37
1.45
0.98
0.67
0.46
0.33
0.25
0.18
0.13
0.10
. . . .
1.80
1.21
0.82
0.56
0.40
0.29
0.21
0.15
0.12
TABLE 2.10 Barometric Latitude-Gravity Table— Metric Units
Smithsonian Tables.
The values in the table below are to be subtracted from the barometric reading for latitudes from 0 to 45⬚
inclusive, and are to be added from 46 to 90⬚.
Barometer readings, millimeters
720
740
Deg.
Lat.
680
700
760
780
mm.
mm.
mm.
mm.
mm.
mm.
0
5
10
15
20
1.82
1.79
1.71
1.58
1.40
1.87
1.85
1.76
1.63
1.44
1.93
1.90
1.81
1.67
1.49
1.98
1.95
1.86
1.72
1.53
2.04
2.00
1.92
1.77
1.57
2.09
2.06
1.97
1.81
1.61
21
22
23
24
25
1.36
1.32
1.28
1.23
1.18
1.40
1.36
1.31
1.27
1.22
1.44
1.40
1.35
1.30
1.25
1.48
1.44
1.39
1.34
1.29
1.52
1.48
1.43
1.37
1.32
1.56
1.51
1.46
1.41
1.36
26
27
28
29
30
1.13
1.08
1.03
0.98
0.93
1.17
1.12
1.06
1.01
0.95
1.20
1.15
1.09
1.04
0.98
1.23
1.18
1.12
1.07
1.01
1.27
1.21
1.15
1.10
1.04
1.30
1.24
1.18
1.12
1.06
31
32
33
34
35
0.87
0.82
0.76
0.70
0.64
0.90
0.84
0.78
0.72
0.66
0.92
0.86
0.80
0.74
0.68
0.95
0.89
0.83
0.76
0.70
0.98
0.91
0.85
0.79
0.72
1.00
0.94
0.87
0.81
0.74
36
37
38
0.58
0.52
0.46
0.60
0.54
0.48
0.62
0.56
0.49
0.64
0.57
0.51
0.65
0.59
0.52
0.67
0.60
0.53
short
standard
long
DEAN #37261
LEFT
(McGHP)
2.76
INTERACTIVE
top of rh
base of rh
SECTION 2
cap height
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TABLE 2.10 Barometric Latitude-Gravity Table— Metric Units (Continued)
Barometer readings, millimeters
720
740
Deg.
Lat.
680
700
760
780
mm.
mm.
mm.
mm.
mm.
mm.
39
40
0.40
0.34
0.42
0.35
0.43
0.36
0.44
0.37
0.45
0.38
0.46
0.39
41
42
43
44
45
0.28
0.22
0.16
0.09
0.03
0.29
0.22
0.16
0.10
0.03
0.30
0.23
0.16
0.10
0.03
0.30
0.24
0.17
0.10
0.03
0.31
0.24
0.17
0.10
0.03
0.32
0.25
0.18
0.11
0.04
46
47
48
49
50
0.03
0.09
0.16
0.22
0.28
0.03
0.10
0.16
0.23
0.29
0.03
0.10
0.17
0.23
0.30
0.03
0.10
0.17
0.24
0.31
0.04
0.10
0.18
0.25
0.31
0.04
0.11
0.18
0.25
0.32
51
52
53
54
55
0.34
0.40
0.46
0.52
0.58
0.35
0.42
0.48
0.54
0.60
0.36
0.43
0.49
0.56
0.62
0.37
0.44
0.51
0.57
0.64
0.38
0.45
0.52
0.59
0.65
0.39
0.46
0.53
0.60
0.67
56
57
58
59
60
0.64
0.70
0.76
0.81
0.87
0.66
0.72
0.78
0.84
0.89
0.68
0.74
0.80
0.86
0.92
0.70
0.76
0.82
0.89
0.94
0.72
0.78
0.85
0.91
0.97
0.74
0.80
0.87
0.93
1.00
61
62
63
64
65
0.92
0.97
1.03
1.08
1.13
0.95
1.00
1.06
1.11
1.16
0.98
1.02
1.09
1.14
1.19
1.00
1.05
1.12
1.17
1.22
1.03
1.08
1.15
1.20
1.26
1.06
1.11
1.18
1.23
1.29
66
67
68
69
70
1.17
1.22
1.26
1.31
1.35
1.21
1.25
1.30
1.34
1.39
1.24
1.29
1.34
1.38
1.43
1.28
1.33
1.37
1.42
1.47
1.31
1.36
1.41
1.46
1.51
1.35
1.40
1.45
1.50
1.55
72
75
80
85
90
1.42
1.53
1.66
1.74
1.77
1.47
1.57
1.71
1.79
1.82
1.51
1.62
1.76
1.84
1.87
1.55
1.66
1.81
1.90
1.93
1.59
1.71
1.86
1.95
1.98
1.63
1.75
1.90
2.00
2.03
short
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DEAN #37261
(McGHP)
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GENERAL INFORMATION, CONVERSION TABLES, AND MATHEMATICS
2.77
top of rh
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base of text
TABLE 2.11 Barometric Correction for Gravity— Metric Units
The values in the table below are to be subtracted from the readings taken on a mercurial barometer to correct
for the decrease in gravity with increase in altitude.
Observed barometer height in millimeters
Height
above
sealevel
meters
400
450
500
550
600
650
700
750
800
mm.
mm.
mm.
mm.
mm.
mm.
mm.
mm.
mm.
100
200
300
400
500
....
....
....
....
....
....
....
....
....
....
....
....
....
....
....
....
....
....
....
....
....
....
....
....
....
....
....
....
....
....
0.02
0.04
0.07
0.09
0.11
0.02
0.05
0.07
0.10
0.12
0.02
0.05
0.07
0.10
0.13
600
700
800
900
1000
....
....
....
....
....
....
....
....
....
....
....
....
....
....
....
....
....
....
....
0.18
....
....
....
....
0.19
0.12
0.14
0.16
0.18
0.20
0.13
0.15
0.18
0.20
0.22
0.14
0.16
0.19
0.22
0.24
....
....
....
....
....
1100
1200
1300
1400
1500
....
....
....
....
....
....
....
....
....
....
....
....
....
....
0.24
0.19
0.21
0.22
0.24
0.26
0.21
0.23
0.24
0.26
0.28
0.22
0.24
0.26
0.28
0.30
0.24
0.26
0.29
0.31
0.33
....
....
....
....
....
....
....
....
....
....
1600
1700
1800
1900
2000
....
....
....
....
....
....
....
....
....
0.28
0.25
0.27
0.28
0.30
0.31
0.28
0.30
0.31
0.33
0.34
0.30
0.32
0.34
0.36
0.38
0.32
0.34
0.36
0.39
0.41
....
....
....
....
....
....
....
....
....
....
....
....
....
....
....
2100
2200
2300
2400
2500
....
....
....
....
0.31
0.30
0.31
0.32
0.34
0.35
0.33
0.35
0.36
0.38
0.39
0.36
0.38
0.40
0.42
0.43
0.40
0.41
0.43
0.45
0.47
....
....
....
....
....
....
....
....
....
....
....
....
....
....
....
....
....
....
....
....
2600
2800
3000
3200
3400
0.33
0.35
0.38
0.40
0.43
0.37
0.40
0.42
0.46
0.48
0.41
0.44
0.47
....
....
....
....
....
....
....
....
....
....
....
....
....
....
....
....
....
....
....
....
....
....
....
....
....
....
....
....
....
....
....
....
short
standard
long
DEAN #37261
LEFT
(McGHP)
2.78
INTERACTIVE
top of rh
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SECTION 2
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base of text
TABLE 2.12 Reduction of the Barometer to Sea Level— Metric Units
A barometer located at an elevation above sea level will show a reading lower than a barometer at sea level
by an amount approximately 2.5 mm (0.1 in) for each 30.5 m (100 ft) of elevation. A closer approximation can
be made by reference to the following tables, which take into account (1) the effect of altitude of the station at
which the barometer is read, (2) the mean temperature of the air column extending from the station down to sea
level, (3) the latitude of the station at which the barometer is read, and (4) the reading of the barometer corrected
for its temperature, a correction which is applied only to mercurial barometers since the aneroid barometers are
compensated for temperature effects.
Example. A barometer which has been corrected for its temperature reads 650 mm at a station whose
altitude is 1350 m above sea level and at a latitude of 30⬚. The mean temperature (outdoor temperature) at the
station is 20⬚C.
Table A (metric units) gives for these conditions a temperature-altitude factor of . . . . . . . . . . . . . . . . . .
The Latitude Factor Table gives for 135.2 at 30⬚ lat. a correction of . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Therefore, the corrected value of the temperature-altitude factor is . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Entering Table B (metric units), with a temperature-altitude factor of 135.37 and a barometric
reading of 650 mm (corrected for temperature), the correction is found to be . . . . . . . . . . . . . . . . . . . . . . . . . .
Accordingly the barometric reading reduced to sea level is 650 ⫹ 109.6 ⫽ 759.6 mm.
135.2
⫹0.17
135.37
109.6
Latitude Factor– English or Metric Units. For latitudes 0⬚– 45⬚ add the latitude factor, for 45⬚– 90⬚ subtract the
latitude factor, from the values obtained in Table A.
Temp.— Alt.
Factor
From Table A
Latitude
0⬚
10⬚
20⬚
30⬚
45⬚
50
100
150
200
250
300
350
0.1
0.3
0.4
0.5
0.7
0.8
0.9
0.1
0.3
0.4
0.5
0.6
0.8
0.9
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.1
0.1
0.2
0.3
0.3
0.4
0.5
0.0
0.0
0.0
0.0
0.0
0.0
0.0
90⬚
80⬚
70⬚
60⬚
45⬚
A. Values of the temperature-altitude factor for use in Table B.*
Altitude
in
Meters
Mean Temperature of Air Column in Centigrade Degrees
⫺16⬚
⫺8⬚
⫺4⬚
0⬚
6⬚
10⬚
14⬚
18⬚
20⬚
22⬚
26⬚
10
50
100
150
1.2
5.8
11.5
17.3
1.1
5.6
11.2
16.7
1.1
5.5
11.0
16.5
1.1
5.4
10.8
16.2
1.1
5.3
10.6
15.9
1.0
5.2
10.4
15.6
1.0
5.1
10.3
15.4
1.0
5.0
10.1
15.1
1.0
5.0
10.0
15.0
1.0
5.0
9.9
14.9
1.0
4.9
9.8
14.7
200
250
300
350
400
23.0
28.8
34.5
40.3
46.0
22.3
27.9
33.5
39.0
44.6
22.0
27.5
33.0
38.5
43.9
21.6
27.0
32.5
37.9
43.3
21.1
26.4
31.7
37.0
42.3
20.8
26.0
31.2
36.4
41.6
20.5
25.6
30.7
35.9
41.0
20.2
25.2
30.3
35.3
40.4
20.0
25.0
30.1
35.1
40.1
19.9
24.9
29.8
34.8
39.8
19.6
24.5
29.4
34.3
39.2
450
500
550
600
650
51.8
57.5
63.3
69.0
74.8
51.3
55.8
61.4
66.9
72.5
49.4
54.9
60.4
65.9
71.4
48.7
54.1
59.5
64.9
70.3
47.6
52.9
58.1
63.4
68.7
46.8
52.0
57.2
62.4
67.6
46.1
51.2
56.4
61.5
66.6
45.4
50.5
55.5
60.6
65.6
45.1
50.1
55.1
60.1
65.1
44.8
49.7
54.7
59.7
64.6
44.1
49.0
53.9
58.8
63.7
short
standard
long
DEAN #37261
(McGHP)
RIGHT
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2.79
GENERAL INFORMATION, CONVERSION TABLES, AND MATHEMATICS
top of rh
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TABLE 2.12 Reduction of the Barometer to Sea Level— Metric Units (Continued)
Altitude
in
Meters
Mean Temperature of Air Column in Centigrade Degrees
⫺16⬚
⫺8⬚
⫺4⬚
0⬚
6⬚
10⬚
14⬚
18⬚
20⬚
22⬚
26⬚
700
750
800
850
900
80.6
86.3
92.1
97.8
103.6
78.1
83.7
89.2
94.8
100.4
76.9
82.4
87.9
93.4
98.9
75.7
81.1
86.5
92.0
97.4
74.0
79.3
84.6
89.8
95.1
72.9
78.1
83.3
88.5
93.7
71.7
76.9
82.0
87.1
92.2
70.7
75.7
80.8
85.8
90.8
70.1
75.1
80.1
85.2
90.2
69.6
74.6
79.6
84.5
89.5
68.6
73.5
78.4
83.3
88.2
950
1000
1050
1100
1150
109.3
115.1
120.8
126.6
132.3
106.0
111.5
117.1
122.7
128.3
104.4
109.8
115.3
120.8
126.3
102.8
108.2
113.6
119.0
124.4
100.4
105.7
111.0
116.3
121.6
98.9
104.1
109.3
114.5
119.7
97.4
102.5
107.6
112.7
117.9
95.9
100.9
106.0
111.0
116.1
95.2
100.2
105.2
110.2
115.2
94.5
99.4
104.4
109.4
114.4
93.1
98.0
102.9
107.8
112.7
1200
1250
1300
1350
1400
138.1
143.8
149.6
155.3
161.1
133.8
139.4
145.0
150.6
156.2
131.8
137.3
142.8
148.3
153.8
129.8
135.2
140.6
146.0
151.4
126.8
132.1
137.4
142.7
148.0
124.9
130.1
135.3
140.5
145.7
123.0
128.1
133.2
138.4
143.5
121.1
126.2
131.2
136.3
141.3
120.2
125.2
130.2
135.2
140.2
119.3
124.3
129.3
134.2
139.2
117.6
122.5
127.4
132.3
137.2
1450
1500
1550
1600
1650
166.8
172.6
178.3
184.1
189.8
161.7
167.3
172.9
178.5
184.0
159.3
164.8
170.2
175.7
181.2
156.8
162.3
167.7
173.1
178.5
153.3
158.5
163.8
169.1
174.4
150.9
156.1
161.3
166.5
171.7
148.6
153.7
158.8
164.0
169.1
146.4
151.4
156.4
161.5
166.5
145.3
150.3
155.3
160.3
165.3
144.2
149.1
154.1
159.1
164.1
142.1
147.0
151.8
156.7
161.6
1700
1750
1800
1850
1900
195.6
201.4
207.1
212.9
218.6
189.6
195.2
200.8
206.3
211.9
186.7
192.2
197.7
203.2
208.7
183.9
189.3
194.7
200.1
205.5
179.7
185.0
190.2
195.5
200.8
176.9
182.1
187.3
192.5
197.7
174.2
179.3
184.5
189.6
194.7
171.6
176.6
181.7
186.7
191.8
170.3
175.3
180.3
185.3
190.3
169.0
174.0
179.0
183.9
188.9
166.5
171.4
176.3
181.2
186.1
1950
2000
2050
2100
2150
224.4
230.1
235.9
241.6
247.4
217.5
223.0
228.6
234.2
239.8
214.2
219.7
225.1
230.6
236.1
210.9
216.3
221.7
227.1
232.5
206.1
211.4
216.7
221.9
227.2
202.9
208.1
213.3
218.5
223.7
199.8
204.9
210.1
215.2
220.3
196.8
201.9
206.9
211.9
217.0
195.3
200.3
205.3
210.4
215.4
193.9
198.8
203.8
208.8
213.8
191.0
195.0
200.8
205.7
210.6
2200
2250
2300
2350
2400
253.1
258.9
264.6
270.4
276.1
245.4
250.9
256.5
262.1
267.7
241.6
247.1
252.6
258.1
263.6
237.9
243.4
248.8
254.2
259.6
232.5
237.8
243.1
248.3
253.6
228.9
234.1
239.3
244.5
249.7
225.4
230.6
235.7
240.8
245.9
222.0
227.1
232.1
237.2
242.2
220.4
225.4
230.4
235.4
240.4
218.7
223.7
228.7
233.6
238.6
215.5
220.4
225.3
230.2
235.1
2450
2500
2550
2600
2650
281.9
287.6
293.4
299.1
304.9
273.2
278.8
284.4
290.0
295.5
269.1
274.5
280.0
285.5
291.0
265.0
270.4
275.8
281.2
286.6
258.9
264.2
269.5
274.8
280.0
254.9
260.1
265.3
270.5
275.7
251.0
256.2
261.3
266.4
271.5
247.3
252.3
257.3
262.4
267.4
245.4
250.4
255.4
260.4
265.4
243.6
248.5
253.5
258.5
263.4
240.0
244.9
249.8
254.7
259.6
2700
2750
2800
2850
2900
310.6
316.4
322.1
327.9
333.6
301.1
306.7
312.3
317.8
323.4
296.5
302.0
307.5
313.0
318.4
292.0
297.4
302.8
308.2
313.6
285.3
290.6
295.9
301.2
306.4
280.9
286.1
291.3
296.5
301.7
276.6
281.8
286.9
292.0
297.1
272.5
277.5
282.6
287.6
292.6
270.4
275.4
280.4
285.4
290.4
268.4
273.4
278.3
283.3
288.3
264.5
269.4
274.3
279.2
284.1
2950
3000
339.4
345.1
329.0
334.5
323.9
329.4
319.0
324.4
311.7
317.0
306.9
312.1
302.2
307.4
297.7
302.7
295.5
300.5
293.3
298.2
289.0
293.8
* From Smithsonian Meteorological Tables, 3d ed., 1907.
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long
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(McGHP)
2.80
INTERACTIVE
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TABLE 2.12 Reduction of the Barometer to Sea Level— Metric Units (Continued)
B. Values in millimeters to be added.*
Temp.
— Alt.
Factor
1
5
10
15
20
25
30
35
40
45
50
55
60
65
70
75
80
85
90
95
100
105
110
115
120
125
130
135
140
145
150
155
160
165
170
175
180
185
190
195
Barometer Reading in Millimeters
790
770
750
730
710
690
670
0.9
4.6
9.1
13.8
18.4
0.9
4.4
8.9
13.4
17.9
22.5
27.1
31.7
36.3
0.9
4.3
8.7
13.1
17.5
21.9
26.4
30.8
35.3
39.9
0.8
4.2
8.5
12.7
17.0
21.3
25.7
30.0
34.4
38.8
0.8
4.1
8.2
12.4
16.5
20.7
25.0
29.2
33.5
37.8
0.8
4.0
8.0
12.0
16.1
20.1
24.2
28.4
32.5
36.7
31.6
35.6
750
730
710
690
670
650
630
44.4
49.0
53.6
58.3
43.3
47.7
52.2
56.7
61.3
65.8
70.4
75.0
42.1
46.4
50.8
55.2
59.6
64.0
68.5
73.0
77.5
82.1
40.9
45.1
49.3
53.6
57.9
62.2
66.6
70.9
75.3
79.7
39.7
43.8
47.9
52.1
56.2
60.4
64.6
68.9
73.1
77.4
62.7
66.8
71.0
75.1
60.8
64.8
68.8
72.8
710
690
670
650
630
610
86.6
91.2
95.9
100.5
84.2
88.7
93.2
97.7
102.2
106.8
111.4
116.0
120.7
81.8
86.1
90.5
94.8
99.3
103.7
108.2
112.7
117.2
121.7
79.3
83.5
87.8
92.0
96.3
100.6
104.9
109.3
113.7
118.1
76.9
81.0
85.1
89.2
93.3
97.5
101.7
105.9
110.2
114.5
94.4
98.5
102.6
106.7
110.8
670
650
630
610
590
570
126.3
130.9
135.5
140.2
122.5
127.0
131.5
136.0
140.5
145.1
149.7
154.3
158.9
118.8
123.1
127.4
131.8
136.2
140.6
145.1
149.5
154.0
158.6
115.0
119.2
123.4
127.6
131.9
136.2
140.5
144.8
149.2
153.5
127.5
131.7
135.9
140.0
144.3
148.5
123.2
127.2
131.3
135.3
139.4
143.5
* From Smithsonian Meteorological Tables, 3d ed., 1907.
short
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RIGHT
INTERACTIVE
GENERAL INFORMATION, CONVERSION TABLES, AND MATHEMATICS
2.81
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TABLE 2.12 Reduction of the Barometer to Sea Level— Metric Units (Continued)
B. Values in millimeters to be added.*
Temp.
— Alt.
Factor
200
205
210
215
220
225
230
235
240
245
250
255
260
265
270
275
280
285
290
295
300
305
310
315
320
325
330
335
340
345
Barometer Reading in Millimeters
630
610
590
570
550
530
163.1
167.7
172.3
176.9
157.9
162.4
166.8
171.3
175.8
180.4
184.9
189.5
194.1
198.8
152.8
157.1
161.4
165.7
170.1
174.5
178.9
183.3
187.8
192.3
147.6
151.7
155.9
160.1
164.3
168.5
172.8
177.1
181.4
185.7
154.5
158.5
162.6
166.7
170.9
175.0
179.2
148.9
152.8
156.7
160.7
164.7
168.7
172.7
590
570
550
530
510
196.8
201.3
205.9
210.5
215.1
219.8
190.1
194.5
198.9
203.3
207.8
212.3
216.8
221.4
225.9
230.5
183.4
187.7
191.9
196.2
200.5
204.9
209.2
213.6
218.0
222.4
176.8
180.8
185.0
189.1
193.2
197.4
201.6
205.8
210.1
214.3
178.0
181.9
185.9
190.0
194.0
198.1
202.1
206.3
570
550
530
510
490
235.1
239.8
226.9
231.4
235.9
240.4
245.0
249.6
254.2
258.8
263.5
218.6
223.0
227.3
231.7
236.1
240.5
244.9
249.4
253.9
258.4
210.4
214.6
218.7
222.9
227.2
231.4
235.7
240.0
244.4
248.7
206.1
210.1
214.2
218.3
222.4
226.5
230.6
234.8
238.9
* From Smithsonian Meteorological Tables, 3d ed., 1907.
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TABLE 2.13 Viscosity Conversion Table
Centistokes to Saybolt, Redwood, and Engler units.
Poise ⫽ cgs unit of absolute viscosity
Centipoise ⫽ 0.01 poise
Stoke ⫽ cgs unit of kinematic viscosity
Centistoke ⫽ 0.01 stoke
Centipoises ⫽ centistokes ⫻ density (at temperature under consideration)
Reyn (1 lb · s per sq in) ⫽ 69 ⫻ 105 centpoises
Cf. Jour. Inst. Pet. Tech., Vol. 22, p. 21 (1936); Reports of A. S. T. M. Committee D-2, 1936 and 1937.
The values of Saybolt Universal Viscosity at 100⬚F and at 210⬚F are taken directly from the comprehensive
ASTM Viscosity Table, Special Technical Publication No. 43A (1953) by permission of the publishers, American
Society for Testing Materials, 1916 Race St., Philadelphia 3, Pa.
Centistokes
100⬚F.
130⬚F.
210⬚F.
70⬚F.
140⬚F.
200⬚F.
Engler
Degrees at
all Temps.
2.0
3.0
4.0
5.0
6.0
7.0
8.0
9.0
10.0
11.0
12.0
14.0
16.0
18.0
20.0
22.0
24.0
26.0
28.0
30.0
32.0
34.0
36.0
38.0
40.0
42.0
44.0
46.0
48.0
50.0
60.0
70.0
80.0
90.0
100.0*
32.62
36.03
39.14
42.35
45.56
48.77
52.09
55.50
58.91
62.43
66.04
73.57
81.30
89.44
97.77
106.4
115.0
123.7
132.5
141.3
150.2
159.2
168.2
177.3
186.3
195.3
204.4
213.7
222.9
232.1
278.3
324.4
370.8
417.1
463.5
32.68
36.10
39.22
42.43
45.65
48.86
52.19
55.61
59.02
62.55
66.17
73.71
81.46
89.61
97.96
106.6
115.2
123.9
132.8
141.6
150.5
159.5
168.5
177.6
186.7
195.7
204.8
214.1
223.3
232.5
278.8
325.0
371.5
417.9
464.4
32.85
36.28
39.41
42.65
45.88
49.11
52.45
55.89
59.32
62.86
66.50
74.09
81.87
90.06
98.45
107.1
115.8
124.5
133.4
142.3
151.2
160.3
169.4
178.5
187.6
196.7
205.9
215.2
224.5
233.8
280.2
326.7
373.4
420.0
466.7
30.2
32.7
35.3
37.9
40.5
43.2
46.0
48.9
51.7
54.8
57.9
64.4
71.0
77.9
85.0
92.4
99.9
107.5
115.3
123.1
131.0
138.9
146.9
155.0
163.0
171.0
179.1
187.1
195.2
203.3
243.5
283.9
323.9
364.4
404.9
31.0
33.5
36.0
38.5
41.0
43.7
46.4
49.1
52.0
55.0
58.1
64.6
71.4
78.5
85.8
93.3
100.9
108.6
116.5
124.4
132.3
140.2
148.2
156.2
164.3
172.3
180.4
188.5
196.6
204.7
245.3
286.0
326.6
367.4
408.2
31.2
33.7
36.3
38.9
41.5
44.2
46.9
49.7
52.6
55.6
58.8
65.3
72.2
79.4
86.9
94.5
102.2
110.0
118.0
126.0
134.1
142.2
150.3
158.3
166.7
175.0
183.3
191.7
200.0
208.3
250.0
291.7
333.4
375.0
416.7
1.14
1.22
1.31
1.40
1.48
1.56
1.65
1.75
1.84
1.93
2.02
2.22
2.43
2.64
2.87
3.10
3.34
3.58
3.82
4.07
4.32
4.57
4.83
5.08
5.34
5.59
5.85
6.11
6.37
6.63
7.90
9.21
10.53
11.84
13.16
Saybolt Universal Viscosity at
Redwood Seconds at
* At higher values use the same ratio as above for 100 centistokes; e.g., 102 centistokes ⫽ 102 ⫻ 4.635 Saybolt seconds at
100⬚F.
To obtain the Saybolt Universal viscosity equivalent to a kinematic viscosity determined at t⬚F., multiply the equivalent
Saybolt Universal viscosity at 100⬚F. by 1 ⫹ (t ⫺ 100) 0.000064; e.g., 10 centistokes at 210⬚F are equivalent to 58.91 ⫻
1.0070, or 59.32 Saybolt Universal Viscosity at 210⬚F.
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GENERAL INFORMATION, CONVERSION TABLES, AND MATHEMATICS
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TABLE 2.14 Conversion of Weighings in Air to Weighings in Vacuo
If the mass of a substance in air is m f , its density ␳m , the density of weights used in making the weighing
␳w , and the density* of air ␳a , the true mass of the substance in vacuo, mvac , is
mvac ⫽ m f ⫹ ␳a m f
冉
1
1
⫺
␳m ␳w
冊
For most purposes it is sufficient to assume a density of 8.4 for brass weights, and a density of 0.0012 for
air under ordinary conditions. The equation then becomes
m vac ⫽ m f ⫹ 0.0012m f
冉
1
1
⫺
␳m 8.4
冊
The table which follows gives the values of k (buoyancy reduction factor), which is the correction necessary
because of the buoyant effect of the air upon the object weighed; the table is computed for air with the density
of 0.0012; m is the weight in grams of the object when weighed in air; weight of object reduced to “in
vacuo” ⫽ m ⫹ km/1000.
Buoyancy reduction factor, k
Density of
object weighed
Brass weights,
density ⫽ 8.4
Pt or Pt-Ir
weights, density ⫽ 21.5
Al or quartz weights,
density ⫽ 2.7
Gold weights,
density ⫽ 17
0.2
0.3
0.4
0.5
0.6
5.89
3.87
2.87
2.26
1.86
5.98
3.96
2.95
2.35
1.95
5.58
3.56
2.55
1.95
1.55
5.97
3.95
2.94
2.34
1.93
0.7
0.75
0.80
0.82
0.84
1.57
1.46
1.36
1.32
1.29
1.66
1.55
1.45
1.41
1.37
1.26
1.15
1.05
1.01
0.98
1.65
1.53
1.43
1.39
1.36
0.86
0.88
0.90
0.92
0.94
1.25
1.22
1.19
1.16
1.13
1.34
1.31
1.28
1.25
1.22
0.94
0.91
0.88
0.85
0.82
1.33
1.29
1.26
1.24
1.21
0.96
0.98
1.00
1.02
1.04
1.11
1.08
1.06
1.03
1.01
1.20
1.17
1.15
1.12
1.10
0.80
0.77
0.75
0.72
0.70
1.18
1.16
1.13
1.11
1.08
1.06
1.08
1.10
1.12
1.14
0.99
0.97
0.95
0.93
0.91
1.08
1.06
1.04
1.02
1.00
0.68
0.66
0.64
0.62
0.60
1.06
1.04
1.02
1.00
0.98
1.16
1.18
1.20
1.25
1.30
0.89
0.87
0.86
0.82
0.78
0.98
0.96
0.95
0.91
0.87
0.58
0.56
0.55
0.51
0.47
0.96
0.95
0.93
0.89
0.85
* See Table 5.15, Specific Gravity of Air at Various Temperatures.
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TABLE 2.14 Conversion of Weighings in Air to Weighings in Vacuo (Continued)
Buoyancy reduction factor, k
Density of
object weighed
Brass weights,
density ⫽ 8.4
Pt or Pt-Ir
weights, density ⫽ 21.5
Al or quartz weights,
density ⫽ 2.7
Gold weights,
density ⫽ 17
1.35
1.40
1.50
1.6
1.7
0.75
0.71
0.66
0.61
0.56
0.83
0.80
0.74
0.69
0.65
0.44
0.40
0.35
0.30
0.25
0.82
0.79
0.73
0.68
0.64
1.8
1.9
2.0
2.2
2.4
0.52
0.49
0.46
0.40
0.36
0.61
0.58
0.54
0.49
0.44
0.21
0.18
0.15
0.09
0.05
0.60
0.56
0.53
0.48
0.43
2.6
2.8
3.0
3.5
4
0.32
0.29
0.26
0.20
0.16
0.41
0.37
0.34
0.29
0.24
0.01
⫺0.02
⫺0.05
⫺0.11
⫺0.15
0.39
0.36
0.33
0.27
0.23
5
6
7
8
9
0.10
0.06
0.03
0.01
⫺0.01
0.18
0.14
0.12
0.09
0.08
⫺0.21
⫺0.25
⫺0.28
⫺0.30
⫺0.32
0.17
0.13
0.10
0.08
0.06
10
12
14
16
18
⫺0.02
⫺0.04
⫺0.06
⫺0.07
⫺0.08
0.06
0.04
0.03
0.02
0.01
⫺0.33
⫺0.35
⫺0.37
⫺0.38
⫺0.39
0.05
0.03
0.02
0.00
0.00
20
22
⫺0.08
⫺0.09
0.00
0.00
⫺0.39
⫺0.40
⫺0.01
⫺0.02
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GENERAL INFORMATION, CONVERSION TABLES, AND MATHEMATICS
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TABLE 2.15 Hydrometer Conversion Table
This table gives the relation between density (c.g.s.) and degrees on the Baumé and Twaddell scales. The
Twaddell scale is never used for densities less than unity. See also Sec. 2.1.2.1, Hydrometers.
Density
Degrees Baumé
(NIST* scale)
Degrees
Baumé
(A.P.I.† scale)
Density
Degrees Baumé
(NIST* scale)
Degrees
Baumé
(A.P.I.† scale)
0.600
0.605
0.610
0.615
0.620
103.33
101.40
99.51
97.64
95.81
104.33
102.38
100.47
98.58
96.73
0.825
0.830
0.835
0.840
0.845
39.70
38.68
37.66
36.67
35.68
40.02
38.98
37.96
36.95
35.96
0.625
0.630
0.635
0.640
0.645
94.00
92.22
90.47
88.75
87.05
94.90
93.10
91.33
89.59
87.88
0.850
0.855
0.860
0.865
0.870
34.71
33.74
32.79
31.85
30.92
34.97
34.00
33.03
32.08
31.14
0.650
0.655
0.660
0.665
0.670
85.38
83.74
82.12
80.52
78.95
86.19
84.53
82.89
81.28
79.69
0.875
0.880
0.885
0.890
30.00
29.09
28.19
27.30
30.21
29.30
28.39
27.49
0.675
0.680
0.685
0.690
0.695
77.41
75.88
74.38
72.90
71.43
78.13
76.59
75.07
73.57
72.10
0.895
0.900
0.905
0.910
0.915
26.42
25.56
24.70
23.85
23.01
26.60
25.72
24.85
23.99
23.14
0.700
0.705
0.710
0.715
0.720
70.00
68.57
67.18
65.80
64.44
70.64
69.21
67.80
66.40
65.03
0.920
0.925
0.930
0.935
0.940
22.17
21.35
20.54
19.73
18.94
22.30
21.47
20.65
19.84
19.03
0.725
0.730
0.735
0.740
0.745
63.10
61.78
60.48
59.19
57.92
63.67
62.34
61.02
59.72
58.43
0.945
0.950
0.955
0.960
0.965
18.15
17.37
16.60
15.83
15.08
18.24
17.45
16.67
15.90
15.13
0.750
0.755
0.760
0.765
0.770
56.67
55.43
54.21
53.01
51.82
57.17
55.92
54.68
53.47
52.27
0.970
0.975
0.980
0.985
0.990
14.33
13.59
12.86
12.13
11.41
14.38
13.63
12.89
12.15
11.43
0.775
0.780
0.785
0.790
0.795
50.65
49.49
48.34
47.22
46.10
51.08
49.91
48.75
47.61
46.49
0.995
1.000
10.70
10.00
10.71
10.00
0.800
0.805
0.810
0.815
0.820
45.00
43.91
42.84
41.78
40.73
45.38
44.28
43.19
42.12
41.06
DENSITIES GREATER THAN UNITY
Density
1.00
1.01
1.02
Degrees Baumé
(NIST* scale)
Degrees
Twaddell
0.00
1.44
2.84
* NIST, National Institute for Science and Technology (formerly the National Bureau of Standards, U.S.).
† A.P.I. is the American Petroleum Institute.
0
2
4
short
standard
long
DEAN #37261
LEFT
(McGHP)
2.86
INTERACTIVE
top of rh
base of rh
SECTION 2
cap height
base of text
TABLE 2.15 Hydrometer Conversion Table (Continued)
Density
Degrees Baumé
(NIST* scale)
Degrees
Twaddell
Density
Degrees Baumé
(NIST* scale)
Degrees
Twaddell
1.03
1.04
1.05
1.06
1.07
4.22
5.58
6.91
8.21
9.49
6
8
10
12
14
1.52
1.53
1.54
1.55
1.56
49.60
50.23
50.84
51.45
52.05
104
106
108
110
112
1.08
1.09
1.10
1.11
1.12
10.78
11.97
13.18
14.37
15.54
16
18
20
22
24
1.57
1.58
1.59
1.60
1.61
52.64
53.23
53.80
54.38
54.94
114
116
118
120
122
1.13
1.14
1.15
1.16
1.17
16.68
17.81
18.91
20.00
21.07
26
28
30
32
34
1.62
1.63
1.64
1.65
1.66
55.49
56.04
56.58
57.12
57.65
124
126
128
130
132
1.18
1.19
1.20
1.21
1.22
22.12
23.15
24.17
25.16
26.15
36
38
40
42
44
1.67
1.68
1.69
1.70
1.71
58.17
58.69
59.20
59.71
60.20
134
136
138
140
142
1.23
1.24
1.25
1.26
1.27
27.11
28.06
29.00
29.92
30.83
46
48
50
52
54
1.72
1.73
1.74
1.75
1.76
60.70
61.18
61.67
62.14
62.61
144
146
148
150
152
1.28
1.29
1.30
1.31
1.32
31.72
32.60
33.46
34.31
35.15
56
58
60
62
64
1.77
1.78
1.79
1.80
1.81
63.08
63.54
63.99
64.44
64.89
154
156
158
160
162
1.33
1.34
1.35
1.36
1.37
35.98
36.79
37.59
38.38
39.16
66
68
70
72
74
1.82
1.83
1.84
1.85
1.86
65.31
65.77
66.20
66.62
67.04
164
166
168
170
172
1.38
1.39
1.40
1.41
1.42
39.93
40.68
41.43
42.16
42.89
76
78
80
82
84
1.87
1.88
1.89
1.90
1.91
67.46
67.87
68.28
68.68
69.08
174
176
178
180
182
1.43
1.44
1.45
1.46
1.47
43.60
44.31
45.00
45.68
46.36
86
88
90
92
94
1.92
1.93
1.94
1.95
1.96
69.48
69.87
70.26
70.64
71.02
184
186
188
190
192
1.48
1.49
1.50
1.51
47.03
47.68
48.33
48.97
96
98
100
102
1.97
1.98
1.99
2.00
71.40
71.77
72.14
72.50
194
196
198
200
* NIST, National Institute for Science and Technology (formerly the National Bureau of Standards, U.S.).
short
standard
long
DEAN #37261
(McGHP)
RIGHT
INTERACTIVE
GENERAL INFORMATION, CONVERSION TABLES, AND MATHEMATICS
2.87
cap height
base of text
TABLE 2.16 Pressure Conversion Chart
psi
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0.80
0.90
1.00
2.00
3.00
4.00
5.00
6.00
7.00
8.00
9.00
10.0
14.7
15.0
20.0
25.0
30.0
40.0
50.0
60.0
70.0
80.0
90.0
100.0
150.0
200.0
250.0
300.0
400.0
500.0
top of rh
base of rh
Inches H2O
at 4⬚C
Inches Hg
at 0⬚C
mmH2O
at 4⬚C
mmHg
at 0⬚C
0.2768
0.5536
0.8304
1.107
1.384
1.661
1.938
2.214
2.491
2.768
5.536
8.304
11.07
13.84
16.61
19.38
22.14
24.91
27.68
55.36
83.04
110.7
138.4
166.1
193.8
221.4
249.1
276.8
406.9
415.2
553.6
692.0
830.4
1 107
1 384
1 661
1 938
2 214
2 491
2 768
4 152
5 536
6 920
8 304
0.0204
0.0407
0.0611
0.0814
0.1018
0.1222
0.1425
0.1629
0.1832
0.2036
0.4072
0.6108
0.8144
1.018
1.222
1.425
1.629
1.832
2.036
4.072
6.108
8.144
10.18
12.22
14.25
16.29
18.32
20.36
29.93
30.54
40.72
50.90
61.08
81.44
101.8
122.2
142.5
162.9
183.2
203.6
305.4
407.2
509.0
610.8
7.031
14.06
21.09
28.12
35.15
42.18
49.22
56.25
63.28
70.31
140.6
210.9
281.2
351.5
421.8
492.2
562.5
632.8
703.1
1 072
2 109
2 812
3 515
4 218
4 922
5 625
6 328
7 031
10 332
10 550
14 060
17 580
21 090
28 120
35 150
42 180
49 220
56 250
63 280
70 307
0.517
1.034
1.551
2.068
2.586
3.103
3.620
4.137
4.654
5.171
10.34
15.51
20.68
25.86
31.03
36.20
41.37
46.54
51.71
103.4
155.1
206.8
258.6
310.3
362.0
413.7
465.4
517.1
760.0
775.7
1 034
1 293
1 551
2 068
2 586
3 103
3 620
4 137
4 654
5 171
7 757
10 343
atm
Pascals
(N · m⫺2)
0.0007
0.0014
0.0020
0.0027
0.0034
0.0041
0.0048
0.0054
0.0061
0.0068
0.0136
0.0204
0.0272
0.0340
0.0408
0.0476
0.0544
0.0612
0.0689
0.1361
0.2041
0.2722
0.3402
0.4083
0.4763
0.5444
0.6124
0.6805
1.000
1.021
1.361
1.701
2.041
2.722
3.402
4.083
4.763
5.444
6.124
6.805
10.21
13.61
17.01
20.41
27.22
34.02
68.95
137.90
206.8
275.8
344.7
413.7
482.6
551.6
620.5
689.5
1 379.9
2 068.5
2 758
3 447
4 137
4 826
5 516
6 205
6 895
13 790
20 684
27 579
34 474
41 369
48 263
55 158
62 053
68 948
101 325
103 421
137 895
172 369
206 843
275 790
344 738
413 685
482 633
551 581
620 528
689 476
1 034 214
1 378 951
1 723 689
2 068 427
2 757 903
3 447 379
1 bar ⫽ 105 pascal.
short
standard
long
DEAN #37261
LEFT
(McGHP)
2.88
INTERACTIVE
top of rh
base of rh
SECTION 2
TABLE 2.17 Corrections to Be Added to Molar Values to Convert to Molal
cap height
base of text
Aqueous solution
Temperature,
⬚C
⌬G⬚
J · mol⫺1
⌬H⬚
J · mol⫺1
⌬S⬚
J · deg⫺1 · mol⫺1
⌬C ⬚p
J · deg⫺1 · mol⫺1
0
10
20
30
40
50
60
0.4
0.8
4.2
10.9
20.1
32.2
46.8
⫺42.7
58.1
148.1
230.5
313.4
397.9
482.4
⫺0.17
0.21
0.50
0.79
1.09
1.34
1.59
55.2
45.6
38.9
35.1
33.0
32.6
32.2
TABLE 2.18 Molar Equivalent of One Liter of Gas at Various Temperatures and Pressures
The values in this table, which give the number of moles in 1 liter of gas, are based on the properties of an
“ideal” gas and were calculated by use of the formula:
Moles/liter ⫽
P
273
1
⫻
⫻
760
T
22.40
where P is the pressure in millimeters of mercury and T is the temperature in kelvins (⫽t⬚C ⫹ 273).
To convert to moles per cubic foot multiply the values in the table by 28.316.
Temperature ⬚C
Pressure
mm of
mercury
10⬚
12⬚
14⬚
16⬚
18⬚
20⬚
655
660
665
670
675
0.03712
3731
3768
3796
3825
0.03686
3714
3742
3770
3798
0.03660
3688
3716
3744
3772
0.03634
3662
3690
3718
3745
0.03610
3637
3665
3692
3720
0.03585
3612
3640
3667
3695
680
685
690
695
700
0.03853
3881
3910
3938
3967
0.03826
3854
3882
3910
3939
0.03800
3827
3855
3883
3911
0.03773
3801
3829
3856
3884
0.03747
3775
3802
3830
3858
0.03694
3749
3776
3804
3831
702
704
706
708
710
0.03978
3989
4000
4012
4023
0.03950
3961
3972
3984
3995
0.03922
3934
3945
3956
3967
0.03895
3906
3917
3929
3940
0.03869
3880
3891
3902
3913
0.03842
3853
3864
3875
3886
712
714
716
718
720
0.04035
4046
4057
4068
4080
0.04006
4018
4029
4040
4051
0.03978
3989
4001
4012
4023
0.03951
3962
3973
3984
3995
0.03924
3935
3946
3957
3968
0.03897
3908
3919
3930
3941
short
standard
long
DEAN #37261
(McGHP)
RIGHT
INTERACTIVE
2.89
GENERAL INFORMATION, CONVERSION TABLES, AND MATHEMATICS
TABLE 2.18 Molar Equivalent of One Liter of Gas at Various Temperatures and Pressures (Continued)
top of rh
base of rh
cap height
base of text
Temperature ⬚C
Pressure
mm of
mercury
10⬚
12⬚
14⬚
16⬚
18⬚
20⬚
722
724
726
728
730
0.04091
4103
4114
4125
4136
0.04063
4074
4085
4096
4108
0.04034
4045
4057
4068
4079
0.04006
4017
4028
4040
4051
0.03979
3990
4001
4012
4023
0.03952
3963
3973
3984
3995
732
734
736
738
740
0.04148
4159
4171
4182
4193
0.04119
4130
4141
4153
4164
0.04090
4101
4112
4124
4135
0.04062
4073
4084
4095
4106
0.04034
4045
4056
4067
4078
0.04006
4017
4028
4039
4050
742
744
746
748
750
0.04204
4216
4227
4239
4250
0.04175
4186
4198
4209
4220
0.04146
4157
4168
4179
4191
0.04117
4128
4139
4151
4162
0.04089
4100
4111
4122
4133
0.04061
4072
4038
4094
4105
752
754
756
758
760
0.04261
4273
4284
4295
4307
0.04231
4243
4254
4265
4276
0.04202
4213
4224
4235
4247
0.04173
4184
4195
4206
4217
0.04144
4155
4166
4177
4188
0.04116
4127
4138
4149
4160
762
764
766
768
770
0.04318
4329
4341
4352
4363
0.04287
4299
4310
4321
4333
0.04258
4269
4280
4291
4302
0.04228
4239
4250
4262
4273
0.04199
4210
4221
4232
4243
0.04171
4181
4192
4203
4214
772
774
776
778
780
0.04375
4386
4397
4409
4420
0.04344
4355
4366
4378
4389
0.04314
4325
4336
4347
4358
0.04284
4295
4306
4317
4328
0.04254
4265
4276
4287
4298
0.04225
4236
4247
4258
4269
Temperature ⬚C
Pressure
mm of
mercury
22⬚
24⬚
26⬚
28⬚
30⬚
32⬚
655
660
665
670
675
0.03561
3588
3614
3642
3669
0.03537
3564
3591
3618
3645
0.03515
3541
3568
3595
3622
0.03490
3516
3543
3569
3596
0.03467
3493
3520
3546
3572
0.03444
3470
3496
3523
3549
680
685
690
695
700
0.03697
3724
3751
3778
3805
0.03672
3699
3726
3753
3780
0.03649
3676
3702
3729
3756
0.03623
3649
3676
3703
3729
0.03599
3625
3652
3678
3705
0.03575
3602
3628
3654
3680
short
standard
long
DEAN #37261
LEFT
(McGHP)
2.90
INTERACTIVE
top of rh
base of rh
SECTION 2
TABLE 2.18 Molar Equivalent of One Liter of Gas at Various Temperatures and Pressures (Continued)
cap height
base of text
Temperature ⬚C
Pressure
mm of
mercury
22⬚
24⬚
26⬚
28⬚
30⬚
32⬚
702
704
706
708
710
0.03816
3827
3838
3849
3860
0.03790
3801
3812
3823
3834
0.03767
3777
3788
3799
3810
0.03740
3750
3761
3772
3783
0.03715
3726
3736
3747
3758
0.03691
3701
3712
3722
3733
712
714
716
718
720
0.03870
3881
3892
3902
3914
0.03844
3855
3866
3877
3888
0.03820
3831
3842
3853
3863
0.03793
3804
3815
3825
3836
0.03768
3779
3789
3800
3811
0.03744
3754
3765
3775
3786
722
724
726
728
730
0.03925
3936
3947
3957
3968
0.03898
3909
3920
3931
3941
0.03874
3885
3896
3906
3917
0.03847
3857
3868
3878
3889
0.03821
3832
3842
3853
3863
0.03796
3807
3817
3828
3838
732
734
736
738
740
0.03979
3990
4001
4012
4023
0.03952
3963
3974
3985
3995
0.03928
3938
3949
3960
3971
0.03900
3910
3921
3932
3942
0.03874
3885
3895
3906
3916
0.03849
3859
3870
3880
3891
742
744
746
748
750
0.04033
4044
4055
4066
4077
0.04006
4017
4028
4039
4049
0.03981
3992
4003
4014
4024
0.03953
3964
3974
3985
3996
0.03927
3938
3948
3959
3969
0.03901
3912
3922
3933
3943
752
754
756
758
760
0.04088
4099
4110
4121
4131
0.04060
4071
4082
4093
4103
0.04035
4046
4056
4067
4078
0.04006
4017
4028
4038
4049
0.03980
3991
4001
4012
4022
0.03954
3964
3975
3985
3996
762
764
766
768
770
0.04142
4153
4164
4175
4186
4114
4125
4136
4147
4158
4089
4099
4110
4121
4132
4060
4070
4081
4092
4102
4033
4043
4054
4065
4075
4006
4017
4027
4038
4048
772
774
776
778
780
0.04197
4207
4218
4229
4240
0.04168
4179
4190
4201
4211
0.04142
4153
4164
4175
4185
0.04113
4124
4134
4145
4155
0.04086
4096
4107
4117
4128
0.04059
4070
4080
4091
4101
short
standard
long
DEAN #37261
(McGHP)
RIGHT
INTERACTIVE
2.91
GENERAL INFORMATION, CONVERSION TABLES, AND MATHEMATICS
TABLE 2.19 Factors for Reducing Gas Volumes to Normal (Standard) Temperature and Pressure (760
mmHg)
top of rh
base of rh
cap height
base of text
Examples: (a) 20 mL of dry gas at 22⬚C and 730 mm ⫽ 20 ⫻ 0.8888 ⫽ 17.78 mL at 0⬚C and 760 mm. (b) 20
mL of a gas over water at 22⬚ and 730 mm ⫽ 20 ⫻ (factor corrected for aqueous tension; i.e., 730 ⫺ 19.8 or
710.2 mm) ⫽ 20 mL of dry gas at 22⬚ and 710.2 mm ⫽ 20 ⫻ 0.86475 ⫽ 17.30 mL at 0⬚C and 760 mm. Mass
in milligrams of 1 mL of gas at S.T.P.: acetylene, 1.173; carbon dioxide, 1.9769; hydrogen, 0.0899; nitric oxide
(NO), 1.3402; nitrogen, 1.25057; oxygen, 1.42904.
Temperature ⬚C
Pressure
mm of
mercury
10⬚
11⬚
12⬚
13⬚
14⬚
15⬚
16⬚
17⬚
670
672
674
676
678
0.8504
0.8530
0.8555
0.8580
0.8606
0.8474
0.8500
0.8525
0.8550
0.8576
0.8445
0.8470
0.8495
0.8520
0.8545
0.8415
0.8440
0.8465
0.8490
0.8516
0.8386
0.8411
0.8436
0.8461
0.8486
0.8357
0.8382
0.8407
0.8431
0.8456
0.8328
0.8353
0.8377
0.8402
0.8427
0.8299
0.8324
0.8349
0.8373
0.8398
680
682
684
686
688
0.8631
0.8657
0.8682
0.8707
0.8733
0.8601
0.8626
0.8651
0.8677
0.8702
0.8571
0.8596
0.8621
0.8646
0.8672
0.8541
0.8566
0.8591
0.8616
0.8641
0.8511
0.8536
0.8561
0.8586
0.8611
0.8481
0.8506
0.8531
0.8556
0.8581
0.8452
0.8477
0.8502
0.8527
0.8551
0.8423
0.8448
0.8472
0.8497
0.8522
690
692
694
696
698
0.8758
0.8784
0.8809
0.8834
0.8860
0.8727
0.8753
0.8778
0.8803
0.8828
0.8697
0.8722
0.8747
0.8772
0.8798
0.8666
0.8691
0.8717
0.8742
0.8767
0.8636
0.8661
0.8686
0.8711
0.8736
0.8606
0.8631
0.8656
0.8681
0.8706
0.8576
0.8601
0.8626
0.8651
0.8676
0.8547
0.8572
0.8596
0.8621
0.8646
700
702
704
706
708
0.8885
0.8910
0.8936
0.8961
0.8987
0.8854
0.8879
0.8904
0.8930
0.8955
0.8823
0.8848
0.8873
0.8898
0.8924
0.8792
0.8817
0.8842
0.8867
0.8892
0.8761
0.8786
0.8811
0.8836
0.8861
0.8731
0.8756
0.8781
0.8806
0.8831
0.8700
0.8725
0.8750
0.8775
0.8800
0.8671
0.8695
0.8720
0.8745
0.8770
710
712
714
716
718
0.9012
0.9037
0.9063
0.9088
0.9114
0.8980
0.9006
0.9031
0.9056
0.9081
0.8949
0.8974
0.8999
0.9024
0.9050
0.8917
0.8943
0.8968
0.8993
0.9018
0.8886
0.8911
0.8936
0.8961
0.8987
0.8856
0.8880
0.8905
0.8930
0.8955
0.8825
0.8850
0.8875
0.8899
0.8924
0.8794
0.8819
0.8844
0.8869
0.8894
720
722
724
726
728
0.9139
0.9164
0.9190
0.9215
0.9241
0.9107
0.9132
0.9157
0.9183
0.9208
0.9075
0.9100
0.9125
0.9151
0.9176
0.9043
0.9068
0.9093
0.9118
0.9144
0.9012
0.9037
0.9062
0.9087
0.9112
0.8980
0.9005
0.9030
0.9055
0.9080
0.8949
0.8974
0.8999
0.9024
0.9049
0.8918
0.8943
0.8968
0.8993
0.9017
730
732
734
736
738
0.9266
0.9291
0.9317
0.9342
0.9368
0.9233
0.9259
0.9284
0.9309
0.9334
0.9201
0.9226
0.9251
0.9277
0.9302
0.9169
0.9194
0.9219
0.9244
0.9269
0.9137
0.9162
0.9187
0.9212
0.9237
0.9105
0.9130
0.9155
0.9180
0.9205
0.9073
0.9098
0.9123
0.9148
0.9173
0.9042
0.9067
0.9092
0.9117
0.9141
740
742
744
746
748
0.9393
0.9418
0.9444
0.9469
0.9494
0.9360
0.9385
0.9410
0.9436
0.9461
0.9327
0.9352
0.9377
0.9403
0.9428
0.9294
0.9319
0.9345
0.9370
0.9395
0.9262
0.9287
0.9312
0.9337
0.9362
0.9230
0.9255
0.9280
0.9305
0.9329
0.9198
0.9223
0.9248
0.9272
0.9297
0.9166
0.9191
0.9216
0.9240
0.9265
short
standard
long
DEAN #37261
LEFT
(McGHP)
2.92
INTERACTIVE
top of rh
base of rh
SECTION 2
cap height
base of text
TABLE 2.19 Factors for Reducing Gas Volumes to Normal (Standard) Temperature and
Pressure (Continued)
Temperature ⬚C
Pressure
mm of
mercury
10⬚
11⬚
12⬚
13⬚
14⬚
15⬚
16⬚
17⬚
750
752
754
756
758
0.9520
0.9545
0.9571
0.9596
0.9621
0.9486
0.9511
0.9537
0.9562
0.9587
0.9453
0.9478
0.9504
0.9529
0.9554
0.9420
0.9445
0.9470
0.9495
0.9520
0.9387
0.9412
0.9437
0.9462
0.9487
0.9354
0.9379
0.9404
0.9429
0.9454
0.9322
0.9347
0.9372
0.9397
0.9422
0.9290
0.9315
0.9339
0.9364
0.9389
760
762
764
766
768
0.9647
0.9672
0.9698
0.9723
0.9748
0.9613
0.9638
0.9663
0.9689
0.9714
0.9579
0.9604
0.9630
0.9655
0.9680
0.9546
0.9571
0.9596
0.9620
0.9646
0.9512
0.9537
0.9562
0.9587
0.9612
0.9479
0.9504
0.9529
0.9554
0.9579
0.9446
0.9471
0.9496
0.9521
0.9546
0.9414
0.9439
0.9463
0.9488
0.9513
770
772
774
776
778
0.9774
0.9799
0.9825
0.9850
0.9875
0.9739
0.9764
0.9790
0.9815
0.9840
0.9705
0.9730
0.9756
0.9781
0.9806
0.9671
0.9696
0.9721
0.9746
0.9772
0.9637
0.9662
0.9687
0.9712
0.9737
0.9604
0.9629
0.9654
0.9679
0.9704
0.9571
0.9596
0.9620
0.9645
0.9670
0.9538
0.9562
0.9587
0.9612
0.9637
780
782
784
786
788
0.9901
0.9926
0.9952
0.9977
1.0002
0.9866
0.9891
0.9916
0.9942
0.9967
0.9831
0.9856
0.9882
0.9907
0.9932
0.9797
0.9822
0.9847
0.9872
0.9897
0.9763
0.9788
0.9813
0.9838
0.9863
0.9729
0.9754
0.9778
0.9803
0.9828
0.9695
0.9720
0.9745
0.9770
0.9794
0.9662
0.9686
0.9711
0.9736
0.9761
Temperature ⬚C
Pressure
mm of
mercury
18⬚
19⬚
20⬚
21⬚
22⬚
23⬚
24⬚
25⬚
670
672
674
676
678
0.8270
0.8295
0.8320
0.8345
0.8369
0.8242
0.8267
0.8291
0.8316
0.8341
0.8214
0.8239
0.8263
0.8288
0.8312
0.8186
0.8211
0.8235
0.8259
0.8284
0.8158
0.8183
0.8207
0.8231
0.8256
0.8131
0.8155
0.8179
0.8204
0.8228
0.8103
0.8128
0.8152
0.8176
0.8200
0.8076
0.8100
0.8124
0.8149
0.8173
680
682
684
686
688
0.8394
0.8419
0.8443
0.8468
0.8493
0.8365
0.8390
0.8414
0.8439
0.8464
0.8337
0.8361
0.8386
0.8410
0.8435
0.8308
0.8333
0.8357
0.8382
0.8406
0.8280
0.8304
0.8329
0.8353
0.8378
0.8252
0.8276
0.8301
0.8325
0.8349
0.8224
0.8249
0.8273
0.8297
0.8321
0.8197
0.8221
0.8245
0.8269
0.8293
690
692
694
696
698
0.8517
0.8542
0.8567
0.8591
0.8616
0.8488
0.8513
0.8537
0.8562
0.8587
0.8459
0.8484
0.8508
0.8533
0.8557
0.8430
0.8455
0.8479
0.8504
0.8528
0.8402
0.8426
0.8451
0.8475
0.8499
0.8373
0.8398
0.8422
0.8446
0.8471
0.8345
0.8369
0.8394
0.8418
0.8442
0.8317
0.8341
0.8366
0.8390
0.8414
700
702
704
706
708
0.8641
0.8665
0.8690
0.8715
0.8740
0.8611
0.8636
0.8660
0.8685
0.8710
0.8582
0.8606
0.8631
0.8655
0.8680
0.8553
0.8577
0.8602
0.8626
0.8650
0.8524
0.8547
0.8572
0.8597
0.8621
0.8495
0.8519
0.8543
0.8568
0.8592
0.8466
0.8490
0.8515
0.8539
0.8563
0.8438
0.8462
0.8486
0.8510
0.8534
short
standard
long
DEAN #37261
(McGHP)
RIGHT
INTERACTIVE
2.93
GENERAL INFORMATION, CONVERSION TABLES, AND MATHEMATICS
top of rh
base of rh
cap height
base of text
TABLE 2.19 Factors for Reducing Gas Volumes to Normal (Standard) Temperature and
Pressure (Continued)
Temperature ⬚C
Pressure
mm of
mercury
18⬚
19⬚
20⬚
21⬚
22⬚
23⬚
24⬚
25⬚
710
712
714
716
718
0.8764
0.8789
0.8814
0.8838
0.8863
0.8734
0.8759
0.8783
0.8808
0.8833
0.8704
0.8729
0.8753
0.8778
0.8802
0.8675
0.8699
0.8724
0.8748
0.8773
0.8645
0.8670
0.8694
0.8718
0.8743
0.8616
0.8640
0.8665
0.8689
0.8713
0.8587
0.8611
0.8636
0.8660
0.8684
0.8558
0.8582
0.8607
0.8631
0.8655
720
722
724
726
728
0.8888
0.8912
0.8937
0.8962
0.8986
0.8857
0.8882
0.8906
0.8931
0.8956
0.8827
0.8852
0.8876
0.8901
0.8925
0.8797
0.8821
0.8846
0.8870
0.8895
0.8767
0.8792
0.8816
0.8840
0.8865
0.8738
0.8762
0.8786
0.8810
0.8835
0.8708
0.8732
0.8757
0.8781
0.8805
0.8679
0.8703
0.8727
0.8751
0.8775
730
732
734
736
738
0.9011
0.9036
0.9060
0.9085
0.9110
0.8980
0.9005
0.9029
0.9054
0.9079
0.8950
0.8974
0.8999
0.9023
0.9048
0.8919
0.8944
0.8968
0.8992
0.9017
0.8889
0.8913
0.8938
0.8962
0.8986
0.8859
0.8883
0.8907
0.8932
0.8956
0.8829
0.8853
0.8877
0.8902
0.8926
0.8799
0.8824
0.8848
0.8872
0.8896
740
742
744
746
748
0.9135
0.9159
0.9184
0.9209
0.9233
0.9103
0.9128
0.9153
0.9177
0.9202
0.9072
0.9097
0.9121
0.9146
0.9170
0.9041
0.9066
0.9090
0.9115
0.9139
0.9011
0.9035
0.9059
0.9084
0.9108
0.8980
0.9005
0.9029
0.9053
0.9077
0.8950
0.8974
0.8998
0.9023
0.9047
0.8920
0.8944
0.8968
0.8992
0.9016
750
752
754
756
758
0.9258
0.9283
0.9307
0.9332
0.9357
0.9226
0.9251
0.9276
0.9300
0.9325
0.9195
0.9219
0.9244
0.9268
0.9293
0.9164
0.9188
0.9212
0.9237
0.9261
0.9132
0.9157
0.9181
0.9206
0.9230
0.9102
0.9126
0.9150
0.9174
0.9199
0.9071
0.9095
0.9119
0.9144
0.9168
0.9041
0.9065
0.9089
0.9113
0.9137
760
762
764
766
768
0.9381
0.9406
0.9431
0.9456
0.9480
0.9349
0.9374
0.9399
0.9423
0.9448
0.9317
0.9342
0.9366
0.9391
0.9415
0.9286
0.9310
0.9335
0.9359
0.9383
0.9254
0.9279
0.9303
0.9327
0.9352
0.9223
0.9247
0.9272
0.9296
0.9320
0.9192
0.9216
0.9240
0.9265
0.9289
0.9161
0.9185
0.9209
0.9233
0.9258
770
772
774
776
778
0.9505
0.9530
0.9554
0.9579
0.9604
0.9472
0.9497
0.9522
0.9546
0.9571
0.9440
0.9464
0.9489
0.9514
0.9538
0.9408
0.9432
0.9457
0.9481
0.9506
0.9376
0.9400
0.9425
0.9449
0.9473
0.9344
0.9369
0.9393
0.9417
0.9441
0.9313
0.9337
0.9361
0.9385
0.9410
0.9282
0.9306
0.9330
0.9354
0.9378
780
782
784
786
788
0.9628
0.9653
0.9678
0.9702
0.9727
0.9595
0.9620
0.9645
0.9669
0.9694
0.9563
0.9587
0.9612
0.9636
0.9661
0.9530
0.9555
0.9579
0.9603
0.9628
0.9498
0.9522
0.9546
0.9571
0.9595
0.9466
0.9490
0.9514
0.9538
0.9563
0.9434
0.9458
0.9482
0.9506
0.9531
0.9402
0.9426
0.9450
0.9474
0.9499
short
standard
long
DEAN #37261
LEFT
(McGHP)
2.94
INTERACTIVE
top of rh
base of rh
SECTION 2
cap height
base of text
TABLE 2.19 Factors for Reducing Gas Volumes to Normal (Standard) Temperature and
Pressure (Continued)
Temperature ⬚C
Pressure
mm of
mercury
26⬚
27⬚
28⬚
29⬚
30⬚
31⬚
32⬚
33⬚
670
672
674
676
678
0.8049
0.8073
0.8097
0.8121
0.8145
0.8022
0.8046
0.8070
0.8094
0.8118
0.7996
0.8020
0.8043
0.8067
0.8091
0.7969
0.7993
0.8017
0.8041
0.8064
0.7943
0.7967
0.7990
0.8014
0.8038
0.7917
0.7940
0.7964
0.7988
0.8011
0.7891
0.7914
0.7938
0.7962
0.7985
0.7865
0.7889
0.7912
0.7936
0.7959
680
682
684
686
688
0.8169
0.8193
0.8217
0.8241
0.8265
0.8142
0.8166
0.8190
0.8214
0.8238
0.8115
0.8139
0.8163
0.8187
0.8211
0.8088
0.8112
0.8136
0.8160
0.8183
0.8061
0.8085
0.8109
0.8133
0.8156
0.8035
0.8059
0.8082
0.8106
0.8129
0.8009
0.8032
0.8056
0.8079
0.8103
0.7982
0.8006
0.8029
0.8053
0.8076
690
692
694
696
698
0.8289
0.8313
0.8338
0.8362
0.8386
0.8262
0.8286
0.8310
0.8334
0.8358
0.8234
0.8258
0.8282
0.8306
0.8330
0.8207
0.8231
0.8255
0.8278
0.8302
0.8180
0.8204
0.8227
0.8251
0.8275
0.8153
0.8177
0.8200
0.8224
0.8248
0.8126
0.8150
0.8174
0.8197
0.8221
0.8100
0.8123
0.8147
0.8170
0.8194
700
702
704
706
708
0.8410
0.8434
0.8458
0.8482
0.8506
0.8382
0.8406
0.8429
0.8453
0.8477
0.8354
0.8378
0.8401
0.8425
0.8449
0.8326
0.8350
0.8374
0.8397
0.8421
0.8299
0.8322
0.8346
0.8370
0.8393
0.8271
0.8295
0.8319
0.8342
0.8366
0.8244
0.8268
0.8291
0.8315
0.8338
0.8217
0.8241
0.8264
0.8288
0.8311
710
712
714
716
718
0.8530
0.8554
0.8578
0.8602
0.8626
0.8501
0.8525
0.8549
0.8573
0.8597
0.8473
0.8497
0.8521
0.8545
0.8569
0.8445
0.8469
0.8493
0.8516
0.8540
0.8417
0.8441
0.8465
0.8488
0.8512
0.8389
0.8413
0.8437
0.8460
0.8484
0.8362
0.8386
0.8409
0.8433
0.8456
0.8335
0.8358
0.8382
0.8405
0.8429
720
722
724
726
728
0.8650
0.8674
0.8698
0.8722
0.8746
0.8621
0.8645
0.8669
0.8693
0.8717
0.8592
0.8616
0.8640
0.8664
0.8688
0.8564
0.8588
0.8612
0.8635
0.8659
0.8536
0.8559
0.8583
0.8607
0.8631
0.8508
0.8531
0.8555
0.8579
0.8602
0.8480
0.8503
0.8527
0.8550
0.8574
0.8452
0.8475
0.8499
0.8522
0.8546
730
732
734
736
738
0.8770
0.8794
0.8818
0.8842
0.8866
0.8741
0.8765
0.8789
0.8813
0.8837
0.8712
0.8736
0.8759
0.8783
0.8807
0.8683
0.8707
0.8730
0.8754
0.8778
0.8654
0.8678
0.8702
0.8725
0.8749
0.8626
0.8649
0.8673
0.8697
0.8720
0.8598
0.8621
0.8645
0.8668
0.8692
0.8569
0.8593
0.8616
0.8640
0.8663
740
742
744
746
748
0.8890
0.8914
0.8938
0.8962
0.8986
0.8861
0.8884
0.8908
0.8932
0.8956
0.8831
0.8855
0.8879
0.8903
0.8927
0.8802
0.8826
0.8849
0.8873
0.8897
0.8773
0.8796
0.8820
0.8844
0.8868
0.8744
0.8768
0.8791
0.8815
0.8838
0.8715
0.8739
0.8762
0.8786
0.8809
0.8687
0.8710
0.8734
0.8757
0.8781
750
752
754
756
758
0.9010
0.9034
0.9058
0.9082
0.9106
0.8980
0.9004
0.9028
0.9052
0.9076
0.8950
0.8974
0.8998
0.9022
0.9046
0.8921
0.8945
0.8968
0.8992
0.9016
0.8891
0.8915
0.8939
0.8962
0.8986
0.8862
0.8886
0.8909
0.8933
0.8957
0.8833
0.8857
0.8880
0.8904
0.8927
0.8804
0.8828
0.8851
0.8875
0.8898
short
standard
long
DEAN #37261
(McGHP)
RIGHT
INTERACTIVE
2.95
GENERAL INFORMATION, CONVERSION TABLES, AND MATHEMATICS
top of rh
base of rh
cap height
base of text
TABLE 2.19 Factors for Reducing Gas Volumes to Normal (Standard) Temperature and
Pressure (Continued)
Temperature ⬚C
Pressure
mm of
mercury
26⬚
27⬚
28⬚
29⬚
30⬚
31⬚
32⬚
33⬚
760
762
764
766
768
0.9130
0.9154
0.9178
0.9202
0.9227
0.9100
0.9124
0.9148
0.9172
0.9196
0.9070
0.9094
0.9118
0.9141
0.9165
0.9040
0.9064
0.9087
0.9111
0.9135
0.9010
0.9034
0.9057
0.9081
0.9105
0.8980
0.9004
0.9028
0.9051
0.9075
0.8951
0.8974
0.8998
0.9021
0.9045
0.8922
0.8945
0.8969
0.8992
0.9015
770
772
774
776
778
0.9251
0.9275
0.9299
0.9323
0.9347
0.9220
0.9244
0.9268
0.9292
0.9316
0.9189
0.9213
0.9237
0.9261
0.9285
0.9159
0.9182
0.9206
0.9230
0.9254
0.9128
0.9152
0.9176
0.9200
0.9223
0.9098
0.9122
0.9146
0.9169
0.9193
0.9069
0.9092
0.9116
0.9139
0.9163
0.9039
0.9062
0.9086
0.9109
0.9133
780
782
784
786
788
0.9371
0.9395
0.9419
0.9443
0.9467
0.9340
0.9363
0.9387
0.9411
0.9435
0.9308
0.9332
0.9356
0.9380
0.9404
0.9278
0.9301
0.9325
0.9349
0.9373
0.9247
0.9271
0.9294
0.9318
0.9342
0.9217
0.9240
0.9264
0.9287
0.9311
0.9186
0.9210
0.9233
0.9257
0.9281
0.9156
0.9180
0.9203
0.9227
0.9250
Temperature ⬚C
Pressure
mm of
mercury
34⬚
35⬚
670
672
674
676
678
0.7839
0.7863
0.7886
0.7910
0.7933
680
682
684
686
688
Temperature ⬚C
36⬚
Pressure
mm of
mercury
34⬚
35⬚
36⬚
0.7814
0.7837
0.7861
0.7884
0.7907
0.7789
0.7812
0.7835
0.7858
0.7882
730
732
734
736
738
0.8541
0.8565
0.8588
0.8612
0.8635
0.8514
0.8537
0.8560
0.8584
0.8607
0.8486
0.8509
0.8533
0.8556
0.8579
0.7956
0.7980
0.8003
0.8027
0.8050
0.7931
0.7954
0.7977
0.8001
0.8024
0.7905
0.7928
0.7951
0.7975
0.7998
740
742
744
746
748
0.8658
0.8682
0.8705
0.8729
0.8752
0.8630
0.8654
0.8677
0.8700
0.8724
0.8602
0.8626
0.8649
0.8672
0.8695
690
692
694
696
698
700
702
704
706
708
0.8073
0.8097
0.8120
0.8144
0.8167
0.8190
0.8214
0.8237
0.8261
0.8284
0.8047
0.8071
0.8094
0.8117
0.8141
0.8164
0.8187
0.8211
0.8234
0.8257
0.8021
0.8044
0.8068
0.8091
0.8114
0.8137
0.8161
0.8184
0.8207
0.8230
750
752
754
756
758
760
762
764
766
768
0.8775
0.8799
0.8822
0.8846
0.8869
0.8892
0.8916
0.8939
0.8963
0.8986
0.8747
0.8770
0.8794
0.8817
0.8840
0.8864
0.8887
0.8910
0.8934
0.8957
0.8719
0.8742
0.8765
0.8788
0.8812
0.8835
0.8858
0.8881
0.8905
0.8928
710
712
714
716
718
0.8307
0.8331
0.8354
0.8378
0.8401
0.8281
0.8304
0.8327
0.8350
0.8374
0.8254
0.8277
0.8300
0.8323
0.8347
770
772
774
776
778
0.9009
0.9033
0.9056
0.9080
0.9103
0.8980
0.9004
0.9027
0.9050
0.9074
0.8951
0.8974
0.8998
0.9021
0.9044
720
722
724
726
728
0.8424
0.8448
0.8471
0.8495
0.8518
0.8397
0.8420
0.8444
0.8467
0.8490
0.8370
0.8393
0.8416
0.8440
0.8463
780
782
784
786
788
0.9127
0.9150
0.9173
0.9197
0.9220
0.9097
0.9120
0.9144
0.9167
0.9190
0.9067
0.9091
0.9114
0.9137
0.9160
short
standard
long
DEAN #37261
LEFT
(McGHP)
2.96
INTERACTIVE
top of rh
base of rh
SECTION 2
cap height
base of text
TABLE 2.20 Values of Absorbance for Percent Absorption
To convert percent absorption (% A) to absorbance, find the present absorption to the nearest whole digit in
the left-hand column; read across to the column located under the tenth of a percent desired, and read the value
of absorbance. The value of absorbance corresponding to 26.8% absorption is thus 0.1355.
%A
.0
.1
.2
.3
.4
.5
.6
.7
.8
.9
0.0
1.0
2.0
3.0
4.0
5.0
.0000
.0044
.0088
.0132
.0177
.0223
.0004
.0048
.0092
.0137
.0182
.0227
.0009
.0052
.0097
.0141
.0186
.0232
.0013
.0057
.0101
.0146
.0191
.0236
.0017
.0061
.0106
.0150
.0195
.0241
.0022
.0066
.0110
.0155
.0200
.0246
.0026
.0070
.0114
.0159
.0205
.0250
.0031
.0074
.0119
.0164
.0209
.0255
.0035
.0079
.0123
.0168
.0214
.0259
.0039
.0083
.0128
.0173
.0218
.0264
6.0
7.0
8.0
9.0
10.0
11.0
.0269
.0315
.0362
.0410
.0458
.0506
.0273
.0320
.0367
.0414
.0462
.0511
.0278
.0325
.0372
.0419
.0467
.0516
.0283
.0329
.0376
.0424
.0472
.0521
.0287
.0334
.0381
.0429
.0477
.0526
.0292
.0339
.0386
.0434
.0482
.0531
.0297
.0343
.0391
.0438
.0487
.0535
.0301
.0348
.0395
.0443
.0491
.0540
.0306
.0353
.0400
.0448
.0496
.0545
.0311
.0357
.0405
.0453
.0501
.0550
12.0
13.0
14.0
15.0
16.0
17.0
.0555
.0605
.0655
.0706
.0757
.0809
.0560
.0610
.0660
.0711
.0762
.0814
.0565
.0615
.0665
.0716
.0768
.0820
.0570
.0620
.0670
.0721
.0773
.0825
.0575
.0625
.0675
.0726
.0778
.0830
.0580
.0630
.0680
.0731
.0783
.0835
.0585
.0635
.0685
.0737
.0788
.0841
.0590
.0640
.0691
.0742
.0794
.0846
.0595
.0645
.0696
.0747
.0799
.0851
.0600
.0650
.0701
.0752
.0804
.0857
18.0
19.0
20.0
21.0
22.0
23.0
.0862
.0915
.0969
.1024
.1079
.1135
.0867
.0921
.0975
.1029
.1085
.1141
.0872
.0926
.0980
.1035
.1090
.1146
.0878
.0931
.0985
.1040
.1096
.1152
.0883
.0937
.0991
.1046
.1101
.1158
.0888
.0942
.0996
.1051
.1107
.1163
.0894
.0947
.1002
.1057
.1113
.1169
.0899
.0953
.1007
.1062
.1118
.1175
.0904
.0958
.1013
.1068
.1124
.1180
.0910
.0964
.1018
.1073
.1129
.1186
24.0
25.0
26.0
27.0
28.0
29.0
.1192
.1249
.1308
.1367
.1427
.1487
.1198
.1255
.1314
.1373
.1433
.1494
.1203
.1261
.1319
.1379
.1439
.1500
.1209
.1267
.1325
.1385
.1445
.1506
.1215
.1273
.1331
.1391
.1451
.1512
.1221
.1278
.1337
.1397
.1457
.1518
.1226
.1284
.1343
.1403
.1463
.1524
.1232
.1290
.1349
.1409
.1469
.1530
.1238
.1296
.1355
.1415
.1475
.1537
.1244
.1302
.1361
.1421
.1481
.1543
30.0
31.0
32.0
33.0
34.0
35.0
.1549
.1612
.1675
.1739
.1805
.1871
.1555
.1618
.1681
.1746
.1811
.1878
.1561
.1624
.1688
.1752
.1818
.1884
.1568
.1630
.1694
.1759
.1824
.1891
.1574
.1637
.1701
.1765
.1831
.1898
.1580
.1643
.1707
.1772
.1838
.1904
.1586
.1649
.1713
.1778
.1844
.1911
.1593
.1656
.1720
.1785
.1851
.1918
.1599
.1662
.1726
.1791
.1858
.1925
.1605
.1669
.1733
.1798
.1864
.1931
36.0
37.0
38.0
39.0
40.0
41.0
.1938
.2007
.2076
.2147
.2218
.2291
.1945
.2013
.2083
.2154
.2226
.2299
.1952
.2020
.2090
.2161
.2233
.2306
.1959
.2027
.2097
.2168
.2240
.2314
.1965
.2034
.2104
.2175
.2248
.2321
.1972
.2041
.2111
.2182
.2255
.2328
.1979
.2048
.2118
.2190
.2262
.2336
.1986
.2055
.2125
.2197
.2269
.2343
.1993
.2062
.2132
.2204
.2277
.2351
.2000
.2069
.2140
.2211
.2284
.2358
short
standard
long
DEAN #37261
(McGHP)
RIGHT
INTERACTIVE
2.97
GENERAL INFORMATION, CONVERSION TABLES, AND MATHEMATICS
top of rh
base of rh
cap height
base of text
TABLE 2.20 Values of Absorbance for Percent Absorption (Continued)
%A
.0
.1
.2
.3
.4
.5
.6
.7
.8
.9
42.0
43.0
44.0
45.0
46.0
47.0
.2366
.2441
.2518
.2596
.2676
.2757
.2373
.2449
.2526
.2604
.2684
.2765
.2381
.2457
.2534
.2612
.2692
.2774
.2388
.2464
.2541
.2620
.2700
.2782
.2396
.2472
.2549
.2628
.2708
.2790
.2403
.2480
.2557
.2636
.2716
.2798
.2411
.2487
.2565
.2644
.2725
.2807
.2418
.2495
.2573
.2652
.2733
.2815
.2426
.2503
.2581
.2660
.2741
.2823
.2434
.2510
.2588
.2668
.2749
.2832
48.0
49.0
50.0
51.0
52.0
53.0
.2840
.2924
.3010
.3098
.3188
.3279
.2848
.2933
.3019
.3107
.3197
.3288
.2857
.2941
.3028
.3116
.3206
.3298
.2865
.2950
.3036
.3125
.3215
.3307
.2874
.2958
.3045
.3134
.3224
.3316
.2882
.2967
.3054
.3143
.3233
.3325
.2890
.2976
.3063
.3152
.3242
.3335
.2899
.2984
.3072
.3161
.3251
.3344
.2907
.2993
.3080
.3170
.3261
.3354
.2916
.3002
.3089
.3179
.3270
.3363
54.0
55.0
56.0
57.0
58.0
59.0
.3372
.3468
.3565
.3665
.3768
.3872
.3382
.3478
.3575
.3675
.3778
.3883
.3391
.3487
.3585
.3686
.3788
.3893
.3401
.3497
.3595
.3696
.3799
.3904
.3410
.3507
.3605
.3706
.3809
.3915
.3420
.3516
.3615
.3716
.3820
.3925
.3429
.3526
.3625
.3726
.3830
.3936
.3439
.3536
.3635
.3737
.3840
.3947
.3449
.3546
.3645
.3747
.3851
.3958
.3458
.3556
.3655
.3757
.3862
.3969
60.0
61.0
62.0
63.0
64.0
65.0
.3979
.4089
.4202
.4318
.4437
.4559
.3990
.4101
.4214
.4330
.4449
.4572
.4001
.4112
.4225
.4342
.4461
.4584
.4012
.4123
.4237
.4353
.4473
.4597
.4023
.4134
.4248
.4365
.4485
.4609
.4034
.4145
.4260
.4377
.4498
.4622
.4045
.4157
.4271
.4389
.4510
.4634
.4056
.4168
.4283
.4401
.4522
.4647
.4067
.4179
.4295
.4413
.4535
.4660
.4078
.4191
.4306
.4425
.4547
.4672
66.0
67.0
68.0
69.0
70.0
71.0
.4685
.4815
.4948
.5086
.5229
.5376
.4698
.4828
.4962
.5100
.5243
.5391
.4711
.4841
.4976
.5114
.5258
.5406
.4724
.4855
.4989
.5129
.5272
.5421
.4737
.4868
.5003
.5143
.5287
.5436
.4750
.4881
.5017
.5157
.5302
.5452
.4763
.4895
.5031
.5171
.5317
.5467
.4776
.4908
.5045
.5186
.5331
.5482
.4789
.4921
.5058
.5200
.5346
.5498
.4802
.4935
.5072
.5214
.5361
.5513
72.0
73.0
74.0
75.0
76.0
77.0
.5528
.5686
.5850
.6021
.6198
.6383
.5544
.5702
.5867
.6038
.6216
.6402
.5560
.5719
.5884
.6055
.6234
.6421
.5575
.5735
.5901
.6073
.6253
.6440
.5591
.5751
.5918
.6091
.6271
.6459
.5607
.5768
.5935
.6108
.6289
.6478
.5622
.5784
.5952
.6126
.6308
.6498
.5638
.5800
.5969
.6144
.6326
.6517
.5654
.5817
.5986
.6162
.6345
.6536
.5670
.5834
.6003
.6180
.6364
.6556
78.0
79.0
80.0
81.0
82.0
83.0
.6576
.6778
.6990
.7212
.7447
.7696
.6596
.6799
.7011
.7235
.7471
.7721
.6615
.6819
.7033
.7258
.7496
.7747
.6635
.6840
.7055
.7282
.7520
.7773
.6655
.6861
.7077
.7305
.7545
.7799
.6676
.6882
.7100
.7328
.7570
.7825
.6696
.6904
.7122
.7352
.7595
.7852
.6716
.6925
.7144
.7375
.7620
.7878
.6737
.6946
.7167
.7399
.7645
.7905
.6757
.6968
.7190
.7423
.7670
.7932
84.0
85.0
86.0
87.0
88.0
89.0
.7959
.8239
.8539
.8861
.9208
.9586
.7986
.8268
.8570
.8894
.9245
.9626
.8013
.8297
.8601
.8928
.9281
.9666
.8041
.8327
.8633
.8962
.9318
.9706
.8069
.8356
.8665
.8996
.9355
.9747
.8097
.8386
.8697
.9031
.9393
.9788
.8125
.8416
.8729
.9066
.9431
.9830
.8153
.8447
.8761
.9101
.9469
.9872
.8182
.8477
.8794
.9136
.9508
.9914
.8210
.8508
.8827
.9172
.9547
.9957
short
standard
long
DEAN #37261
LEFT
(McGHP)
2.98
INTERACTIVE
top of rh
base of rh
SECTION 2
cap height
base of text
TABLE 2.21 Transmittance-Absorbance Conversion Table
From Meites, Handbook of Analytical Chemistry, 1963, McGraw-Hill Book Company; by permission.
This table gives absorbance values to four significant figures corresponding to % transmittance values which
are given to three significant figures. The values of % transmittance are given in the left-hand column and in the
top row. For example, 8.4% transmittance corresponds to an absorbance of 1.076.
Interpolation is facilitated and accuracy is maximized if the % transmittance is between 1 and 10, by multiplying its value by 10, finding the absorbance corresponding to the result, and adding 1. For example, to find
the absorbance corresponding to 8.45% transmittance, note that 84.5% transmittance corresponds to an absorbance of 0.0731, so that 8.45% transmittance corresponds to an absorbance of 1.0731. For % transmittance values
between 0.1 and 1, multiply by 100, find the absorbance corresponding to the result, and add 2.
Conversely, to find the % transmittance corresponding to an absorbance between 1 and 2, subtract 1 from
the absorbance, find the % transmittance corresponding to the result, and divide by 10. For example, an absorbance of 1.219 can best be converted to % transmittance by noting that an absorbance of 0.219 would correspond
to 60.4% transmittance; dividing this by 10 gives the desired value, 6.04% transmittance. For absorbance values
between 2 and 3, subtract 2 from the absorbance, find the % transmittance corresponding to the result, and divide
by 100.
%
Transmittance
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0
1
2
3
4
5
......
2.000
1.699
1.523
1.398
1.301
3.000
1.959
1.678
1.509
1.387
1.292
2.699
1.921
1.658
1.495
1.377
1.284
2.523
1.886
1.638
1.481
1.367
1.276
2.398
1.854
1.620
1.469
1.357
1.268
2.301
1.824
1.602
1.456
1.347
1.260
2.222
1.796
1.585
1.444
1.337
1.252
2.155
1.770
1.569
1.432
1.328
1.244
2.097
1.745
1.553
1.420
1.319
1.237
2.046
1.721
1.538
1.409
1.310
1.229
6
7
8
9
10
1.222
1.155
1.097
1.046
1.000
1.215
1.149
1.092
1.041
0.9957
1.208
1.143
1.086
1.036
0.9914
1.201
1.137
1.081
1.032
0.9872
1.194
1.131
1.076
1.027
0.9830
1.187
1.125
1.071
1.022
0.9788
1.180
1.119
1.066
1.018
0.9747
1.174
1.114
1.060
1.013
0.9706
1.167
1.108
1.056
1.009
0.9666
1.161
1.102
1.051
1.004
0.9626
11
12
13
14
15
0.9586
0.9208
0.8861
0.8539
0.8239
0.9547
0.9172
0.8827
0.8508
0.8210
0.9508
0.9136
0.8794
0.8477
0.8182
0.9469
0.9101
0.8761
0.8447
0.8153
0.9431
0.9066
0.8729
0.8416
0.8125
0.9393
0.9031
0.8697
0.8386
0.8097
0.9355
0.8996
0.8665
0.8356
0.8069
0.9318
0.8962
0.8633
0.8327
0.8041
0.9281
0.8928
0.8601
0.8297
0.8013
0.9245
0.8894
0.8570
0.8268
0.7986
16
17
18
19
20
0.7959
0.7696
0.7447
0.7212
0.6990
0.7932
0.7670
0.7423
0.7190
0.6968
0.7905
0.7645
0.7399
0.7167
0.6946
0.7878
0.7620
0.7375
0.7144
0.6925
0.7852
0.7595
0.7352
0.7122
0.6904
0.7825
0.7570
0.7328
0.7100
0.6882
0.7799
0.7545
0.7305
0.7077
0.6861
0.7773
0.7520
0.7282
0.7055
0.6840
0.7747
0.7496
0.7258
0.7033
0.6819
0.7721
0.7471
0.7235
0.7011
0.6799
21
22
23
24
25
0.6778
0.6576
0.6383
0.6198
0.6021
0.6757
0.6556
0.6364
0.6180
0.6003
0.6737
0.6536
0.6345
0.6162
0.5986
0.6716
0.6517
0.6326
0.6144
0.5969
0.6696
0.6498
0.6308
0.6126
0.5952
0.6676
0.6478
0.6289
0.6108
0.5935
0.6655
0.6459
0.6271
0.6091
0.5918
0.6635
0.6440
0.6253
0.6073
0.5901
0.6615
0.6421
0.6234
0.6055
0.5884
0.6596
0.6402
0.6216
0.6038
0.5867
26
27
28
29
30
0.5850
0.5686
0.5528
0.5376
0.5229
0.5834
0.5670
0.5513
0.5361
0.5214
0.5817
0.5654
0.5498
0.5346
0.5200
0.5800
0.5638
0.5482
0.5331
0.5186
0.5784
0.5622
0.5467
0.5317
0.5171
0.5766
0.5607
0.5452
0.5302
0.5157
0.5751
0.5591
0.5436
0.5287
0.5143
0.5735
0.5575
0.5421
0.5272
0.5129
0.5719
0.5560
0.5406
0.5258
0.5114
0.5702
0.5544
0.5391
0.5243
0.5100
short
standard
long
DEAN #37261
(McGHP)
RIGHT
INTERACTIVE
2.99
GENERAL INFORMATION, CONVERSION TABLES, AND MATHEMATICS
top of rh
base of rh
cap height
base of text
TABLE 2.21 Transmittance-Absorbance Conversion Table (Continued)
%
Transmittance
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
31
32
33
34
35
0.5086
0.4949
0.4815
0.4685
0.4559
0.5072
0.4935
0.4802
0.4672
0.4547
0.5058
0.4921
0.4789
0.4660
0.4535
0.5045
0.4908
0.4776
0.4647
0.4522
0.5031
0.4895
0.4763
0.4634
0.4510
0.5017
0.4881
0.4750
0.4622
0.4498
0.5003
0.4868
0.4737
0.4609
0.4486
0.4989
0.4855
0.4724
0.4597
0.4473
0.4976
0.4841
0.4711
0.4584
0.4461
0.4962
0.4828
0.4698
0.4572
0.4449
36
37
38
39
40
0.4437
0.4318
0.4202
0.4089
0.3979
0.4425
0.4306
0.4191
0.4078
0.3969
0.4413
0.4295
0.4179
0.4067
0.3958
0.4401
0.4283
0.4168
0.4056
0.3947
0.4389
0.4271
0.4157
0.4045
0.3936
0.4377
0.4260
0.4145
0.4034
0.3925
0.4365
0.4248
0.4134
0.4023
0.3915
0.4353
0.4237
0.4123
0.4012
0.3904
0.4342
0.4225
0.4112
0.4001
0.3893
0.4330
0.4214
0.4101
0.3989
0.3883
41
42
43
44
45
0.3872
0.3768
0.3665
0.3565
0.3468
0.3862
0.3757
0.3655
0.3556
0.3458
0.3851
0.3747
0.3645
0.3546
0.3449
0.3840
0.3737
0.3635
0.3536
0.3439
0.3830
0.3726
0.3625
0.3526
0.3429
0.3820
0.3716
0.3615
0.3516
0.3420
0.3809
0.3706
0.3605
0.3507
0.3410
0.3799
0.3696
0.3595
0.3497
0.3401
0.3788
0.3686
0.3585
0.3487
0.3391
0.3778
0.3675
0.3575
0.3478
0.3382
46
47
48
49
50
0.3372
0.3279
0.3188
0.3098
0.3010
0.3363
0.3270
0.3179
0.3089
0.3002
0.3354
0.3261
0.3170
0.3080
0.2993
0.3344
0.3251
0.3161
0.3072
0.2984
0.3335
0.3242
0.3152
0.3063
0.2976
0.3325
0.3233
0.3143
0.3054
0.2967
0.3316
0.3224
0.3134
0.3045
0.2958
0.3307
0.3215
0.3125
0.3036
0.2950
0.3298
0.3206
0.3116
0.3028
0.2941
0.3288
0.3197
0.3107
0.3019
0.2933
51
52
53
54
55
0.2924
0.2840
0.2757
0.2676
0.2596
0.2916
0.2832
0.2749
0.2668
0.2588
0.2907
0.2823
0.2741
0.2660
0.2581
0.2899
0.2815
0.2733
0.2652
0.2573
0.2890
0.2807
0.2725
0.2644
0.2565
0.2882
0.2798
0.2716
0.2636
0.2557
0.2874
0.2790
0.2708
0.2628
0.2549
0.2865
0.2782
0.2700
0.2620
0.2541
0.2857
0.2774
0.2692
0.2612
0.2534
0.2848
0.2765
0.2684
0.2604
0.2526
56
57
58
59
60
0.2518
0.2441
0.2366
0.2291
0.2218
0.2510
0.2434
0.2358
0.2284
0.2211
0.2503
0.2426
0.2351
0.2277
0.2204
0.2495
0.2418
0.2343
0.2269
0.2197
0.2487
0.2411
0.2336
0.2262
0.2190
0.2480
0.2403
0.2328
0.2255
0.2182
0.2472
0.2396
0.2321
0.2248
0.2175
0.2464
0.2388
0.2314
0.2240
0.2168
0.2457
0.2381
0.2306
0.2233
0.2161
0.2449
0.2373
0.2299
0.2226
0.2154
61
62
63
64
65
0.2147
0.2076
0.2007
0.1938
0.1871
0.2140
0.2069
0.2000
0.1931
0.1864
0.2132
0.2062
0.1993
0.1925
0.1858
0.2125
0.2055
0.1986
0.1918
0.1851
0.2118
0.2048
0.1979
0.1911
0.1844
0.2111
0.2041
0.1972
0.1904
0.1838
0.2104
0.2034
0.1965
0.1898
0.1831
0.2097
0.2027
0.1959
0.1891
0.1824
0.2090
0.2020
0.1952
0.1884
0.1818
0.2083
0.2013
0.1945
0.1878
0.1811
66
67
68
69
70
0.1805
0.1739
0.1675
0.1612
0.1549
0.1798
0.1733
0.1669
0.1605
0.1543
0.1791
0.1726
0.1662
0.1599
0.1537
0.1785
0.1720
0.1656
0.1593
0.1530
0.1778
0.1713
0.1649
0.1586
0.1524
0.1772
0.1707
0.1643
0.1580
0.1518
0.1765
0.1701
0.1637
0.1574
0.1512
0.1759
0.1694
0.1630
0.1568
0.1506
0.1752
0.1688
0.1624
0.1561
0.1500
0.1746
0.1681
0.1618
0.1555
0.1494
71
72
73
0.1487
0.1427
0.1367
0.1481
0.1421
0.1361
0.1475
0.1415
0.1355
0.1469
0.1409
0.1349
0.1463
0.1403
0.1343
0.1457
0.1397
0.1337
0.1451
0.1391
0.1331
0.1445
0.1385
0.1325
0.1439
0.1379
0.1319
0.1433
0.1373
0.1314
short
standard
long
DEAN #37261
LEFT
(McGHP)
2.100
INTERACTIVE
top of rh
base of rh
SECTION 2
cap height
base of text
TABLE 2.21 Transmittance-Absorbance Conversion Table (Continued)
%
Transmittance
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
74
75
0.1308
0.1249
0.1302
0.1244
0.1296
0.1238
0.1290
0.1232
0.1284
0.1226
0.1278
0.1221
0.1273
0.1215
0.1267
0.1209
0.1261
0.1203
0.1255
0.1198
76
77
78
79
80
0.1192
0.1135
0.1079
0.1024
0.0969
0.1186
0.1129
0.1073
0.1018
0.0964
0.1180
0.1124
0.1068
0.1013
0.0958
0.1175
0.1118
0.1062
0.1007
0.0953
0.1169
0.1113
0.1057
0.1002
0.0947
0.1163
0.1107
0.1051
0.0996
0.0942
0.1158
0.1101
0.1046
0.0991
0.0937
0.1152
0.1096
0.1040
0.0985
0.0931
0.1146
0.1090
0.1035
0.0980
0.0926
0.1141
0.1085
0.1029
0.0975
0.0921
81
82
83
84
85
0.0915
0.0862
0.0809
0.0757
0.0706
0.0910
0.0857
0.0804
0.0752
0.0701
0.0904
0.0851
0.0799
0.0747
0.0696
0.0899
0.0846
0.0794
0.0742
0.0691
0.0894
0.0841
0.0788
0.0737
0.0685
0.0888
0.0835
0.0783
0.0731
0.0680
0.0883
0.0830
0.0778
0.0726
0.0675
0.0878
0.0825
0.0773
0.0721
0.0670
0.0872
0.0820
0.0768
0.0716
0.0665
0.0867
0.0814
0.0762
0.0711
0.0660
86
87
88
89
90
0.0655
0.0605
0.0555
0.0506
0.0458
0.0650
0.0600
0.0550
0.0501
0.0453
0.0645
0.0595
0.0545
0.0496
0.0448
0.0640
0.0590
0.0540
0.0491
0.0443
0.0635
0.0585
0.0535
0.0487
0.0438
0.0630
0.0580
0.0531
0.0482
0.0434
0.0625
0.0575
0.0526
0.0477
0.0429
0.0620
0.0570
0.0521
0.0472
0.0424
0.0615
0.0565
0.0516
0.0467
0.0419
0.0610
0.0560
0.0511
0.0462
0.0414
91
92
93
94
95
0.0410
0.0362
0.0315
0.0269
0.0223
0.0405
0.0357
0.0311
0.0264
0.0218
0.0400
0.0353
0.0306
0.0259
0.0214
0.0395
0.0348
0.0301
0.0255
0.0209
0.0391
0.0343
0.0297
0.0250
0.0205
0.0386
0.0339
0.0292
0.0246
0.0200
0.0381
0.0334
0.0287
0.0241
0.0195
0.0376
0.0329
0.0283
0.0237
0.0191
0.0372
0.0325
0.0278
0.0232
0.0186
0.0367
0.0320
0.0273
0.0227
0.0182
96
97
98
99
0.0177
0.0132
0.0088
0.0044
0.0173
0.0128
0.0083
0.0039
0.0168
0.0123
0.0079
0.0035
0.0164
0.0119
0.0074
0.0031
0.0159
0.0114
0.0070
0.0026
0.0155
0.0110
0.0066
0.0022
0.0150
0.0106
0.0061
0.0017
0.0146
0.0101
0.0057
0.0013
0.0141
0.0097
0.0052
0.0009
0.0137
0.0092
0.0048
0.0004
short
standard
long
DEAN #37261
(McGHP)
RIGHT
INTERACTIVE
2.101
GENERAL INFORMATION, CONVERSION TABLES, AND MATHEMATICS
top of rh
base of rh
cap height
base of text
TABLE 2.22 Wavenumber/Wavelength Conversion Table
This table is based on the conversion: wavenumber (in cm⫺1) ⫽ 10 000/wavelength (in ␮m). For example,
15.4 ␮m is equal to 649 cm⫺1.
Wavelength
(␮m)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
cm⫺1
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
9.0
10.0
11.0
12.0
13.0
14.0
15.0
16.0
17.0
18.0
19.0
20.0
21.0
22.0
23.0
24.0
25.0
26.0
27.0
28.0
29.0
30.0
31.0
32.0
33.0
34.0
35.0
36.0
37.0
38.0
39.0
40.0
10000
5000
3333
2500
2000
1667
1429
1250
1111
1000
909
833
769
714
667
625
588
556
526
500
476
455
435
417
400
385
370
357
345
333
323
313
303
294
286
278
270
263
256
250
9091
4762
3226
2439
1961
1639
1408
1235
1099
990
901
826
763
709
662
621
585
552
524
498
474
452
433
415
398
383
369
356
344
332
322
312
302
293
285
277
270
262
256
8333
4545
3125
2381
1923
1613
1389
1220
1087
980
893
820
758
704
658
617
581
549
521
495
472
450
431
413
397
382
368
355
342
331
321
311
301
292
284
276
269
262
255
7692
4348
3030
2326
1887
1587
1370
1205
1075
971
885
813
752
699
654
613
578
546
518
493
469
448
429
412
395
380
366
353
341
330
319
310
300
292
283
275
268
261
254
7143
4167
2941
2273
1852
1563
1351
1190
1064
962
877
806
746
694
649
610
575
543
515
490
467
446
427
410
394
379
365
352
340
329
318
309
299
291
282
274
267
260
254
6667
4000
2857
2222
1818
1538
1333
1176
1053
952
870
800
741
690
645
606
571
541
513
488
465
444
426
408
392
377
364
351
339
328
317
308
299
290
282
274
267
260
253
6250
3846
2778
2174
1786
1515
1316
1163
1042
943
862
794
735
685
641
602
568
538
510
485
463
442
424
407
391
376
362
350
338
327
316
307
298
289
281
273
266
259
253
5882
3704
2703
2128
1754
1493
1299
1149
1031
935
855
787
730
680
637
599
565
535
508
483
461
441
422
405
389
375
361
348
337
326
315
306
297
288
280
272
265
258
252
5556
3571
2632
2083
1724
1471
1282
1136
1020
926
847
781
725
676
633
595
562
532
505
481
459
439
420
403
388
373
360
347
336
325
314
305
296
287
279
272
265
258
251
5263
3448
2564
2041
1695
1449
1266
1124
1010
917
840
775
719
671
629
592
559
529
503
478
457
437
418
402
386
372
358
346
334
324
313
304
295
287
279
271
264
257
251
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DEAN #37261
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2.102
2.2
2.2.1
INTERACTIVE
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MATHEMATICAL TABLES
Logarithms
2.2.1.1 Properties and Uses
Definition of Logarithm. The logarithm x of the number N to the base b is the exponent of the
power to which b must be raised to give N. That is,
logb N ⫽ x
bx ⫽ N
or
The number N is positive and b may be any positive number except 1.
Properties of Logarithms
1. The logarithm of a product is equal to the sum of the logarithms of the factors; thus,
logb M · N ⫽ logb M ⫹ logb N
2. The logarithm of a quotient is equal to the logarithm of the numerator minus the logarithm of
the denominator; thus,
logb
M
⫽ logb M ⫺ logb N
N
3. The logarithm of a power of a number is equal to the logarithm of the base multiplied by the
exponent of the power; thus,
logb Mp ⫽ p · logb M
4. The logarithm of a root of a number is equal to the logarithm of the number divided by the index
of the root; thus
q
logb p M ⫽
1
logb M
q
Other properties of logarithms:
q
p
log b M
q
logb b ⫽ 1
logb p Mp ⫽
logb 1 ⫽ 0
logb N ⫽ loga N · logb a ⫽
logb (bN) ⫽ N
loga N
loga b
blogbN ⫽ N
Systems of Logarithms. There are two common systems of logarithms in use: (1) the natural
(Napierian or hyperbolic) system which uses the base e ⫽ 2.71828 . . . ; (2) the common (Briggsian) system which uses the base 10.
We shall use the abbreviation log N ⬅ log10 N in this section.
Unless otherwise stated, tables of logarithms are always tables of common logarithms.
Characteristic of a Common Logarithm of a Number. Every real positive number has a real
common logarithm such that if a ⬍ b, log a ⬍ log b. Neither zero nor any negative number has a
real logarithm.
A common logarithm, in general, consists of an integer, which is called the characteristic, and
a decimal (usually endless), which is called the mantissa. The characteristic of any number may be
determined from the following rules:
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DEAN #37261
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GENERAL INFORMATION, CONVERSION TABLES, AND MATHEMATICS
2.103
Rule I. The characteristic of any number greater than 1 is one less than the number of digits
before the decimal point.
Rule II.* The characteristic of a number less than 1 is found by subtracting from 9 the number
of ciphers between the decimal point and the first significant digit, and writing ⫺10 after the result.
Thus the characteristic of log 936 is 2; the characteristic of log 9.36 is 0; of log 0.936 is 9 ⫺ 10;
of log 0.00936 is 7 ⫺ 10.
Mantissa of a Common Logarithm of a Number. An important consequence of the use of base
10 is that the mantissa of a number is independent of the position of the decimal point. Thus
93 600, 93.600, 0.000 936, all have the same mantissa. Hence in Tables of Common Logarithms
only mantissas are given. A five-place table gives the values of the mantissa correct to five places
of decimals.
Since it is possible to obtain logarithms by using hand calculators, this Handbook contains no
logarithm tables.
Helpful Hints
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1. When connecting numbers to logarithms, use as many decimal places in the mantissa as there
are significant digits in the number.
2. When finding the antilogarithm, keep as many significant digits as there are decimal places in
the mantissa.
Examples: log 10.35 ⫽ 1.0149; antilog 0.065 ⫽ 1.16.
* Some writers use a dash over the characteristic to indicate a negative value; for example,
log 0.004657 ⫽ 7.6681 ⫺ 10 ⫽ 3.6681
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DEAN #37261
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2.104
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TABLE 2.23 Derivatives and Differentiation
Rules for differentiation.
From Baumeister and Marks, Standard Handbook for Mechanical Engineers, 7th ed., McGraw-Hill Book
Company, New York (1967); by permission.
To find the derivative of a given function at a given point: (1) If the function is given only by a curve,
measure graphically the slope of the tangent at the point in question; (2) if the function is given by a mathematical
expression, use the following rules for differentiation. These rules give, directly, the differential, dy, in terms of
dx; to find the derivative, dy/dx, divide through by dx.
Here u, v, w, . . . represents any functions of a variable x, or may themselves be independent variables. a
is a constant which does not change in values in the same discussion; e ⫽ 2.71828.
1. d(a ⫹ u) ⫽ du
3. d(u ⫹ v ⫹ w ⫹ · · ·) ⫽ du ⫹ dv ⫹ dw ⫹ · · ·
4. d(uv) ⫽ u dv ⫹ v du
du dv dw
5. d(uvw . . .) ⫽ (uvw . . .)
⫹
⫹
⫹ ···
u
v
w
u v du ⫺ u dv
6. d ⫽
v
v2
Thus, d(u2) ⫽ 2u
du
8. d p u ⫽
2 pu
10. d(eu) ⫽ eu du
du
12. d ln u ⫽
u
14. d sin u ⫽ cos u du
16. d cos u ⫽ ⫺sin u du
18. d tan u ⫽ sec2 u du
du
20. d sin⫺1 u ⫽
2
p1 ⫺ u
du
22. d cos⫺1 u ⫽ ⫺
2
p1 ⫺ u
du
24. d tan⫺1 u ⫽
1 ⫹ u2
冉
26. d ln sin u ⫽ cot u du
2. d(au) ⫽ a du
冊
7. d(um) ⫽ mum ⫺ 1 du
du; d(u3) ⫽ 3u2 du; etc.
1
du
9. d
⫽⫺ 2
u
u
11. d(au) ⫽ (ln a)au du
冉冊
du
du
⫽ (0.4343 . . .)
u
u
d csc u ⫽ ⫺cot u csc u du
d sec u ⫽ tan u sec u du
d cot u ⫽ ⫺csc2 u du
du
d csc⫺1 u ⫽ ⫺
u p u2 ⫺ 1
du
d sec⫺1 u ⫽
u p u2 ⫺ 1
du
d cot⫺1 u ⫽ ⫺
1 ⫹ u2
2 du
d ln tan u ⫽
sin 2u
2 du
d ln cot u ⫽ ⫺
sin 2u
d csch u ⫽ ⫺csch u coth u du
d sech u ⫽ ⫺sech u tanh u du
d coth u ⫽ ⫺csch2 u du
du
d csch⫺1 u ⫽ ⫺
u p u2 ⫹ 1
du
d sech⫺1 u ⫽ ⫺
u p 1 ⫺ u2
du
d coth⫺1 u ⫽
1 ⫺ u2
13. d log10 u ⫽ (log10 e)
15.
17.
19.
21.
23.
25.
27.
28. d ln cos u ⫽ ⫺tan u du
29.
30. d sinh u ⫽ cosh u du
32. d cosh u ⫽ sinh u du
34. d tanh u ⫽ sech2 u du
du
36. d sinh⫺1 u ⫽
2
pu ⫹ 1
du
38. d cosh⫺1 u ⫽
2
pu ⫺ 1
du
40. d tanh⫺1 u ⫽
1 ⫺ u2
42. d(uv) ⫽ (uv ⫺ 1)(u ln u dv ⫹ v du)
31.
33.
35.
37.
39.
41.
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DEAN #37261
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2.105
GENERAL INFORMATION, CONVERSION TABLES, AND MATHEMATICS
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TABLE 2.24 Integrals
From Baumeister and Marks, Standard Handbook for Mechanical Engineers, 7th ed., McGraw-Hill Book
Company, New York (1967); by permission.
An integral of f(x) dx is any function whose differential is f(x) dx, and is denoted by 兰 f(x) dx. All the integrals
of f(x) dx are included in the expression 兰 f(x) dx ⫹ C, where 兰 f(x) dx is any particular integral, and C is an
arbitrary constant. The process of finding (when possible) an integral of a given function consists in recognizing
by inspection a function which, when differentiated, will produce the given function; or in transforming the
given function into a form in which such recognition is easy. The most common integrable forms are collected
in the following brief table; for a more extended list, see Peirce, Table of Integrals, Ginn, or Dwight, Table of
Integrals and other Mathematical Data, Macmillan.
General formulas
1.
3.
5.
冕
冕
冕
冕
冕冕
冕冕
a du ⫽ a
du ⫽ au ⫹ C
u dv ⫽ uv ⫺
dy
2.
v du
f (x, y) dx ⫽
dx
4.
冕
冕
(u ⫹ v) dx ⫽
冕
f (x) dx ⫽
f [F(y)]F⬘(y) dy, x ⫽ F(y)
冕
u dx ⫹
冕
v dx
f (x, y) dy
Fundamental integrals
6.
7.
9.
11.
13.
14.
冕
冕
冕
冕
冕
冕
x n dx ⫽
x n⫹1
⫹ C, when n ⫽ ⫺1
n⫹1
dx
⫽ ln x ⫹ C ⫽ ln cx
x
8.
sin x dx ⫽ ⫺cos x ⫹ C
10.
dx
⫽ ⫺cot x ⫹ C
sin2 x
dx
⫽ sin⫺1 x ⫹ C ⫽ ⫺cos⫺1 x ⫹ c
2
p1 ⫺ x
dx
⫽ tan⫺1 x ⫹ C ⫽ ⫺cot ⫺1 x ⫹ c
1 ⫹ x2
12.
冕
冕
冕
ex dx ⫽ ex ⫹ C
cos x dx ⫽ sin x ⫹ C
dx
⫽ tan x ⫹ C
cos2 x
Rational functions
15.
16.
17.
19.
20.
21.
22.
冕
冕
冕
冕
冕
冕
冕
(a ⫹ bx)n⫹1
⫹C
(n ⫹ 1)b
dx
1
1
⫽ ln(a ⫹ bx) ⫹ C ⫽ ln c(a ⫹ bx)
a ⫹ bx b
b
1
1
dx
1
dx ⫽ ⫺ ⫹ C
18.
⫽⫺
⫹C
x2
x
(a ⫹ bx)2
b(a ⫹ bx)
dx
1⫹x
⫽ 1⁄2 ln
⫹ C ⫽ tanh⫺1 x ⫹ C, when x ⬍ 1
1 ⫺ x2
1⫺x
dx
x⫺1
⫽ 1⁄2 ln
⫹ C ⫽ ⫺coth⫺1 x ⫹ C, when x ⬎ 1
x2 ⫺ 1
x⫹1
dx
1
b
⫽
tan⫺1
x ⫹C
a ⫹ bx2 p ab
a
(a ⫹ bx)n dx ⫽
冕
冉q 冊
dx
1
p ab ⫹ bx
⫽
ln
⫹C
a ⫺ bx2 2 p ab p ab ⫺ bx
1
b
⫽
tanh⫺1
x ⫹C
a
p ab
冉q 冊
冎
[a ⬎ 0, b ⬎ 0]
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DEAN #37261
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2.106
INTERACTIVE
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SECTION 2
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TABLE 2.24 Integrals (Continued)
Rational functions (continued)
冕
冎
dx
1
b ⫹ cx
⫽
tan⫺1
⫹ C [ac ⫺ b2 ⬎ 0]
2
a ⫹ 2bx ⫹ cx2 p ac ⫺ b2
p ac ⫺ b
2
1
p b ⫺ ac ⫺ b ⫺ cx
⫽
ln
⫹C
2 p b2 ⫺ ac p b2 ⫺ ac ⫹ b ⫹ cx
[b2 ⫺ ac ⬎ 0]
1
b ⫹ cx
⫽⫺
tanh⫺1
⫹C
2
2
p b ⫺ ac
p b ⫺ ac
dx
1
24.
⫽⫺
⫹ C, when b2 ⫽ ac
a ⫹ 2bx ⫹ cx2
b ⫹ cx
(m ⫹ nx) dx
n
mc ⫺ nb
dx
25.
⫽
ln (a ⫹ 2bx ⫹ cx2) ⫹
a ⫹ 2bx ⫹ cx2 2c
c
a ⫹ 2bx ⫹ cx2
f (x) dx
26. In
, if f (x) is a polynomial of higher than the first degree, divide
a ⫹ 2bx ⫹ cx2
by the denominator before integrating.
dx
1
b ⫹ cx
27.
⫽
⫻
(a ⫹ 2bx ⫹ cx2) p 2(ac ⫺ b2)(p ⫺ 1) (a ⫹ 2bx ⫹ cx2) p ⫺ 1
23.
冕
冕
冕
冕
冕
28.
冕
冕
冕
(2p ⫺ 3)c
dx
2(ac ⫺ b2)(p ⫺ 1)
(a ⫹ 2bx ⫹ cx2) p ⫺ 1
(m ⫹ nx) dx
n
1
⫽⫺
⫻
(a ⫹ 2bx ⫹ cx2) p
2c(p ⫺ 1) (a ⫹ 2bx ⫹ cx2) p ⫺ 1
⫹
⫹
29.
冎
x m ⫺ 1(a ⫹ bx)n dx ⫽
⫽
mc ⫺ nb
c
冕
dx
(a ⫹ 2bx ⫹ cx2) p
x m ⫺ 1(a ⫹ bx)n⫹1 (m ⫺ 1)a
⫺
(m ⫹ n)b
(m ⫹ n)b
x m(a ⫹ bx)n
na
⫹
m⫹n
m⫹n
冕
冕
x m ⫺ 2(a ⫹ bx)n dx
xm ⫺ 1(a ⫹ bx)n ⫺ 1 dx
Irrational functions
30.
32.
33.
34.
35.
36.
37.
冕
冕
冕
冕
冕
冕
冕
冕
2
dx
2
3
31.
⫽ p a ⫹ bx ⫹ C
p (a ⫹ bx) ⫹ C
3b
b
a
⫹
bx
p
(m ⫹ nx) dx
2
⫽ 2 (3mb ⫺ 2an ⫹ nbx) p a ⫹ bx ⫹ C
3b
p a ⫹ bx
dx
; substitute y ⫽ p a ⫹ bx, and use 21 and 22
(m ⫹ nx) p a ⫹ bx
p a ⫹ bx dx ⫽
n
f (x, p
a ⫹ bx)
n
dx; substitute p
a ⫹ bx ⫽ y
n
F(x, p
a ⫹ bx)
dx
x
x
⫽ sin⫺1 ⫹ C ⫽ ⫺cos⫺1 ⫹ c
2
2
a
a
pa ⫺ x
dx
x
⫽ ln (x ⫹ p a2 ⫹ x2) ⫹ C ⫽ sinh⫺1 ⫹ c
2
2
a
a
⫹
x
p
dx
x
⫽ ln (x ⫹ p x2 ⫺ a2) ⫹ C ⫽ cosh⫺1 ⫹ c
2
2
a
px ⫺ a
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DEAN #37261
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RIGHT
INTERACTIVE
GENERAL INFORMATION, CONVERSION TABLES, AND MATHEMATICS
2.107
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TABLE 2.24 Integrals (Continued)
Irrational functions (continued)
38.
冕
dx
2
p a ⫹ 2bx ⫹ cx
⫽
⫽
⫽
39.
40.
41.
42.
43.
44.
冕
冕
冕
冕
冕
冕
1
pc
1
pc
1
pc
ln (b ⫹ cx ⫹ p c p a ⫹ 2bx ⫹ cx2) ⫹ C, when c ⬎ 0
sinh⫺1
cosh⫺1
⫹ C, when ac ⫺ b2 ⬎ 0
2
p ac ⫺ b
b ⫹ cx
2
p b ⫺ ac
⫹ C, when b2 ⫺ ac ⬎ 0
b ⫹ cx
sin⫺1
⫹ C, when c ⬍ 0
2
⫺c
p
p b ⫺ ac
(m ⫹ nx) dx
n
mc ⫺ nb
dx
⫽ p a ⫹ 2bx ⫹ cx2 ⫹
2
2
c
c
p a ⫹ 2bx ⫹ cx
p a ⫹ 2bx ⫹ cx
⫽
x m dx
p a ⫹ 2bx ⫹ cx
2
⫽
⫺1
b ⫹ cx
x m ⫺ 1 X (m ⫺ 1)a
⫺
mc
mc
冕
冕
x m ⫺ 2 dx (2m ⫺ 1)b
⫺
X
mc
冕
x m ⫺ 1 dx
,
X
when X ⫽ p a ⫹ 2bx ⫹ cx2
x
a2
2
2
2
2
ln (x ⫹ p a2 ⫹ x2) ⫹ C
p a ⫹ x dx ⫽ p a ⫹ x ⫹
2
2
x
a2
x
⫽ p a2 ⫹ x2 ⫹ sinh⫺1 ⫹ c
2
2
a
x
a2 ⫺1 x
2
2
2
2
sin
⫹C
p a ⫺ x dx ⫽ p a ⫺ x ⫹
2
2
a
2
x
a
2
2
2
2
ln (x ⫹ p x2 ⫺ a2) ⫹ C
p x ⫺ a dx ⫽ p x ⫺ a ⫺
2
2
x
a2
x
⫽ p x2 ⫺ a2 ⫺ cosh⫺1 ⫹ c
2
2
a
b ⫹ cx
2
2
p a ⫹ 2bx ⫹ cx dx ⫽
p a ⫹ 2bx ⫹ cx
2c
ac ⫺ b2
dx
⫹
⫹C
2
2c
p a ⫹ 2bx ⫹ cx
冕
Transcendental functions
45.
46.
47.
48.
49.
50.
51.
52.
冕
冕
冕
冕
冕
冕
冕
冕
ax
⫹C
ln a
x neax
n
n(n ⫺ 1)
n!
x neax dx ⫽
1⫺
⫹
⫺ ··· ⫾ n n
a
ax
a2x2
ax
ax dx ⫽
冋
册
⫹C
ln x dx ⫽ x ln x ⫺ x ⫹ C
ln x
ln x 1
dx ⫽ ⫺
⫺ ⫹C
x2
x
x
(ln x)n
1
dx ⫽
(ln x)n ⫹ 1 ⫹ C
x
n⫹1
sin2 x dx ⫽ ⫺ 1⁄4 sin 2x ⫹ 1⁄2x ⫹ C ⫽ ⫺ 1⁄2 sin x cos x ⫹ 1⁄2x ⫹ C
cos2 x dx ⫽ 1⁄4 sin 2x ⫹ 1⁄2x ⫹ C ⫽ 1⁄2 sin x cos x ⫹ 1⁄2x ⫹ C
sin mx dx ⫽ ⫺
cos mx
⫹C
m
53.
冕
cos mx dx ⫽
sin mx
⫹C
m
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DEAN #37261
LEFT
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2.108
INTERACTIVE
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SECTION 2
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TABLE 2.24 Integrals (Continued)
Transcendental functions (continued)
冕
冕
冕
冕
冕
冕
冕
冕
冕
冕
冕
冕
冕
冕
cos(m ⫹ n)x cos(m ⫺ n)x
⫺
⫹C
2(m ⫹ n)
2(m ⫺ n)
sin mx cos nx dx ⫽ ⫺
54.
sin mx sin nx dx ⫽
55.
sin(m ⫺ n)x sin(m ⫹ n)x
⫺
⫹C
2(m ⫺ n)
2(m ⫹ n)
cos mx cos nx dx ⫽
56.
sin(m ⫺ n)x sin(m ⫹ n)x
⫹
⫹C
2(m ⫺ n)
2(m ⫹ n)
tan x dx ⫽ ⫺ln cos x ⫹ C
57.
59.
61.
65.*
66.*
67.
68.
69.
62.
64.
冕
冕
cos x sinn ⫺ 1 x n ⫺ 1
⫹
sinn ⫺ 2 x dx
n
n
sin x cosn ⫺ 1 x n ⫺ 1
cosn x dx ⫽
⫹
cosn ⫺ 2 x dx
n
n
tann ⫺ 1 x
tann x dx ⫽
⫺
tann ⫺ 2 x dx
n⫺1
cotn ⫺ 1 x
cotn x dx ⫽ ⫺
⫺
cotn ⫺ 2 x dx
n⫺1
dx
cos x
n⫺2
dx
⫽⫺
⫹
sinn x
(n ⫺ 1) sinn ⫺ 1 x n ⫺ 1
sinn ⫺ 2 x
sinn x dx ⫽ ⫺
冕
冕
dx
sin x
n⫺2
⫽
⫹
cosn x (n ⫺ 1) cosn ⫺ 1 x n ⫺ 1
70.
71.*
sin p x cosq x dx ⫽
72.*
73.*
冕
冕
冕
冕
冕
sin p ⫺ 1 x cos q ⫹ 1 p ⫺ 1
⫹
p⫹q
p⫹q
sin⫺p x cos q x dx ⫽ ⫺
sin p x cos⫺q x dx ⫽
冕
冕
冕
冕
cot x dx ⫽ ln sin x ⫹ C
冉冊
dx
␲ x
⫽ ln tan
⫹
⫹C
cos x
4
2
dx
x
⫽ ⫺cot ⫹ C
1 ⫺ cos x
2
dx
⫽ ln tan x ⫹ C
sin x cos x
dx
cosn ⫺ 2 x
sin p ⫹ 1 x cos q ⫺ 1 x q ⫺ 1
⫹
p⫹q
p⫹q
⫽⫺
冕
冕
sin p x cos q ⫺ 2 x dx
sin p ⫺ 2 x cos q x dx
冕
冕
sin⫺p ⫹ 1 x cosq ⫹ 1 x p ⫺ q ⫺ 2
⫹
p⫺1
p⫺1
sin p ⫹ 1 x cos⫺q ⫹ 1 x q ⫺ p ⫺ 2
⫹
q⫺1
q⫺1
冉q
冊
sin⫺p ⫹ 2 x cos q x dx
sin p x cos⫺q ⫹ 2 x dx
dx
2
a⫺b
⫽
tan⫺1
tan 1⁄2x ⫹ C, when a2 ⬎ b2
a ⫹ b cos x p a2 ⫺ b2
a⫹b
1
b ⫹ a cos x ⫹ sin x p b2 ⫺ a2
⫽
ln
⫹C
2
2
a ⫹ b cos x
pb ⫺ a
[a2 ⬍ b2]
⫽
75.
60.
sin x cos x dx ⫽ 1⁄2 sin2 x ⫹ C
63.
74.
58.
dx
x
⫽ ln tan ⫹ C
sin x
2
dx
x
⫽ tan ⫹ C
1 ⫹ cos x
2
冕
2
2
2
pb ⫺ a
cos x dx
x a
⫽ ⫺
a ⫹ b cos x b b
冕
tanh⫺1
冉q
冊
b⫺a
tan 1⁄2x
b⫹a
⫹C
冎
dx
⫹C
a ⫹ b cos x
* If n, p, or q is an odd number, substitute cos x ⫽ z or sin x ⫽ z
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long
DEAN #37261
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RIGHT
INTERACTIVE
GENERAL INFORMATION, CONVERSION TABLES, AND MATHEMATICS
2.109
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TABLE 2.24 Integrals (Continued)
Transcendental functions (continued)
76.
77.
冕
冕
sin x dx
1
⫽ ⫺ ln (a ⫹ b cos x) ⫹ C
a ⫹ b cos x
b
A ⫹ B cos x ⫹ C sin x
dx ⫽ A
a ⫹ b cos x ⫹ c sin x
⫹ (B cos u ⫹ C sin u)
冕
冕
dy
a ⫹ p cos y
cos y dy
⫺ (B sin u ⫺ C cos u)
a ⫹ p cos y
冕
sin y dy
,
a ⫹ p ⫹ cos y
where b ⫽ p cos u, c ⫽ p sin u, and x ⫺ u ⫽ y
78.
79.
80.
81.
82.
83.
84.
86.
88.
90.
91.
92.
冕
冕
冕
冕
冕
冕
冕
冕
冕
冕
冕
冕
2.2.2
eax sin bx dx ⫽
a sin bx ⫺ b cos bx ax
e ⫹C
a2 ⫹ b 2
eax cos bx dx ⫽
a cos bx ⫹ b sin bx ax
e ⫹C
a2 ⫹ b 2
sin⫺1 x dx ⫽ x sin⫺1 x ⫹ p 1 ⫺ x2 ⫹ C
cos⫺1 x dx ⫽ x cos⫺1 x ⫺ p 1 ⫺ x2 ⫹ C
tan⫺1 x dx ⫽ x tan⫺1 x ⫺ 1⁄2 ln(1 ⫹ x2) ⫹ C
cot⫺1 x dx ⫽ x cot⫺1 x ⫹ 1⁄2 ln(1 ⫹ x2) ⫹ C
sinh x dx ⫽ cosh x ⫹ C
85.
cosh x dx ⫽ sinh x ⫹ C
87.
sech x dx ⫽ 2 tan⫺1 (ex) ⫹ C
89.
冕
冕
冕
tanh x dx ⫽ ln cosh x ⫹ C
coth x dx ⫽ ln sinh x ⫹ C
csch x dx ⫽ ln tanh
冉冊
x
2
⫹C
sinh2 x dx ⫽ 1⁄2 sinh x cosh x ⫺ 1⁄2x ⫹ C
cosh2 x dx ⫽ 1⁄2 sinh x cosh ⫹ 1⁄2x ⫹ C
sech2 x dx ⫽ tanh x ⫹ C
93.
冕
csch2 x dx ⫽ ⫺coth x ⫹ C
Surface Areas and Volumes*
Let a, b, c, d, and s denote lengths, A denote areas, and V denote volumes.
Triangle. A ⫽ bh/2, where b denotes the base and h the altitude.
Rectangle. A ⫽ ab, where a and b denote the lengths of the sides.
Parallelogram (opposite sides parallel). A ⫽ ah ⫽ ab sin ␪, where a and b denote the sides, h
the altitude, and ␪ the angle between the sides.
Trapezoid (four sides, two parallel). A ⫽ 1⁄2h(a ⫹ b), where a and b are the sides and h the
altitude.
* Adapted by permission from Burington, Handbook of Mathematical Tables and Formulas, 3d. ed., McGraw-Hill Book
Company, New York (1959).
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DEAN #37261
LEFT
(McGHP)
2.110
INTERACTIVE
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base of rh
SECTION 2
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Regular Polygon of n Sides (Fig. 2.1)
A⫽
1 2
180⬚
na ctn
4
n
R⫽
a
180⬚
csc
2
n
where R is radius of circumscribed circle
r⫽
a
180⬚
ctn
2
n
where r is radius of inscribed circle
␣⫽
360⬚ 2␲
⫽
radians
n
n
␤⫽
冉冊
n⫺2
n
a ⫽ 2r tan
where a is length of side
· 180⬚ ⫽
FIGURE 2.1
冉冊
n⫺2
n
␲ radians
where ␣ and ␤ are the angles indicated in
Fig. 2.1
␣
␣
⫽ 2R sin
2
2
Circle (Fig. 2.2). Let
C ⫽ circumference
S ⫽ length of arc subtended by ␪
R ⫽ radius
l ⫽ chord subtended by arc S
D ⫽ diameter
h ⫽ rise
A ⫽ area
␪ ⫽ central angle in radians
C ⫽ 2␲R ⫽ ␲D
␲ ⫽ 3.14159 . . .
S ⫽ R␪ ⫽ 1⁄2D␪ ⫽ D cos⫺1
d
R
␪
␪
l ⫽ 2p R2 ⫺ d 2 ⫽ 2R sin ⫽ 2d tan
2
2
d ⫽ 1⁄2p 4R2 ⫺ l2 ⫽ R cos
␪ 1
␪
⫽ ⁄2l ctn
2
2
FIGURE 2.2
h⫽R⫺d
␪⫽
S
2S
d
l
l
⫽
⫽ 2 cos⫺1 ⫽ 2 tan⫺1
⫽ 2 sin⫺1
R
D
R
2d
D
A (circle) ⫽ ␲R2 ⫽ 1⁄4␲D2
A (sector) ⫽ 1⁄2Rs ⫽ 1⁄2R2␪
A (segment) ⫽ A (sector) ⫺ A (triangle) ⫽ 1⁄2R2 (␪ ⫺ sin ␪)
⫽ R2 cos⫺1
R⫺h
⫺ (R ⫺ h)p 2Rh ⫺ h2
R
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standard
long
DEAN #37261
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RIGHT
INTERACTIVE
2.111
GENERAL INFORMATION, CONVERSION TABLES, AND MATHEMATICS
Perimeter of n-sided regular polygon inscribed in a circle ⫽ 2nR sin
Area of inscribed polygon ⫽ 1⁄2nR2 sin
␲
n
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2␲
n
Perimeter of n-sided regular polygon circumscribed about a circle ⫽ 2nR tan
Area of circumscribed polygon ⫽ nR2 tan
␲
n
␲
n
Radius of circle inscribed in a triangle of sides a, b, and c is
r⫽
q
(s ⫺ a)(s ⫺ b)(s ⫺ c)
s
s ⫽ 1⁄2(a ⫹ b ⫹ c)
Radius of circle circumscribed about a triangle is
R⫽
abc
4p s(s ⫺ a)(s ⫺ b)(s ⫺ c)
Ellipse (Fig. 2.3). A ⫽ ␲ab, where a and b are lengths of semimajor
and semiminor axes, respectively.
Parabola (Fig. 2.4)
A⫽
2ld
3
FIGURE 2.3
d
Height of d1 ⫽ 2 (l 2 ⫺ l 21)
l
Width of l1 ⫽ l
Length of arc ⫽ l
q
d ⫺ d1
d
冋冉冊冉冊
1⫹
2 2d
3 l
2
⫺
2 2d
5 l
4
册
⫹ ···
FIGURE 2.4
Area by Approximation (Fig. 2.5). If y0 , y1 , y2 , . . . , yn are the length of a series of equally
spaced parallel chords, and if h is their distance apart, the area enclosed by the boundary is given
approximately by any one of the following formulae:
AT ⫽ h[1⁄2(y0 ⫹ yn) ⫹ y1 ⫹ y2 ⫹ · · · ⫹ yn⫺ 1]
FIGURE 2.5
(Trapezoidal Rule)
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standard
long
DEAN #37261
LEFT
(McGHP)
2.112
INTERACTIVE
top of rh
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SECTION 2
AD ⫽ h[0.4(y0 ⫹ yn) ⫹ 1.1(y1 ⫹ yn⫺ 1) ⫹ y2 ⫹ y3 ⫹ · · · ⫹ yn⫺ 2]
(Durand’s Rule)
cap height
base of text
AS ⫽ 1⁄3h[(y0 ⫹ yn) ⫹ 4(y1 ⫹ y3 ⫹ · · · ⫹ yn⫺ 1) ⫹ 2(y2 ⫹ y4 ⫹ · · · ⫹ yn⫺ 2)]
(n even, Simpson’s Rule)
In general, AS gives the most accurate approximation.
The greater the value of n, the greater the accuracy of approximation.
Cube. V ⫽ a3; d ⫽ ap 2; total surface area ⫽ 6a2, where a is length of side and d is length of
diagonal.
Rectangular Parallelopiped. V ⫽ abc; d ⫽ p a2 ⫹ b2 ⫹ c2; total surface area ⫽ 2(ab ⫹ bc ⫹
ca), where a, b, and c are the lengths of the sides and d is length of diagonal.
Prism or Cylinder
V ⫽ (area of base) · (altitude)
Lateral area ⫽ (perimeter of right section) · (lateral edge)
Pyramid or Cone
V ⫽ 1⁄3(area of base) · (altitude)
Lateral area of regular pyramid ⫽ 1⁄2(perimeter of base) · (slant height)
Frustum of Pyramid or Cone. V ⫽ 1⁄3(A1 ⫹ A2 ⫹ p A1 · A2)h, where h is the altitude and A1 and
A2 are the areas of the bases.
Lateral area of a regular figure ⫽ 1⁄2(sum of perimeters of base) · (slant height)
Prismoid
V⫽
h
(A1 ⫹ A2 ⫹ 4A3)
6
where h ⫽ altitude, A1 and A2 are the areas of the bases, and A3 is the area of the midsection parallel
to bases.
Area of Surface and Volume of Regular Polyhedra of Edge l
Name
Type of surface
Area of surface
Volume
Tetrahedron
Hexahedron (cube)
Octahedron
Dodecahedron
Icosahedron
4 equilateral triangles
6 squares
8 equilateral triangles
12 pentagons
20 equilateral triangles
1.73205l 2
6.00000l 2
3.46410l 2
20.64578l 2
8.66025l 2
0.11785l 3
1.00000l 3
0.47140l 3
7.66312l 3
2.18170l 3
Sphere (Fig. 2.6)
A (sphere) ⫽ 4␲R2 ⫽ ␲D2
A (zone) ⫽ 2␲Rh1 ⫽ ␲Dh1
V (sphere) ⫽ 4⁄3␲R3 ⫽ 1⁄8␲D3
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DEAN #37261
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RIGHT
INTERACTIVE
GENERAL INFORMATION, CONVERSION TABLES, AND MATHEMATICS
2.113
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V (spherical sector) ⫽ 2⁄3␲R2h1 ⫽ 1⁄6␲D2h1
V (spherical segment of one base) ⫽ 1⁄6␲h3(3r32 ⫹ h32)
V (spherical segment of two bases) ⫽ 1⁄6␲h2(3r23 ⫹ 3r22 ⫹ h22)
A (lune) ⫽ 2R2␪
where ␪ is angle
in radians of lune
Ellipsoid. V ⫽ 4⁄3␲abc, where a, b, and c are the lengths
of the semiaxes.
Torus (Fig. 2.7)
V ⫽ 2␲2Rr2
FIGURE 2.6
Area of surface ⫽ S ⫽ 4␲ Rr
2
2.2.3
Trigonometric Functions of an Angle ␣
Let x be any angle whose initial side lies on the positive x axis and
whose vertex is at the origin, and (x, y) be any point on the terminal
side of the angle. (x is positive if measured along OX to the right, from
the y axis; and negative, if measured along OX⬘ to the left from the y
axis. Likewise, y is positive if measured parallel to OY, and negative
if measured parallel to OY⬘.) Let r be the positive distance from the
origin to the point. The trigonometric functions of an angle are defined
as follows:
sine ␣
⫽ sin ␣
⫽
y
r
cosine ␣
⫽ cos ␣
⫽
x
r
⫽ tan ␣
⫽
y
x
cotangent ␣ ⫽ ctn ␣
⫽ cot ␣ ⫽
x
y
⫽ sec ␣
⫽
r
x
cosecant x ⫽ csc ␣
⫽
r
y
exsecant x ⫽ exsec ␣
⫽ sec ␣ ⫺ 1
versine ␣
⫽ 1 ⫺ cos ␣
tangent ␣
secant ␣
⫽ vers ␣
coversine ␣ ⫽ covers ␣
⫽ 1 ⫺ sin ␣
haversine ␣ ⫽ hav ␣
⫽ 1⁄2 vers ␣
FIGURE 2.7
FIGURE 2.8
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DEAN #37261
LEFT
(McGHP)
2.114
INTERACTIVE
top of rh
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SECTION 2
2.2.3.1
cap height
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Signs of the Functions
Quadrant
sin
cos
tan
ctn
sec
csc
I
II
III
IV
⫹
⫹
⫺
⫺
⫹
⫺
⫺
⫹
⫹
⫺
⫹
⫺
⫹
⫺
⫹
⫺
⫹
⫺
⫺
⫹
⫹
⫹
⫺
⫺
2.2.3.2
Relations between the Functions of a Single Angle*
sin2 x ⫹ cos2 x ⫽ 1
tan x ⫽
sin x
cos x
cot x ⫽
1
cos x
⫽
tan x
sin x
1 ⫹ tan2 x ⫽ sec2 x ⫽
1
cos2 x
1 ⫹ cot2 x ⫽ csc2 x ⫽
1
sin2 x
sin x ⫽ p 1 ⫺ cos2 x ⫽
cos x ⫽ p 1 ⫺ sin2 x ⫽
FIGURE 2.9
tan x
p 1 ⫹ tan x
2
1
2
p 1 ⫹ tan x
⫽
⫽
1
2
p 1 ⫹ cot x
cot x
2
p 1 ⫹ cot x
2.2.3.3 Functions of Negative Angles. sin(⫺x) ⫽ ⫺sin x; cos(⫺x) ⫽ cos x; tan(⫺x) ⫽
⫺tan x.
2.2.3.4
Functions of the Sum and Difference of Two Angles
sin(x ⫹ y) ⫽ sin x cos y ⫹ cos x sin y
cos(x ⫹ y) ⫽ cos x cos y ⫺ sin x sin y
tan(x ⫹ y) ⫽
tan x ⫹ tan y
1 ⫺ tan x tan y
cot(x ⫹ y) ⫽
cot x cot y ⫺ 1
cot x ⫹ cot y
sin(x ⫺ y) ⫽ sin x cos y ⫺ cos x sin y
* From Baumeister and Marks, Standard Handbook for Mechanical Engineers, 7th ed., McGraw-Hill Book Company, New
York (1967); by permission.
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DEAN #37261
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RIGHT
INTERACTIVE
GENERAL INFORMATION, CONVERSION TABLES, AND MATHEMATICS
2.115
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cos(x ⫺ y) ⫽ cos x cos y ⫹ sin x sin y
tan(x ⫺ y) ⫽
tan x ⫺ tan y
1 ⫹ tan x tan y
cot(x ⫺ y) ⫽
cot x cot y ⫹ 1
cot y ⫺ cot x
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base of rh
sin x ⫹ sin y ⫽ 2 sin 1⁄2(x ⫹ y) cos 1⁄2(x ⫺ y)
sin x ⫺ sin y ⫽ 2 cos 1⁄2(x ⫹ y) sin 1⁄2(x ⫺ y)
cos x ⫹ cos y ⫽ 2 cos 1⁄2(x ⫹ y) cos 1⁄2(x ⫺ y)
cos x ⫺ cos y ⫽ ⫺2 sin 1⁄2(x ⫹ y) sin 1⁄2(x ⫺ y)
tan x ⫹ tan y ⫽
sin(x ⫹ y)
cos x cos y
cot x ⫹ cot y ⫽
sin(x ⫹ y)
sin x sin y
tan x ⫺ tan y ⫽
sin(x ⫺ y)
cos x cos y
cot x ⫺ cot y ⫽
sin(y ⫺ x)
sin x sin y
sin2 x ⫺ sin2 y ⫽ cos2 y ⫺ cos2 x ⫽ sin(x ⫹ y) sin(x ⫺ y)
cos2 x ⫺ sin2 y ⫽ cos2 y ⫺ sin2 x ⫽ cos(x ⫹ y) cos(x ⫺ y)
sin(45⬚ ⫹ x) ⫽ cos(45⬚ ⫺ x)
tan(45⬚ ⫹ x) ⫽ cot(45⬚ ⫺ x)
sin(45⬚ ⫺ x) ⫽ cos(45⬚ ⫹ x)
tan(45⬚ ⫺ x) ⫽ cot(45⬚ ⫹ x)
In the following transformations, a and b are supposed to be positive, c ⫽ p a2 ⫹ b2, A ⫽
the positive acute angle for which B ⫽ b/a:
a cos x ⫹ b sin x ⫽ c sin(A ⫹ x) ⫽ c cos(B ⫺ x)
a cos x ⫺ b sin x ⫽ c sin(A ⫺ x) ⫽ c cos(B ⫹ x)
2.2.4
Expansion in Series*
The range of values of x for which each of the series is convergent is stated at the right of the series.
2.2.4.1
Exponential and Logarithmic Series
ex ⫽ 1 ⫹
x
x2
x3
x4
⫹ ⫹ ⫹ ⫹ ···
1! 2! 3! 4!
ax ⫽ emx ⫽ 1 ⫹
(⫺⬁ ⬍ x ⬍ ⫹⬁)
m
m2 2 m3 3
x⫹
x ⫹
x ⫹ ···
1!
2!
3!
(a ⬎ 0, ⫺⬁ ⬍ x ⬍ ⫹⬁)
where m ⫽ ln a ⫽ 2.3026 log10 a.
* From Baumeister and Marks, Standard Handbook for Mechanical Engineers, 7th ed., McGraw-Hill Book Company, New
York (1967); by permission.
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DEAN #37261
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2.116
INTERACTIVE
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SECTION 2
ln(1 ⫹ x) ⫽ x ⫺
x2 x3 x4 x5
⫹ ⫺ ⫹ ···
2
3
4
5
ln(1 ⫺ x) ⫽ ⫺x ⫺
冉冊冉
冉冊冉
冋
x2 x3 x4 x5
⫺ ⫺ ⫺ ⫺ ···
2
3
4
5
ln
1⫹x
1⫺x
⫽2 x⫹
ln
x⫹1
x⫺1
⫽2
ln x ⫽ 2
冊
x3 x5 x7
⫹ ⫹ ⫹ ···
3
5
7
(⫺1 ⬍ x ⬍ ⫹1)
(⫺1 ⬍ x ⬍ ⫹1)
冊
冉冊冉冊册
冋
冉冊冉冊
1
1
1
1
⫹
⫹
⫹
⫹ ···
x 3x3 5x5 7x7
x⫺1 1 x⫺1
⫹
x⫹1 3 x⫹1
ln(a ⫹ x) ⫽ ln a ⫹ 2
3
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(⫺1 ⬍ x ⬍ ⫹1)
⫹
(x ⬍ ⫺1 or ⫹ 1 ⬍ x)
1 x⫺1
5 x⫹1
x
1
x
⫹
2a ⫹ x 3 2a ⫹ x
3
⫹
5
⫹ ···
1
x
5 2a ⫹ x
(0 ⬍ x ⬍ ⬁)
5
册
⫹ ···
(0 ⬍ a ⬍ ⫹⬁, ⫺a ⬍ x ⬍ ⫹⬁)
Series for the Trigonometric Functions. In the following formulas, all angles must be expressed
in radians. If D ⫽ the number of degrees in the angle, and x ⫽ its radian measure, then x ⫽
0.017453D.
sin x ⫽ x ⫺
x3
x5
x7
⫹ ⫺ ⫹ ···
3! 5! 7!
cos x ⫽ 1 ⫺
x2
x4
x6
x8
⫹ ⫺ ⫹ ⫺ ···
2! 4! 6! 8!
tan x ⫽ x ⫹
x3 2x5 17x7
62x9
⫹
⫹
⫹
⫹ ···
3
15
315
2835
cot x ⫽
(⫺⬁ ⬍ x ⬍ ⫹⬁)
(⫺⬁ ⬍ x ⬍ ⫹⬁)
1 x
x3
2x5
x7
⫺ ⫺
⫺
⫺
⫺ ···
x 3 45 945 4725
sin⫺1 y ⫽ y ⫹
y3 3y5
5y7
⫹
⫹
⫹ ···
6
40
112
tan⫺1 y ⫽ y ⫺
y3 y5 y7
⫹ ⫺ ⫹ ···
3
5
7
cos⫺1 y ⫽ 1⁄2␲ ⫺ sin⫺1 y
冉
⫺
␲
␲
⬍x⬍⫹
2
2
冊
(⫺ ␲ ⬍ x ⬍ ⫹ ␲)
(⫺1 ⬉ y ⬉ ⫹1)
(⫺1 ⬉ y ⬉ ⫹1)
cot⫺1 y ⫽ 1⁄2␲ ⫺ tan⫺1 y
Reversing a Series. If y ⫽ x ⫹ bx2 ⫹ cx3 ⫹ dx4 ⫹ ex5 ⫹ · · · , then x ⫽ y ⫺ by2 ⫹ (2b2 ⫺
c)y3 ⫺ (5b3 ⫺ 5bc ⫹ d)y4 ⫹ (14b4 ⫺ 21b2c ⫹ 6bd ⫹ 3c2 ⫺ e)y5 ⫹ · · · , provided the latter series
is convergent.
Fourier’s Series. Let f (x) be a function which is finite in the interval from x ⫽ ⫺c to x ⫽
⫹c and whose graph has finite arc length in that interval.* Then, for any value of x between ⫺c
and c,
* If x ⫽ x0 is a point of discontinuity, f (x0) is to be defined as 1⁄2[f1(x0) ⫹ f2(x0)], where f1(x0) is the limit of f (x) when x
approaches x0 from below, and f2(x0) is the limit of f(x) when x approaches x0 from above.
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f(x) ⫽ 1⁄2a0 ⫹ a1 cos
2.117
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␲x
2␲c
3␲x
⫹ a2 cos
⫹ a3 cos
⫹ ···
c
c
c
␲x
2␲x
3␲x
⫹ b2 sin
⫹ b3 sin
⫹ ···
c
c
c
⫹ b1 sin
where the constant coefficients are determined as follows:
an ⫽
1
c
冕
c
⫺c
f(t) cos
n␲t
dt
c
bn ⫽
1
c
冕
c
⫺c
f(t) sin
n␲t
dt
c
In case the curve y ⫽ f (x) is symmetrical with respect to the origin, the a’s are all zero, and the
series is a sine series. In case the curve is symmetrical with respect to the y axis, the b’s are all zero,
and a cosine series results. (In this case, the series will be valid not only for values of x between
⫺c and c, but also for x ⫽ ⫺c and x ⫽ c .) A Fourier series can always be integrated term by term;
but the result of differentiating term by term may not be a convergent series.
TABLE 2.25 Some Constants
Constant
Number
Log10 of Number
Pi (␲)
Napierian Base (e)
M ⫽ log10e
1 ⫼ M ⫽ loge 10
180 ⫼ ␲ ⫽ degrees in 1 radian
␲ ⫼ 180 ⫽ radians in 1⬚
␲ ⫼ 10800 ⫽ radians in 1⬘
␲ ⫼ 648000 ⫽ radians in 1⬙
3.14159 26535 89793 23846
2.71828 18284 59045 23536
0.43429 44819 03251 82765
2.30258 50929 94045 68402
57.2957 795
0.01745 329
0.00029 08882
0.00000 48481 36811 095
0.49714 98726 94133 85435
0.43429 448
9.63778 43113 00536 78912 ⫺ 10
0.36221 569
1.75812 263
8.24187 737 ⫺ 10
6.46372 612 ⫺ 10
4.68557 487 ⫺ 10
2.3
2.3.1
STATISTICS IN CHEMICAL ANALYSIS
Introduction
Each observation in any branch of scientific investigation is inaccurate to some degree. Often the
accurate value for the concentration of some particular constituent in the analyte cannot be determined. However, it is reasonable to assume the accurate value exists, and it is important to estimate
the limits between which this value lies. It must be understood that the statistical approach is concerned with the appraisal of experimental design and data. Statistical techniques can neither detect
nor evaluate constant errors (bias); the detection and elimination of inaccuracy are analytical problems. Nevertheless, statistical techniques can assist considerably in determining whether or not
inaccuracies exist and in indicating when procedural modifications have reduced them.
By proper design of experiments, guided by a statistical approach, the effects of experimental
variables may be found more efficiently than by the traditional approach of holding all variables
constant but one and systematically investigating each variable in turn. Trends in data may be sought
to track down nonrandom sources of error.
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Errors in Quantitative Analysis
Two broad classes of errors may be recognized. The first class, determinate or systematic errors, is
composed of errors that can be assigned to definite causes, even though the cause may not have been
located. Such errors are characterized by being unidirectional. The magnitude may be constant from
sample to sample, proportional to sample size, or variable in a more complex way. An example is
the error caused by weighing a hygroscopic sample. This error is always positive in sign; it increases
with sample size but varies depending on the time required for weighing, with humidity and temperature. An example of a negative systematic error is that caused by solubility losses of a precipitate.
The second class, indeterminate or random errors, is brought about by the effects of uncontrolled
variables. Truly random errors are as likely to cause high as low results, and a small random error
is much more probable than a large one. By making the observation coarse enough, random errors
would cease to exist. Every observation would give the same result, but the result would be less
precise than the average of a number of finer observations with random scatter.
The precision of a result is its reproducibility; the accuracy is its nearness to the truth. A systematic error causes a loss of accuracy, and it may or may not impair the precision depending upon
whether the error is constant or variable. Random errors cause a lowering of reproducibility, but
by making sufficient observations it is possible to overcome the scatter within limits so that the
accuracy may not necessarily be affected. Statistical treatment can properly be applied only to
random errors.
2.3.3
Representation of Sets of Data
Raw data are collected observations that have not been organized numerically. An average is a value
that is typical or representative of a set of data. Several averages can be defined, the most common
being the arithmetic mean (or briefly, the mean), the median, the mode, and the geometric mean.
The mean of a set of N numbers, x1, x2, x3, . . . , xN, is denoted by x and is defined as:
x⫽
x1 ⫹ x2 ⫹ x3 ⫹ · · · ⫹ xN
N
(2.4)
It is an estimation of the unknown true value ␮ of an infinite population. We can also define the
sample variance s2 as follows:
冘(x ⫺ x)
s ⫽
N
2
i⫽1
i
2
N⫺1
(2.5)
The values of x and s2 vary from sample set to sample set. However, as N increases, they may be
expected to become more and more stable. Their limiting values, for very large N, are numbers
characteristic of the frequency distribution, and are referred to as the population mean and the
population variance, respectively.
The median of a set of numbers arranged in order of magnitude is the middle value or the
arithmetic mean of the two middle values. The median allows inclusion of all data in a set without
undue influence from outlying values; it is preferable to the mean for small sets of data.
The mode of a set of numbers is that value which occurs with the greatest frequency (the most
common value). The mode may not exist, and even if it does exist it may not be unique. The empirical
relation that exists between the mean, the mode, and the median for unimodal frequency curves
which are moderately asymmetrical is:
Mean ⫺ mode ⫽ 3(mean ⫺ median)
(2.6)
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The geometric mean of a set of N numbers is the Nth root of the product of the numbers:
N
p x1x2x3 . . . xN
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(2.7)
The root mean square (RMS) or quadratic mean of a set of numbers is defined by:
RMS ⫽ p x 2 ⫽
2.3.4
q 冘 x /N
N
i⫽1
2
i
(2.8)
The Normal Distribution of Measurements
The normal distribution of measurements (or the normal law of error) is the fundamental starting
point for analysis of data. When a large number of measurements are made, the individual measurements are not all identical and equal to the accepted value ␮, which is the mean of an infinite
population or universe of data, but are scattered about ␮, owing to random error. If the magnitude
of any single measurement is the abscissa and the relative frequencies (i.e., the probability) of
occurrence of different-sized measurements are the ordinate, the smooth curve drawn through the
points (Fig. 2.10) is the normal or Gaussian distribution curve (also the error curve or probability
curve). The term error curve arises when one considers the distribution of errors (x ⫺ ␮) about the
true value.
FIGURE 2.10 The Normal Distribution Curve.
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The breadth or spread of the curve indicates the precision of the measurements and is determined
by and related to the standard deviation, a relationship that is expressed in the equation for the
normal curve (which is continuous and infinite in extent):
Y⫽
1
␴ p 2␲
exp
冋冉冊册
⫺
1
2
x⫺␮
␴
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2
(2.9)
where ␴ is the standard deviation of the infinite population. The population mean ␮ expresses the
magnitude of the quantity being measured. In a sense, ␴ measures the width of the distribution, and
thereby also expresses the scatter or dispersion of replicate analytical results. When (x ⫺ ␮)/␴ is
replaced by the standardized variable z, then:
Y⫽
1
p 2␲
2
e⫺(1/2)z
(2.10)
The standardized variable (the z statistic) requires only the probability level to be specified. It measures the deviation from the population mean in units of standard deviation. Y is 0.399 for the most
probable value, ␮. In the absence of any other information, the normal distribution is assumed to
apply whenever repetitive measurements are made on a sample, or a similar measurement is made
on different samples.
Table 2.26a lists the height of an ordinate (Y) as a distance z from the mean, and Table 2.26b
the area under the normal curve at a distance z from the mean, expressed as fractions of the total
area, 1.000. Returning to Fig. 2.10, we note that 68.27% of the area of the normal distribution curve
lies within 1 standard deviation of the center or mean value. Therefore, 31.73% lies outside those
limits and 15.86% on each side. Ninety-five percent (actually 95.43%) of the area lies within 2
standard deviations, and 99.73% lies within 3 standard deviations of the mean. Often the last two
areas are stated slightly different; viz. 95% of the area lies within 1.96␴ (approximately 2␴) and
99% lies within approximately 2.5␴. The mean falls at exactly the 50% point for symmetric normal
distributions.
Example 5 The true value of a quantity is 30.00, and ␴ for the method of measurement is 0.30.
What is the probability that a single measurement will have a deviation from the mean greater than
0.45; that is, what percentage of results will fall outside the range 30.00 ⫾ 0.45?
z⫽
x ⫺ ␮ 0.45
⫽
⫽ 1.5
␴
0.30
From Table 2.26b the area under the normal curve from ⫺1.5␴ to ⫹1.5␴ is 0.866, meaning that
86.6% of the measurements will fall within the range 30.00 ⫾ 0.45 and 13.4% will lie outside this
range. Half of these measurements, 6.7%, will be less than 29.55; and a similar percentage will
exceed 30.45. In actuality the uncertainty in z is about 1 in 15; therefore, the value of z could lie
between 1.4 and 1.6; the corresponding areas under the curve could lie between 84% and 89%.
Example 6 If the mean value of 500 determinations is 151 and ␴ ⫽ 15, how many results lie
between 120 and 155 (actually any value between 119.5 and 155.5)?
z⫽
119.5 ⫺ 151
⫽ ⫺2.10
15
z⫽
155.5 ⫺ 151
⫽ 0.30
15
Area: 0.482
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Total area: 0.600
500(0.600) ⫽ 300 results
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TABLE 2.26a
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Ordinates (Y) of the Normal Distribution Curve at Values of z
z
0
1
2
3
4
5
6
7
8
9
0.0
0.1
0.2
0.3
0.4
0.3989
0.3970
0.3910
0.3814
0.3683
0.3989
0.3965
0.3902
0.3802
0.3668
0.3989
0.3961
0.3894
0.3790
0.3653
0.3988
0.3956
0.3885
0.3778
0.3637
0.3986
0.3951
0.3876
0.3765
0.3621
0.3984
0.3945
0.3867
0.3752
0.3605
0.3982
0.3939
0.3857
0.3739
0.3589
0.3980
0.3932
0.3847
0.3725
0.3572
0.3977
0.3925
0.3836
0.3712
0.3555
0.3973
0.3918
0.3825
0.3697
0.3538
0.5
0.6
0.7
0.8
0.9
0.3521
0.3332
0.3123
0.2897
0.2661
0.3503
0.3312
0.3101
0.2874
0.2637
0.3485
0.3292
0.3079
0.2850
0.2613
0.3467
0.3271
0.3056
0.2827
0.2589
0.3448
0.3251
0.3034
0.2803
0.2565
0.3429
0.3230
0.3011
0.2780
0.2541
0.3410
0.3209
0.2989
0.2756
0.2516
0.3391
0.3187
0.2966
0.2732
0.2492
0.3372
0.3166
0.2943
0.2709
0.2468
0.3352
0.3144
0.2920
0.2685
0.2444
1.0
1.1
1.2
1.3
1.4
0.2420
0.2179
0.1942
0.1714
0.1497
0.2396
0.2155
0.1919
0.1691
0.1476
0.2371
0.2131
0.1895
0.1669
0.1456
0.2347
0.2107
0.1872
0.1647
0.1435
0.2323
0.2083
0.1849
0.1626
0.1415
0.2299
0.2059
0.1826
0.1604
0.1394
0.2275
0.2036
0.1804
0.1582
0.1374
0.2251
0.2012
0.1781
0.1561
0.1354
0.2227
0.1989
0.1758
0.1539
0.1334
0.2203
0.1965
0.1736
0.1518
0.1315
1.5
1.6
1.7
1.8
1.9
0.1295
0.1109
0.0940
0.0790
0.0656
0.1276
0.1092
0.0925
0.0775
0.0644
0.1257
0.1074
0.0909
0.0761
0.0632
0.1238
0.1057
0.0893
0.0748
0.0620
0.1219
0.1040
0.0878
0.0734
0.0608
0.1200
0.1023
0.0863
0.0721
0.0596
0.1182
0.1006
0.0848
0.0707
0.0584
0.1163
0.0989
0.0833
0.0694
0.0573
0.1145
0.0973
0.0818
0.0681
0.0562
0.1127
0.0957
0.0804
0.0669
0.0551
2.0
2.1
2.2
2.3
2.4
0.0540
0.0440
0.0355
0.0283
0.0224
0.0529
0.0431
0.0347
0.0277
0.0219
0.0519
0.0422
0.0339
0.0270
0.0213
0.0508
0.0413
0.0332
0.0264
0.0208
0.0498
0.0404
0.0325
0.0258
0.0203
0.0488
0.0396
0.0317
0.0252
0.0198
0.0478
0.0387
0.0310
0.0246
0.0194
0.0468
0.0379
0.0303
0.0241
0.0189
0.0459
0.0371
0.0297
0.0235
0.0184
0.0449
0.0363
0.0290
0.0229
0.0180
2.5
2.6
2.7
2.8
2.9
0.0175
0.0136
0.0104
0.0079
0.0060
0.0171
0.0132
0.0101
0.0077
0.0058
0.0167
0.0129
0.0099
0.0075
0.0056
0.0163
0.0126
0.0096
0.0073
0.0055
0.0158
0.0122
0.0093
0.0071
0.0053
0.0154
0.0119
0.0091
0.0069
0.0051
0.0151
0.0116
0.0088
0.0067
0.0050
0.0147
0.0113
0.0086
0.0065
0.0048
0.0143
0.0110
0.0084
0.0063
0.0047
0.0139
0.0107
0.0081
0.0061
0.0046
3.0
3.1
3.2
3.3
3.4
0.0044
0.0033
0.0024
0.0017
0.0012
0.0043
0.0032
0.0023
0.0017
0.0012
0.0042
0.0031
0.0022
0.0016
0.0012
0.0040
0.0030
0.0022
0.0016
0.0011
0.0039
0.0029
0.0021
0.0015
0.0011
0.0038
0.0028
0.0020
0.0015
0.0010
0.0037
0.0027
0.0020
0.0014
0.0010
0.0036
0.0026
0.0019
0.0014
0.0010
0.0035
0.0025
0.0018
0.0013
0.0009
0.0034
0.0025
0.0018
0.0013
0.0009
3.5
3.6
3.7
3.8
3.9
0.0009
0.0006
0.0004
0.0003
0.0002
0.0008
0.0006
0.0004
0.0003
0.0002
0.0008
0.0006
0.0004
0.0003
0.0002
0.0008
0.0005
0.0004
0.0003
0.0002
0.0008
0.0005
0.0004
0.0003
0.0002
0.0007
0.0005
0.0004
0.0002
0.0002
0.0007
0.0005
0.0003
0.0002
0.0002
0.0007
0.0005
0.0003
0.0002
0.0002
0.0007
0.0005
0.0003
0.0002
0.0001
0.0006
0.0004
0.0003
0.0002
0.0001
2.3.5
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Standard Deviation as a Measure of Dispersion
Several ways may be used to characterize the spread or dispersion in the original data. The range
is the difference between the largest value and the smallest value in a set of observations. However,
almost always the most efficient quantity for characterizing variability is the standard deviation (also
called the root mean square).
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TABLE 2.26b Areas Under the Normal Distribution Curve from 0 to z
z
0
1
2
3
4
5
6
7
8
9
0.0
0.1
0.2
0.3
0.4
0.0000
0.0398
0.0793
0.1179
0.1554
0.0040
0.0438
0.0832
0.1217
0.1591
0.0080
0.0478
0.0871
0.1255
0.1628
0.0120
0.0517
0.0910
0.1293
0.1664
0.0160
0.0557
0.0948
0.1331
0.1700
0.0199
0.0596
0.0987
0.1368
0.1736
0.0239
0.0636
0.1026
0.1406
0.1772
0.0279
0.0675
0.1064
0.1443
0.1808
0.0319
0.0714
0.1103
0.1480
0.1844
0.0359
0.0754
0.1141
0.1517
0.1879
0.5
0.6
0.7
0.8
0.9
0.1915
0.2258
0.2580
0.2881
0.3159
0.1950
0.2291
0.2612
0.2910
0.3186
0.1985
0.2324
0.2642
0.2939
0.3212
0.2019
0.2357
0.2673
0.2967
0.3238
0.2054
0.2389
0.2704
0.2996
0.3264
0.2088
0.2422
0.2734
0.3023
0.3289
0.2123
0.2454
0.2764
0.3051
0.3315
0.2157
0.2486
0.2794
0.3078
0.3340
0.2190
0.2518
0.2823
0.3106
0.3365
0.2224
0.2549
0.2852
0.3133
0.3389
1.0
1.1
1.2
1.3
1.4
0.3413
0.3643
0.3849
0.4032
0.4192
0.3438
0.3665
0.3869
0.4049
0.4207
0.3461
0.3686
0.3888
0.4066
0.4222
0.3485
0.3708
0.3907
0.4082
0.4236
0.3508
0.3729
0.3925
0.4099
0.4251
0.3531
0.3749
0.3944
0.4115
0.4265
0.3554
0.3770
0.3962
0.4131
0.4279
0.3577
0.3790
0.3980
0.4147
0.4292
0.3599
0.3810
0.3997
0.4162
0.4306
0.3621
0.3830
0.4015
0.4177
0.4319
1.5
1.6
1.7
1.8
1.9
0.4332
0.4452
0.4554
0.4641
0.4713
0.4345
0.4463
0.4564
0.4649
0.4719
0.4357
0.4474
0.4573
0.4656
0.4726
0.4370
0.4484
0.4582
0.4664
0.4732
0.4382
0.4495
0.4591
0.4671
0.4738
0.4394
0.4505
0.4599
0.4678
0.4744
0.4406
0.4515
0.4608
0.4686
0.4750
0.4418
0.4525
0.4616
0.4693
0.4756
0.4429
0.4535
0.4625
0.4699
0.4761
0.4441
0.4545
0.4633
0.4706
0.4767
2.0
2.1
2.2
2.3
2.4
0.4772
0.4821
0.4861
0.4893
0.4918
0.4778
0.4826
0.4864
0.4896
0.4920
0.4783
0.4830
0.4868
0.4898
0.4922
0.4788
0.4834
0.4871
0.4901
0.4925
0.4793
0.4838
0.4875
0.4904
0.4927
0.4798
0.4842
0.4878
0.4906
0.4929
0.4803
0.4846
0.4881
0.4909
0.4931
0.4808
0.4850
0.4884
0.4911
0.4932
0.4812
0.4854
0.4887
0.4913
0.4934
0.4817
0.4857
0.4890
0.4916
0.4936
2.5
2.6
2.7
2.8
2.9
0.4938
0.4953
0.4965
0.4974
0.4981
0.4940
0.4955
0.4966
0.4975
0.4982
0.4941
0.4956
0.4967
0.4976
0.4982
0.4943
0.4957
0.4968
0.4977
0.4983
0.4945
0.4959
0.4969
0.4977
0.4984
0.4946
0.4960
0.4970
0.4978
0.4984
0.4948
0.4961
0.4971
0.4979
0.4985
0.4949
0.4962
0.4972
0.4979
0.4985
0.4951
0.4963
0.4973
0.4980
0.4986
0.4952
0.4964
0.4974
0.4981
0.4986
3.0
3.1
3.2
3.3
3.4
0.4987
0.4990
0.4993
0.4995
0.4997
0.4987
0.4991
0.4993
0.4995
0.4997
0.4987
0.4991
0.4994
0.4995
0.4997
0.4988
0.4991
0.4994
0.4996
0.4997
0.4988
0.4992
0.4994
0.4996
0.4997
0.4989
0.4992
0.4994
0.4996
0.4997
0.4989
0.4992
0.4994
0.4996
0.4997
0.4989
0.4992
0.4995
0.4996
0.4997
0.4990
0.4993
0.4995
0.4996
0.4997
0.4990
0.4993
0.4995
0.4997
0.4998
3.5
3.6
3.7
3.8
3.9
0.4998
0.4998
0.4999
0.4999
0.5000
0.4998
0.4998
0.4999
0.4999
0.5000
0.4998
0.4999
0.4999
0.4999
0.5000
0.4998
0.4999
0.4999
0.4999
0.5000
0.4998
0.4999
0.4999
0.4999
0.5000
0.4998
0.4999
0.4999
0.4999
0.5000
0.4998
0.4999
0.4999
0.4999
0.5000
0.4998
0.4999
0.4999
0.4999
0.5000
0.4998
0.4999
0.4999
0.4999
0.5000
0.4998
0.4999
0.4999
0.4999
0.5000
The standard deviation is the square root of the average squared differences between the individual observations and the population mean:
冘 (x ⫺ ␮)
N
␴⫽
q
i⫽1
i
N
2
(2.11)
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The standard deviation ␴ may be estimated by calculating the standard deviation s drawn from a
small sample set as follows:
冘 (x ⫺ x)
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N
s⫽
q
i⫽1
i
2
s⫽
or
N
q
x 21 ⫹ x 22 ⫹ · · · ⫺ [(x1 ⫹ x2 ⫹ · · ·)2]/N
N⫺1
(2.12)
where xi ⫺ x represents the deviation of each number in the array from the arithmetic mean. Since
two pieces of information, namely s and x, have been extracted from the data, we are left with
N ⫺ 1 degrees of freedom (df); that is, independent data points available for measurement of precision. If a relatively large sample of data corresponding to N ⬎ 30 is available, its mean can be
taken as a measure of ␮, and s as equal to ␴.
So basic is the notion of a statistical estimate of a physical parameter that statisticians use Greek
letters for the parameters and Latin letters for the estimates. For many purposes, one uses the
variance, which for the sample is s2 and for the entire populations is ␴ 2. The variance s2 of a finite
sample is an unbiased estimate of ␴ 2, whereas the standard deviation s is not an unbiased estimate
of ␴.
Because the standard deviation ␴ for the universe is a characteristic of the measuring procedure,
it is possible to get a good estimate not only from a long series of repeated analyses of the same
sample, but also by taking together several short series measured with slightly different samples of
the same type. When a series of observations can be logically arranged into k subgroups, the variance
is calculated by summing the squares of the deviations for each subgroup, and then adding all the
k sums and dividing by N ⫺ k because one degree of freedom is lost in each subgroup. It is not
required that the number of repeated analyses in the different groups be the same. For two groups
of observations consisting of NA and NB members of standard deviations sA and sB, respectively, the
variance is given by:
s2 ⫽
(NA ⫺ 1)sA2 ⫹ (NB ⫺ 1)sB2
NA ⫹ NB ⫺ 2
(2.13)
Another measure of dispersion is the coefficient of variation, which is merely the standard deviation expressed as a fraction of the arithmetic mean, viz., s/x . It is useful mainly to show whether
the relative or the absolute spread of values is constant as the values are changed.
2.3.6
Student’s Distribution or t Test
In the next several sections, the theoretical distributions and tests of significance will be examined
beginning with Student’s distribution or t test. If the data contained only random (or chance) errors,
the cumulative estimates x and s would gradually approach the limits ␮ and ␴. The distribution of
results would be normally distributed with mean ␮ and standard deviation ␴. Were the true mean
of the infinite population known, it would also have some symmetrical type of distribution centered
around ␮. However, it would be expected that the dispersion or spread of this dispersion about the
mean would depend on the sample size.
The standard deviation of the distribution of means equals ␴/N1/2. Since ␴ is not usually known,
its approximation for a finite number of measurements is overcome by the Student t test. It is a
measure of error between ␮ and x. The Student t takes into account both the possible variation of
the value of x from ␮ on the basis of the expected variance ␴ 2/N1/2 and the reliability of using s in
place of ␴. The distribution of the statistic is:
⫾t ⫽
x⫺␮
s/p N
or
␮⫽x⫾
ts
pN
(2.14)
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The distribution of the t-statistic (x ⫺ ␮)s is symmetrical about zero and is a function of the degrees
of freedom. Limits assigned to the distance on either side of ␮ are called confidence limits. The
percentage probability that ␮ lies within this interval is called the confidence level. The level of
significance or error probability (100 ⫺ confidence level or 100 ⫺ ␣) is the percent probability that
␮ will lie outside the confidence interval, and represents the chances of being incorrect in stating
that ␮ lies within the confidence interval. Values of t are in Table 2.27 for any desired degrees of
freedom and various confidence levels.
An analytical procedure is often tested on materials of known composition. These materials may
be pure substances, standard samples, or materials analyzed by some other more accurate method.
Repeated determinations on a known material furnish data for both an estimate of the precision and
a test for the presence of a constant error in the results. The standard deviation is found from Equation
12 (with the known composition replacing ␮). A calculated value for t (Eq. 14) in excess of the
appropriate value in Table 2.27 is interpreted as evidence of the presence of a constant error at the
indicated level of significance.
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TABLE 2.27 Percentile Values for Student t Distribution
df
t 0.995
t 0.99
t 0.975
t 0.95
t 0.90
t 0.80
t 0.75
t 0.70
t 0.60
t 0.55
1
2
3
4
63.66
9.92
5.84
4.60
31.82
6.96
4.54
3.75
12.71
4.30
3.18
2.78
6.31
2.92
2.35
2.13
3.08
1.89
1.64
1.53
1.376
1.061
0.978
0.941
1.000
0.816
0.765
0.741
0.727
0.617
0.584
0.569
0.325
0.289
0.277
0.271
0.158
0.142
0.137
0.134
5
6
7
8
9
4.03
3.71
3.50
3.36
3.25
3.36
3.14
3.00
2.90
2.82
2.57
2.45
2.36
2.31
2.26
2.02
1.94
1.90
1.86
1.83
1.48
1.44
1.42
1.40
1.38
0.920
0.906
0.896
0.889
0.883
0.727
0.718
0.711
0.706
0.703
0.559
0.553
0.549
0.546
0.543
0.267
0.265
0.263
0.262
0.261
0.132
0.131
0.130
0.130
0.129
10
11
12
13
14
3.17
3.11
3.06
3.01
2.98
2.76
2.72
2.68
2.65
2.62
2.23
2.20
2.18
2.16
2.14
1.81
1.80
1.78
1.77
1.76
1.37
1.36
1.36
1.35
1.34
0.879
0.876
0.873
0.870
0.868
0.700
0.697
0.695
0.694
0.692
0.542
0.540
0.539
0.538
0.537
0.260
0.260
0.259
0.259
0.258
0.129
0.129
0.128
0.128
0.128
15
16
17
18
19
2.95
2.92
2.90
2.88
2.86
2.60
2.58
2.57
2.55
2.54
2.13
2.12
2.11
2.10
2.09
1.75
1.75
1.74
1.73
1.73
1.34
1.34
1.33
1.33
1.33
0.866
0.865
0.863
0.862
0.861
0.691
0.690
0.689
0.688
0.688
0.536
0.535
0.534
0.534
0.533
0.258
0.258
0.257
0.257
0.257
0.128
0.128
0.128
0.127
0.127
20
21
22
23
24
2.84
2.83
2.82
2.81
2.80
2.53
2.52
2.51
2.50
2.49
2.09
2.08
2.07
2.07
2.06
1.72
1.72
1.72
1.71
1.71
1.32
1.32
1.32
1.32
1.32
0.860
0.859
0.858
0.858
0.857
0.687
0.686
0.686
0.685
0.685
0.533
0.532
0.532
0.532
0.531
0.257
0.257
0.256
0.256
0.256
0.127
0.127
0.127
0.127
0.127
25
26
27
28
29
2.79
2.78
2.77
2.76
2.76
2.48
2.48
2.47
2.47
2.46
2.06
2.06
2.05
2.05
2.04
1.71
1.71
1.70
1.70
1.70
1.32
1.32
1.31
1.31
1.31
0.856
0.856
0.855
0.855
0.854
0.684
0.684
0.684
0.683
0.683
0.531
0.531
0.531
0.530
0.530
0.256
0.256
0.256
0.256
0.256
0.127
0.127
0.127
0.127
0.127
30
40
60
120
⬁
2.75
2.70
2.66
2.62
2.58
2.46
2.42
2.39
2.36
2.33
2.04
2.02
2.00
2.98
1.96
1.70
1.68
1.67
1.66
1.645
1.31
1.30
1.30
1.29
1.28
0.854
0.851
0.848
0.845
0.842
0.683
0.681
0.679
0.677
0.674
0.530
0.529
0.527
0.526
0.524
0.256
0.255
0.254
0.254
0.253
0.127
0.126
0.126
0.126
0.126
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Example 7 A new method for the analysis of iron using pure FeO was replicated with five
samples giving these results (in % Fe): 76.95, 77.02, 76.90, 77.20, and 77.50. Does a systematic
error exist?
From Equation 4, x is 77.11; and from Equation 5, s is 0.24 for 4 degrees of freedom. Because
␴ is not known, the Student t0.975 (2.78 for 4 degrees of freedom) is used to calculate the confidence
interval at the 95% probability level.
␮⫽x⫾
ts
pN
⫽ 77.11 ⫾
(2.78) (0.24)
p5
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⫽ 77.11 ⫾ 0.30
We used a two-tailed test. Upon rereading the problem, we realize that this was pure FeO whose
iron content was 77.60% so that ␮ ⫽ 77.60 and the confidence interval does not include the known
value. Since the FeO was a standard, a one-tailed test should have been used since only random
values would be expected to exceed 77.60%. Now the Student t value of 2.13 (for ⫺t0.05) should
have been used, and now the confidence interval becomes 77.11 ⫾ 0.23. A systematic error is
presumed to exist.
The t test can be applied to differences between pairs of observations. Perhaps only a single pair
can be performed at one time, or possibly one wishes to compare two methods using samples of
differing analytical content. It is still necessary that the two methods possess the same inherent
standard deviation. An average difference d calculated, and individual deviations from d are used to
evaluate the variance of the differences.
Example 8 From the following data do the two methods actually give concordant results?
Sample
Method A
Method B
1
2
3
4
33.27
51.34
23.91
47.04
33.04
50.96
23.77
46.79
Difference
d1
d2
d3
d4
⫽ 0.23
⫽ 0.38
⫽ 0.14
⫽ 0.25
d ⫽ 0.25
sd ⫽
t⫽
p
冘 (d ⫺ d) ⫽ 0.099
2
N⫺1
0.25
4 ⫺ 1 ⫽ 4.30
0.099 p
From Table 2.27, t0.975 ⫽ 3.18 (at 95% probability) and t0.995 ⫽ 5.84 (at 99% probability). The
difference between the two methods is probably significant.
If the t-value falls short of the formal significance level, this is not to be interpreted as proving
the absence of a systematic error. Perhaps the data were insufficient in precision or in number to
establish the presence of a constant error. Especially when the calculated value for t is only slightly
short of the tabulated value, some additional data may suffice to build up the evidence for a constant
error (or the lack thereof).
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Should there be more than one known material, a weighted average of the individual differences
(x) should be taken. The value of s should be based on the combined estimate from the two or more
materials (perhaps different primary standards for bases). Should the materials differ markedly in
composition, a plot of the individual constant errors against composition should be made. If the
constant error appear to depend upon the composition, they should not be pooled in a weighted
average.
The t test is also used to judge whether a given lot of material conforms to a particular specification. If both plus and minus departures from the known value are to be guarded against, a twotailed test is involved. If departures in only one direction are undesirable, then the 10% level values
for t are appropriate for the 5% level in one direction. Similarly, the 2% level should be used to
obtain the 1% level to test the departure from the known value in one direction only; these constitute
a one-tailed test. More on this subject will be in the next section.
Sometimes just one determination is available on each of several known materials similar in
composition. A single determination by each of two procedures (or two analysts) on a series of
material may be used to test for a relative bias between the two methods, as in Example 2.4. Of
course, the average difference does not throw any light on which procedure has the larger constant
error. It only supplies a test as to whether the two procedures are in disagreement.
2.3.7
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Hypotheses About Means
Statistical methods are frequently used to give a “yes” or “no” answer to a particular question
concerning the significance of data. When performing hypothesis tests on real data, we cannot set
an absolute cutoff as to where we can expect to find no values from the population against which
we are testing data, but we can set a limit beyond which we consider it very unlikely to find a
member of the population. If a measurement is made that does in fact fall outside the specified range,
the probability of its happening by chance alone can be rejected; something beyond the randomness of the reference population must be operating. In other words, hypothesis testing is an attempt
to determine whether a given measured statistic could have come from some hypothesized population.
In attempting to reach decisions, it is useful to make assumptions or guesses about the populations
involved. Such assumptions, which may or may not be true, are called statistical hypotheses and in
general are statements about the probability distributions of the populations. A common procedure
is to set up a null hypothesis, denoted by H0, which states that there is no significant difference
between two sets of data or that a variable exerts no significant effect. Any hypothesis which differs
from a null hypothesis is called an alternative hypothesis, denoted by H1.
Our answer is qualified by a confidence level (or level of significance) indicating the degree of
certainty of the answer. Generally confidence levels of 95% and 99% are chosen to express the
probability that the answer is correct. These are also denoted as the 0.05 and 0.01 level of significance, respectively. When the hypothesis can be rejected at the 0.05 level of significance, but not at
the 0.01 level, we can say that the sample results are probably significant. If, however, the hypothesis
is also rejected at the 0.01 level, the results become highly significant.
The abbreviated table on the next page, which gives critical values of z for both one-tailed and
two-tailed tests at various levels of significance, will be found useful for purposes of reference.
Critical values of z for other levels of significance are found by the use of Table 2.26b. For a small
number of samples we replace z, obtained from above or from Table 2.26b, by t from Table 2.27,
and we replace ␴ by:
[ p N/(N ⫺ 1)] s
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Level of significance, ␣
0.10
⫺1.28
1.28
Critical values of z for
one-tailed tests
or
Critical values of z for
two-tailed tests
⫺1.645
and 1.645
0.05
or
⫺1.645
1.645
⫺1.96
and 1.96
0.01
or
⫺2.33
2.33
⫺2.58
and 2.58
0.005
or
⫺2.58
2.58
⫺2.81
and 2.81
0.002
or
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⫺2.88
2.88
⫺3.08
and 3.08
Procedures which enable us to decide whether to accept or reject hypotheses or to determine whether
observed samples differ significantly from expected results are called tests of hypotheses, tests of
significance, or rules of decision. For example, a set of z values outside the range ⫺1.96 to 1.96 (at
the 0.05 level of significance for a two-tailed test), constitute what is called the critical region or
region of rejection of the hypothesis. The set of z results inside the range ⫺1.96 to 1.96 could then
be called the region of acceptance of the hypothesis.
Example 9 In the past a method gave ␮ ⫽ 0.050%. A recent set of 10 results gave
x ⫽ 0.053% and s ⫽ 0.003%. Is everything satisfactory at a level of significance of 0.05? Of 0.01?
We wish to decide between the hypotheses:
H0: ␮ ⫽ 0.050%
and the method is working properly, and
H1: ␮ ⫽ 0.050%
and the method is not working properly.
A two-tailed test is required; that is, both tails on the distribution curve are involved:
t⫽
0.053 ⫺ 0.050
p 10 ⫺ 1 ⫽ ⫺3.00
0.003
Enter Table 2.27 for nine degrees of freedom under the column headed t0.975 for the 0.05 level of
significance, and the column t0.995 for the 0.01 level of significance. At the 0.05 level, accept H0 if
t lies inside the interval ⫺t0.975 to t0.975, that is, within ⫺2.26 and 2.26; reject otherwise. Since
t ⫽ ⫺3.00, we reject H0. At the 0.01 level of significance, the corresponding interval is ⫺3.25 to
3.25, which t lies within, indicating acceptance of H0. Because we can reject H0 at the 0.05 level
but not at the 0.01 level of significance, we can say that the sample results are probably significant
and that the method is working properly.
Let us digress a moment and consider when a two-tailed test is needed, and what a one-tailed
test implies. We “assume” that the measurements can be described by the curve shown in Fig. 2.10.
If so, then 95% of the time a sample from the specified population will fall within the indicated
range and 5% of the time it will fall outside; 2.5% of the time it is outside on the high side of the
range, and 2.5% of the time it is below the low side of the range. Our assumption implies that if ␮
does not equal the hypothesized value, the probability of its being above the hypothesized value is
equal to the probability of its being below the hypothesized value.
There will be incidences when the foregoing assumptions for a two-tailed test will not be true.
Perhaps some physical situation prevents ␮ from ever being less than the hypothesized value; it can
only be equal or greater. No results would ever fall below the low end of the confidence interval;
only the upper end of the distribution is operative. Now random samples will exceed the upper bound
only 2.5% of the time, not the 5% specified in two-tail testing. Thus, where the possible values are
restricted, what was supposed to be a hypothesis test at the 95% confidence level is actually being
performed at a 97.5% confidence level. Stated in another way, 95% of the population data lie within
the interval below ␮ ⫹ 1.65␴ and 5% lie above. Of course, the opposite situation might also occur
and only the lower end of the distribution is operative.
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Example 10 Six samples from a bulk chemical shipment averaged 77.50% active ingredient
with s ⫽ 1.45%. The manufacturer claimed 80.00%. Can this claim be supported?
A one-tailed test is required since the alternative hypothesis states that the population parameter
is equal to or less than the hypothesized value.
t⫽
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77.50 ⫺ 80.00
p 6 ⫺ 1 ⫽ 3.86
1.45
Since t0.95 ⫽ ⫺2.01, and t0.99 ⫽ ⫺3.36, the hypothesis is rejected at both the 0.05 and the 0.01
levels of significance. It is extremely unlikely that the claim is justified.
2.3.8
The Chi-square (␹ 2 ) Distribution
The ␹2 distribution describes the behavior of variances. Actually there is not a single ␹2 distribution
but a whole set of distributions. Each distribution depends upon the number of degrees of freedom
(designated variously as df, d.f., or f ) in that distribution. Table 2.28 is laid out so that the horizontal
axis is labeled with probability levels, while the vertical axis is listed in descending order of increasing number of degrees of freedom. The entries increase both as you read down and across the
table. Although Table 2.28 does not display the values for the mid-range of the distributions, at the
50% point of each distribution, the expected value of ␹2 is equal to the degrees of freedom. Estimates
of the variance are uncertain when based only on a few degrees of freedom. With the 10 samples
in Example 11, the standard deviation can vary by a large factor purely by random chance alone.
Even 31 samples gives a spread of standard deviation of 2.6 at the 95% confidence level.
Understanding the ␹2 distribution allows us to calculate the expected values of random variables
that are normally and independently distributed. In least squares multiple regression, or in calibration
work in general, there is a basic assumption that the error in the response variable is random and
normally distributed, with a variance that follows a ␹2 distribution.
Confidence limits for an estimate of the variance may be calculated as follows. For each group
of samples a standard deviation is calculated. These estimates of ␴ possess a distribution called the
␹2 distribution:
␹2 ⫽
s2
␴ /df
2
(2.15)
The upper and lower confidence limits for the standard deviation are obtained by dividing
(N ⫺ 1)s2 by two entries taken from Table 2.28. The estimate of variance at the 90% confidence
limits is for use in the entries ␹ 20.05 and ␹ 20.95 (for 5% and 95%) with N degrees of freedom.
Example 11 The variance obtained for 10 samples is (0.65)2. ␴ 2 is known to be (0.75)2. How
reliable is s2 as an estimate of ␴ 2 ?
s2 (N ⫺ 1)
s2 (N ⫺ 1)
⬍ ␴2 ⬍
2
2
␹ 0.975
␹ 0.025
(0.65)2 (10 ⫺ 1)
(0.65)2 (10 ⫺ 1)
⬍ ␴2 ⬍
19.02
2.70
0.20 ⬍ ␴ 2 ⬍ 1.43
Thus, only one time in 40 will 9s2/␴ 2 be less than 2.70 by chance alone. Similarly, only one time
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DEAN #37261
(McGHP)
RIGHT
INTERACTIVE
2.129
GENERAL INFORMATION, CONVERSION TABLES, AND MATHEMATICS
cap height
base of text
TABLE 2.28 Percentile Values for the Chi-square (␹ 2) Distribution
␹ 20.995
␹ 20.99
2
␹ 0.975
1
2
3
4
7.88
10.6
12.8
14.9
6.63
9.21
11.3
13.3
5.02
7.38
9.35
11.1
5
6
7
8
9
16.7
18.5
20.3
22.0
23.6
15.1
16.8
18.5
20.1
21.7
12.8
14.4
16.0
17.5
19.0
10
11
12
13
14
25.2
26.8
28.3
29.8
31.3
23.2
24.7
26.2
27.7
29.1
15
16
17
18
19
32.8
34.3
35.7
37.2
38.6
20
21
22
23
24
df
␹ 20.95
3.84
5.99
7.81
9.49
2
␹ 0.90
2
␹ 0.75
2
␹ 0.50
2
␹ 0.25
2
␹ 0.10
top of rh
base of rh
␹ 20.05
2
␹ 0.025
2
␹ 0.01
2
␹ 0.005
2.71
4.61
6.25
7.78
1.32
2.77
4.11
5.39
0.455
1.39
2.37
3.36
0.102
0.575
1.21
1.92
0.0158
0.211
0.584
1.06
0.0039
0.103
0.352
0.711
0.0010
0.0506
0.216
0.484
0.0002
0.0201
0.115
0.297
0.0000
0.0100
0.072
0.207
11.1
12.6
14.1
15.5
16.9
9.24
10.6
12.0
13.4
14.7
6.63
7.84
9.04
10.2
11.4
4.35
5.35
6.35
7.34
8.34
2.67
3.45
4.25
5.07
5.90
1.61
2.20
2.83
3.49
4.17
1.15
1.64
2.17
2.73
3.33
0.831
1.24
1.69
2.18
2.70
0.554
0.872
1.24
1.65
2.09
0.412
0.676
0.989
1.34
1.73
20.5
21.9
23.3
24.7
26.1
18.3
19.7
21.0
22.4
23.7
16.0
17.3
18.5
19.8
21.1
12.5
13.7
14.8
16.0
17.1
9.34
10.3
11.3
12.3
13.3
6.74
7.58
8.44
9.30
10.2
4.87
5.58
6.30
7.04
7.79
3.94
4.57
5.23
5.89
6.57
3.25
3.82
4.40
5.01
5.63
2.56
3.05
3.57
4.11
4.66
2.16
2.60
3.07
3.57
4.07
30.6
32.0
33.4
34.8
36.2
27.5
28.8
30.2
31.5
32.9
25.0
26.3
27.6
28.9
30.1
22.3
23.5
24.8
26.0
27.2
18.2
19.4
20.5
21.6
22.7
14.3
15.3
16.3
17.3
18.3
11.0
11.9
12.8
13.7
14.6
8.55
9.31
10.1
10.9
11.7
7.26
7.96
8.67
9.39
10.1
6.26
6.91
7.56
8.23
8.91
5.23
5.81
6.41
7.01
7.63
4.60
5.14
5.70
6.26
6.84
40.0
41.4
42.8
44.2
45.6
37.6
38.9
40.3
41.6
43.0
34.2
35.5
36.8
38.1
39.4
31.4
32.7
33.9
35.2
36.4
28.4
29.6
30.8
32.0
33.2
23.8
24.9
26.0
27.1
28.2
19.3
20.3
21.3
22.3
23.3
15.5
16.3
17.2
18.1
19.0
12.4
13.2
14.0
14.8
15.7
10.9
11.6
12.3
13.1
13.8
9.59
10.3
11.0
11.7
12.4
8.26
8.90
9.54
10.2
10.9
7.43
8.03
8.64
9.26
9.89
25
26
27
28
29
46.9
48.3
49.6
51.0
52.3
44.3
45.6
47.0
48.3
49.6
40.6
41.9
43.2
44.5
45.7
37.7
38.9
40.1
41.3
42.6
34.4
35.6
36.7
37.9
39.1
29.3
30.4
31.5
32.6
33.7
24.3
25.3
26.3
27.3
28.3
19.9
20.8
21.7
22.7
23.6
16.5
17.3
18.1
18.9
19.8
14.6
15.4
16.2
16.9
17.7
13.1
13.8
14.6
15.39
16.0
11.5
12.2
12.9
13.6
14.3
10.5
11.2
11.8
12.5
13.1
30
40
50
60
53.7
66.8
79.5
92.0
50.9
63.7
76.2
88.4
47.0
59.3
71.4
83.3
43.8
55.8
67.5
79.1
40.3
51.8
63.2
74.4
34.8
45.6
56.3
67.0
29.3
39.3
49.3
59.3
24.5
33.7
42.9
52.3
20.6
29.1
37.7
46.5
18.5
26.5
34.8
43.2
16.8
24.4
32.4
40.5
15.0
22.2
29.7
37.5
13.8
20.7
28.0
35.5
70
80
90
100
104.2
116.3
128.3
140.2
100.4
112.3
124.1
135.8
95.0
106.6
118.1
129.6
90.5
101.9
113.1
124.3
85.5
96.6
107.6
118.5
77.6
88.1
98.6
109.1
69.3
79.3
89.3
99.3
61.7
71.1
80.6
90.1
55.3
64.3
73.3
82.4
51.7
60.4
69.1
77.9
48.8
57.2
65.6
74.2
45.4
53.5
61.8
70.1
43.3
51.2
59.2
67.3
in 40 will 9s2/␴ 2 be greater than 19.02. Consequently, it is not unlikely that s2 is a reliable estimate
of ␴ 2.
Stated differently:
Upper limit:
␴ 2 ⫽ 9s2/2.7 ⫽ 3.3s2
Lower limit:
␴ 2 ⫽ 9s2/19.02 ⫽ 0.48s2
Ten measurements give an estimate of ␴ 2 that may be as much as 3.3 times or only about one-half
the true variance.
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DEAN #37261
LEFT
(McGHP)
2.130
2.3.9
INTERACTIVE
top of rh
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SECTION 2
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base of text
The F Statistic
The F statistic, along with the z, t, and ␹2 statistics, constitute the group that are thought of as
fundamental statistics. Collectively they describe all the relationships that can exist between means
and standard deviations. To perform an F test, we must first verify the randomness and independence
of the errors. If ␴ 21 ⫽ ␴ 22 , then s21 /s22 will be distributed properly as the F statistic. If the calculated
F is outside the confidence interval chosen for that statistic, then this is evidence that ␴ 21 ⫽ ␴ 22 .
The F statistic describes the distribution of the ratios of variances of two sets of samples. It
requires three table labels: the probability level and the two degrees of freedom. Since the F distribution requires a three-dimensional table which is effectively unknown, the F tables are presented
as large sets of two-dimensional tables. The F distribution in Table 2.29 has the different numbers
of degrees of freedom for the denominator variance placed along the vertical axis, while in each
table the two horizontal axes represent the numerator degrees of freedom and the probability level.
Only two probability levels are given in Table 2.29: the upper 5% points (F0.95) and the upper 1%
points (F0.99). More extensive tables of statistics will list additional probability levels, and they should
be consulted when needed.
It is possible to compare the means of two relatively small sets of observations when the variances
within the sets can be regarded as the same, as indicated by the F test. One can consider the
distribution involving estimates of the true variance. With s 21 determined from a group of N1 observations and s 22 from a second group of N2 observations, the distribution of the ratio of the sample
variances is given by the F statistic:
F⫽
s 21 /␴ 21
s 22 /␴ 22
(2.16)
The larger variance is placed in the numerator. For example, the F test allows judgment regarding
the existence of a significant difference in the precision between two sets of data or between two
analysts. The hypothesis assumed is that both variances are indeed alike and a measure of the
same ␴.
The fact that each sample variance is related to its own population variance means that the sample
variance being used for the calculation need not come from the same population. This is a significant
departure from the assumptions inherent in the z, t, and ␹2 statistics.
Example 12 Suppose Analyst A made five observations and obtained a standard deviation of
0.06, where Analyst B with six observations obtained sB ⫽ 0.03. The experimental variance
ratio is:
F⫽
(0.06)2
⫽ 4.00
(0.03)2
From Table 2.28 with four degrees of freedom for A and five degrees of freedom for B, the value
of F would exceed 5.19 five percent of the time. Therefore, the null hypothesis is valid, and comparable skills are exhibited by the two analysts.
As applied in Example 12, the F test was one-tailed. The F test may also be applied as a twotailed test in which the alternative to the null hypothesis is ␴ 21 ⫽ ␴ 22. This doubles the probability
that the null hypothesis is invalid and has the effect of changing the confidence level, in the above
example, from 95% to 90%.
If improvement in precision is claimed for a set of measurements, the variance for the set against
which comparison is being made should be placed in the numerator, regardless of magnitude. An
experimental F smaller than unity indicates that the claim for improved precision cannot be supported. The technique just given for examining whether the precision varies with the two different
analytical procedures, also serves to compare the precision with different materials, or with different
operators, laboratories, or sets of equipment.
short
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DEAN #37261
TABLE 2.29 F Distribution
Interpolation should be performed using reciprocals of the degrees of freedom.
Upper 5% points (F0.95)
2.131
1
2
3
4
5
6
7
8
9
10
12
15
20
24
30
40
60
⬁
120
INTERACTIVE
161
200
216
225
230
234
237
239
241
242
244
246
248
249
250
251
252
253
254
18.5
19.0
19.2
19.2
19.3
19.3
19.4
19.4
19.4
19.4
19.4
19.4
19.4
19.5
19.5
19.5
19.5
19.5
19.5
10.1
9.55
9.28
9.12
9.01
8.94
8.89
8.85
8.81
8.79
8.74
8.70
8.66
8.64
8.62
8.59
8.57
8.55
8.53
7.71
6.94
6.59
6.39
6.26
6.16
6.09
6.04
6.00
5.96
5.91
5.86
5.80
5.77
5.75
5.72
5.69
5.66
5.63
6.61
5.79
5.41
5.19
5.05
4.95
4.88
4.82
4.77
4.74
4.68
4.62
4.56
4.53
4.50
4.46
4.43
4.40
4.37
5.99
5.14
4.76
4.53
4.39
4.28
4.21
4.15
4.10
4.06
4.00
3.94
3.87
3.84
3.81
3.77
3.74
3.70
3.67
5.59
4.74
4.35
4.12
3.97
3.87
3.79
3.73
3.68
3.64
3.57
3.51
3.44
3.41
3.38
3.34
3.30
3.27
3.23
5.32
4.46
4.07
3.84
3.69
3.58
3.50
3.44
3.39
3.35
3.28
3.22
3.15
3.12
3.08
3.04
3.01
2.97
2.93
5.12
4.26
3.86
3.63
3.48
3.37
3.29
3.23
3.18
3.14
3.07
3.01
2.94
2.90
2.86
2.83
2.79
2.75
2.71
4.96
4.10
3.71
3.48
3.33
3.22
3.14
3.07
3.02
2.98
2.91
2.85
2.77
2.74
2.70
2.66
2.62
2.58
2.54
4.84
3.98
3.59
3.36
3.20
3.09
3.01
2.95
2.90
2.85
2.79
2.72
2.65
2.61
2.57
2.53
2.49
2.45
2.40
4.75
3.89
3.49
3.26
3.11
3.00
2.91
2.85
2.80
2.75
2.69
2.62
2.54
2.51
2.47
2.43
2.38
2.34
2.30
4.67
3.81
3.41
3.18
3.03
2.92
2.83
2.77
2.71
2.67
2.60
2.53
2.46
2.42
2.38
2.34
2.30
2.25
2.21
4.60
3.74
3.34
3.11
2.96
2.85
2.76
2.70
2.65
2.60
2.53
2.46
2.39
2.35
2.31
2.27
2.22
2.18
2.13
4.54
3.68
3.29
3.06
2.90
2.79
2.71
2.64
2.59
2.54
2.48
2.40
2.33
2.29
2.25
2.20
2.16
2.11
2.07
4.49
3.63
3.24
3.01
2.85
2.74
2.66
2.59
2.54
2.49
2.42
2.35
2.28
2.24
2.19
2.15
2.11
2.06
2.01
4.45
3.59
3.20
2.96
2.81
2.70
2.61
2.55
2.49
2.45
2.38
2.31
2.23
2.19
2.15
2.10
2.06
2.01
1.96
4.41
3.55
3.16
2.93
2.77
2.66
2.58
2.51
2.46
2.41
2.34
2.27
2.19
2.15
2.11
2.06
2.02
1.97
1.92
4.38
3.52
3.13
2.90
2.74
2.63
2.54
2.48
2.42
2.38
2.31
2.23
2.16
2.11
2.07
2.03
1.98
1.93
1.88
4.35
3.49
3.10
2.87
2.71
2.60
2.51
2.45
2.39
2.35
2.28
2.20
2.12
2.08
2.04
1.99
1.95
1.90
1.84
4.32
3.47
3.07
2.84
2.68
2.57
2.49
2.42
2.37
2.32
2.25
2.18
2.10
2.05
2.01
1.96
1.92
1.87
1.81
4.30
3.44
3.05
2.82
2.66
2.55
2.46
2.40
2.34
2.30
2.23
2.15
2.07
2.03
1.98
1.94
1.89
1.84
1.78
4.28
3.42
3.03
2.80
2.64
2.53
2.44
2.37
2.32
2.27
2.20
2.13
2.05
2.01
1.96
1.91
1.86
1.81
1.76
4.26
3.40
3.01
2.78
2.62
2.51
2.42
2.36
2.30
2.25
2.18
2.11
2.03
1.98
1.94
1.89
1.84
1.79
1.73
4.24
3.39
2.99
2.76
2.60
2.49
2.40
2.34
2.28
2.24
2.16
2.09
2.01
1.96
1.92
1.87
1.82
1.77
1.71
4.17
3.32
2.92
2.69
2.53
2.42
2.33
2.27
2.21
2.16
2.09
2.01
1.93
1.89
1.84
1.79
1.74
1.68
1.62
4.08
3.23
2.84
2.61
2.45
2.34
2.25
2.18
2.12
2.08
2.00
1.92
1.84
1.79
1.74
1.69
1.64
1.58
1.51
4.00
3.15
2.76
2.53
2.37
2.25
2.17
2.10
2.04
1.99
1.92
1.84
1.75
1.70
1.65
1.59
1.53
1.47
1.39
3.92
3.07
2.68
2.45
2.29
2.18
2.09
2.02
1.96
1.91
1.83
1.75
1.66
1.61
1.55
1.50
1.43
1.35
1.25
3.84
3.00
2.60
2.37
2.21
2.10
2.01
1.94
1.88
1.83
1.75
1.67
1.57
1.52
1.46
1.39
1.32
1.22
1.00
RIGHT
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
30
40
60
120
⬁
Degrees of freedom for numerator
(McGHP)
Degrees of
freedom for
denominator
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DEAN #37261
TABLE 2.29 F Distribution (Continued)
Upper 1% points (F0.99)
Degrees of
freedom for
denominator
1
2
3
4
5
6
7
8
9
10
12
15
20
24
30
40
60
⬁
120
LEFT
INTERACTIVE
4052 5000 5403 5625 5764 5859 5928 5982 6023 6056 6106 6157 6209 6235 6261 6287 6313 6339 6366
98.5
99.0
99.2
99.2
99.3
99.3
99.4
99.4
99.4
99.4
99.4
99.4
99.4
99.5
99.5
99.5
99.5
99.5
99.5
34.1
30.8
29.5
28.7
28.2
27.9
27.7
27.5
27.3
27.2
27.1
26.9
26.7
26.6
26.5
26.4
26.3
26.2
26.1
21.2
18.0
16.7
16.0
15.5
15.2
15.0
14.8
14.7
14.5
14.4
14.2
14.0
13.9
13.8
13.7
13.7
13.6
13.5
16.3
13.3
12.1
11.4
11.0
10.7
10.5
10.3
10.2
10.1
9.89
9.72
9.55
9.47
9.38
9.29
9.20
9.11
9.02
13.7
10.9
9.78
9.15
8.75
8.47
8.26
8.10
7.98
7.87
7.72
7.56
7.40
7.31
7.23
7.14
7.06
6.97
6.88
12.2
9.55
8.45
7.85
7.46
7.19
6.99
6.84
6.72
6.62
6.47
6.31
6.16
6.07
5.99
5.91
5.82
5.74
5.65
11.3
8.65
7.59
7.01
6.63
6.37
6.18
6.03
5.91
5.81
5.67
5.52
5.36
5.28
5.20
5.12
5.03
4.95
4.86
10.6
8.02
6.99
6.42
6.06
5.80
5.61
5.47
5.35
5.26
5.11
4.96
4.81
4.73
4.65
4.57
4.48
4.40
4.31
10.0
7.56
6.55
5.99
5.64
5.39
5.20
5.06
4.94
4.85
4.71
4.56
4.41
4.33
4.25
4.17
4.08
4.00
3.91
9.65
7.21
6.22
5.67
5.32
5.07
4.89
4.74
4.63
4.54
4.40
4.25
4.10
4.02
3.94
3.86
3.78
3.69
3.60
9.33
6.93
5.95
5.41
5.06
4.82
4.64
4.50
4.39
4.30
4.16
4.01
3.86
3.78
3.70
3.62
3.54
3.45
3.36
9.07
6.70
5.74
5.21
4.86
4.62
4.44
4.30
4.19
4.10
3.96
3.82
3.66
3.59
3.51
3.43
3.34
3.25
3.17
8.86
6.51
5.56
5.04
4.70
4.46
4.28
4.14
4.03
3.94
3.80
3.66
3.51
3.43
3.35
3.27
3.18
3.09
3.00
8.68
6.36
5.42
4.89
4.56
4.32
4.14
4.00
3.89
3.80
3.67
3.52
3.37
3.29
3.21
3.13
3.05
2.96
2.87
8.53
6.23
5.29
4.77
4.44
4.20
4.03
3.89
3.78
3.69
3.55
3.41
3.26
3.18
3.10
3.02
2.93
2.84
2.75
8.40
6.11
5.19
4.67
4.34
4.10
3.93
3.79
3.68
3.59
3.46
3.31
3.16
3.08
3.00
2.92
2.83
2.75
2.65
8.29
6.01
5.09
4.58
4.25
4.01
3.84
3.71
3.60
3.51
3.37
3.23
3.08
3.00
2.92
2.84
2.75
2.66
2.57
8.19
5.93
5.01
4.50
4.17
3.94
3.77
3.63
3.52
3.43
3.30
3.15
3.00
2.92
2.84
2.76
2.67
2.58
2.49
8.10
5.85
4.94
4.43
4.10
3.87
3.70
3.56
3.46
3.37
3.23
3.09
2.94
2.86
2.78
2.69
2.61
2.52
2.42
8.02
5.78
4.87
4.37
4.04
3.81
3.64
3.51
3.40
3.31
3.17
3.03
2.88
2.80
2.72
2.64
2.55
2.46
2.36
7.95
5.72
4.82
4.31
3.99
3.76
3.59
3.45
3.35
3.26
3.12
2.98
2.83
2.75
2.67
2.58
2.50
2.40
2.31
7.88
5.66
4.76
4.26
3.94
3.71
3.54
3.41
3.30
3.21
3.07
2.93
2.78
2.70
2.62
2.54
2.45
2.35
2.26
7.82
5.61
4.72
4.22
3.90
3.67
3.50
3.36
3.26
3.17
3.03
2.89
2.74
2.66
2.58
2.49
2.40
2.31
2.21
7.77
5.57
4.68
4.18
3.86
3.63
3.46
3.32
3.22
3.13
2.99
2.85
2.70
2.62
2.53
2.45
2.36
2.27
2.17
7.56
5.39
4.51
4.02
3.70
3.47
3.30
3.17
3.07
2.98
2.84
2.70
2.55
2.47
2.39
2.30
2.21
2.11
2.01
7.31
5.18
4.31
3.83
3.51
3.29
3.12
2.99
2.89
2.80
2.66
2.52
2.37
2.29
2.20
2.11
2.02
1.92
1.80
7.08
4.98
4.13
3.65
3.34
3.12
2.95
2.82
2.72
2.63
2.50
2.35
2.20
2.12
2.03
1.94
1.84
1.73
1.60
6.85
4.79
3.95
3.48
3.17
2.96
2.79
2.66
2.56
2.47
2.34
2.19
2.03
1.95
1.86
1.76
1.66
1.53
1.38
6.63
4.61
3.78
3.32
3.02
2.80
2.64
2.51
2.41
2.32
2.18
2.04
1.88
1.79
1.70
1.59
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2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
30
40
60
120
⬁
Degrees of freedom for numerator
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Curve Fitting
Very often in practice a relationship is found (or known) to exist between two or more variables. It
is frequently desirable to express this relationship in mathematical form by determining an equation
connecting the variables.
The first step is the collection of data showing corresponding values of the variables under
consideration. From a scatter diagram, a plot of Y (ordinate) versus X (abscissa), it is often possible
to visualize a smooth curve approximating the data. For purposes of reference, several types of
approximating curves and their equations are listed. All letters other than X and Y represent constants.
1. Y ⫽ a0 ⫹ a1X
Straight line
2. Y ⫽ a0 ⫹ a1X ⫹ a2X2
Parabola or quadratic curve
3. Y ⫽ a0 ⫹ a1X ⫹ a2X ⫹ a3X
2
3
Cubic curve
4. Y ⫽ a0 ⫹ a1X ⫹ a2 ⫹ · · · ⫹ an Xn
nth degree curve
As other possible equations (among many) used in practice, these may be mentioned:
5. Y ⫽ (a0 ⫹ a1X)⫺1 or 1/Y ⫽ a0 ⫹ a1X
Hyperbola
6. Y ⫽ ab
or
log Y ⫽ log a ⫹ (log b)X
Exponential curve
7. Y ⫽ aXb or
log Y ⫽ log a ⫹ b log X
Geometric curve
X
8. Y ⫽ ab ⫹ g
Modified exponential curve
9. Y ⫽ aXn ⫹ g
Modified geometric curve
X
When we draw a scatter plot of all X versus Y data, we see that some sort of shape can be
described by the data points. From the scatter plot we can take a basic guess as to which type of
curve will best describe the X 9Y relationship. To aid in the decision process, it is helpful to obtain
scatter plots of transformed variables. For example, if a scatter plot of log Y versus X shows a linear
relationship, the equation has the form of number 6 above, while if log Y versus log X shows a linear
relationship, the equation has the form of number 7. To facilitate this we frequently employ special
graph paper for which one or both scales are calibrated logarithmically. These are referred to as
semilog or log-log graph paper, respectively.
2.3.10.1 The Least Squares or Best-fit Line. The simplest type of approximating curve is a
straight line, the equation of which can be written as in form number 1 above. It is customary to
employ the above definition when X is the independent variable and Y is the dependent variable.
To avoid individual judgment in constructing any approximating curve to fit sets of data, it is
necessary to agree on a definition of a best-fit line. One could construct what would be considered
the best-fit line through the plotted pairs of data points. For a given value of X1, there will be a
difference D1 between the value Y1 and the constituent value Ŷ as determined by the calibration
model. Since we are assuming that all the errors are in Y, we are seeking the best-fit line that
minimizes the deviations in the Y direction between the experimental points and the calculated line.
This condition will be met when the sum of squares for the differences, called residuals (or the sum
of squares due to error),
冘 (Y ⫺ Yˆ ) ⬅ 冘 (D ⫹ D ⫹ · · · ⫹ D )
N
i⫽1
i
i
2
2
1
2
2
2
N
is the least possible value when compared to all other possible lines fitted to that data. If the sum
of squares for residuals is equal to zero, the calibration line is a perfect fit to the data. With a
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mathematical treatment known as linear regression, one can find the “best” straight line through
these real world points by minimizing the residuals.
This calibration model for the best-fit fit line requires that the line pass through the “centroid”
of the points (X, Y). It can be shown that:
冘 (X ⫺ X) (Y ⫺ Y)
冘 (X ⫺ X)
i
b⫽
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i
i
(2.17)
2
i
i
a ⫽ Y ⫺ bX
(2.18)
The line thus calculated is known as the line of regression of Y on X, that is, the line indicating how
Y varies when X is set to chosen values.
If X is the dependent variable, the definition is modified by considering horizontal instead of
vertical deviations. In general these two definitions lead to different least square curves.
Example 13 The following data were recorded for the potential E of an electrode, measured
against the saturated calomel electrode, as a function of concentration C (moles liter⫺1).
⫺log C
E, mV
⫺log C
E, mV
1.00
1.10
1.20
1.50
1.70
1.90
106
115
121
139
153
158
2.10
2.20
2.40
2.70
2.90
3.00
174
182
187
211
220
226
Fit the best straight line to these data; Xi represents ⫺log C, and Yi represents E. We will perform
the calculation manually, using the following tabular lay-out.
Xi
Yi
(Xi ⫺ X)
(Xi ⫺ X)2
(Yi ⫺ Y)
(Xi ⫺ X)(Yi ⫺ Y)
1.00
1.10
1.20
1.50
1.70
1.90
2.10
2.20
2.40
2.70
2.90
3.00
106
115
121
139
153
158
174
182
187
211
220
226
⫺0.975
⫺0.875
⫺0.775
⫺0.475
⫺0.275
⫺0.075
⫹0.125
0.225
0.425
0.725
0.925
1.025
0.951
0.766
0.600
0.226
0.076
0.006
0.016
0.051
0.181
0.526
0.856
1.051
⫺60
⫺51
⫺45
⫺27
⫺13
⫺8
8
16
21
45
54
60
58.5
44.6
34.9
12.8
3.6
0.6
1.0
3.6
8.9
32.6
50.0
61.5
兺Xi 23.7
兺Yi 1992
兺0
兺 5.306
兺0
兺 312.6
X ⫽ 1.975
Y ⫽ 166
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Now substituting the proper terms into Equation 17, the slope is:
b⫽
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312.6
⫽ 58.91
5.306
and from Equation 18, and substituting the “centroid” values of the points (X, Y), the intercept is:
a ⫽ 166 ⫺ 58.91(1.975) ⫽ 49.64
The best-fit equation is therefore:
E ⫽ 49.64 ⫺ 58.91 log C
2.3.10.2 Errors in the Slope and Intercept of the Best-fit Line. Upon examination of the plot
of pairs of data points, the calibration line, it will be obvious that the precision involved in analyzing
an unknown sample will be considerably poorer than that indicated by replicate error alone. The
scatter of these original points about the calibration line is a good measure of the error to be expected
in analyzing an unknown sample. And this same error is considerably larger than the replication
error because it will include other sources of variability due to a variety of causes. One possible
source of variability might be the presence of different amounts of an extraneous material in the
various samples used to establish the calibration curve. While this variability causes scatter about
the calibration curve, it will not be reflected in the replication error of any one sample if the sample
is homogeneous.
The scatter of the points around the calibration line or random errors are of importance since the
best-fit line will be used to estimate the concentration of test samples by interpolation. The method
used to calculate the random errors in the values for the slope and intercept is now considered. We
must first calculate the standard deviation sY/X , which is given by:
sY/X ⫽
q
冘 (Y ⫺ Y)ˆ
2
i
i
(2.19)
N⫺2
Equation 19 utilizes the Y-residuals, Yi ⫺ Ŷ, where Ŷi are the points on the calculated best-fit line
or the fitted Yi values. The appropriate number of degrees of freedom is N ⫺ 2; the minus 2 arises
from the fact that linear calibration lines are derived from both a slope and an intercept which leads
to a loss of two degrees of freedom.
Now we can calculate the standard deviations for the slope and the intercept. These are
given by:
sb ⫽
sY/X
冘 (X ⫺ X)
q
(2.20)
2
i
i
sa ⫽ sY/X
q
冘X
N 冘 (X ⫺ X)
2
i
i
i
2
(2.21)
i
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The confidence limits for the slope are given by b ⫾ tb , where the t-value is taken at the desired
confidence level and (N ⫺ 2) degrees of freedom. Similarly, the confidence limits for the intercept
are given by a ⫾ tsa . The closeness of x̂ to xi is answered in terms of a confidence interval for x0
that extends from an upper confidence (UCL) to a lower confidence (LCL) level. Let us choose 95%
for the confidence interval. Then, remembering that this is a two-tailed test (UCL and LCL), we
obtain from a table of Student’s t distribution the critical value of tc (t0.975) and the appropriate
number of degrees of freedom.
Example 14 For the best-fit line found in Example 13, express the result in terms of confidence
intervals for the slope and intercept. We will choose 95% for the confidence interval.
The standard deviation sY/X is given by Equation 19, but first a supplementary table must be
constructed for the Y residuals and other data which will be needed in subsequent equations.
Ŷ
(Yi ⫺ Ŷ)
(Yi ⫺ Ŷ)2
X 2i
108.6
114.4
120.3
138.0
149.8
161.6
173.4
179.2
191.0
208.7
220.5
226.4
2.55
⫺0.56
⫺0.67
⫺1.00
⫺3.21
3.57
⫺0.65
⫺2.76
4.02
⫺2.30
0.48
0.40
6.50
0.31
0.45
1.00
10.32
12.94
0.42
7.61
16.16
5.30
0.23
0.16
1.00
1.21
1.44
2.25
2.89
3.61
4.41
4.84
5.76
7.29
8.41
9.00
兺 61.20
兺 52.11
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Now substitute the appropriate values into Equation 19 where there are 12 ⫺ 2 ⫽ 10 degrees of
freedom:
sX/Y ⫽
q
61.20
⫽ 2.47
10
We can now calculate sb and sa from Equations 20 and 21, respectively:
sb ⫽
sY/X
p 5.31
⫽ 1.07
and
sa ⫽ 2.47
q
52.11
⫽ 2.23
12(5.306)
Now, using a two-tailed value for Student’s t:
b ⫾ tsb ⫽ 58.91 ⫾ 2.23(1.07) ⫽ 58.91 ⫾ 2.39
a ⫾ tsa ⫽ 49.64 ⫾ 2.23(2.23) ⫽ 49.64 ⫾ 4.97
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The best-fit equation expressed in terms of the confidence intervals for the slope and intercept is:
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E ⫽ (49.64 ⫾ 5.0) ⫺ (58.91 ⫾ 2.43) log C
To conclude the discussion about the best-fit line, the following relationship can be shown to
exist among Y, Ŷ, and Y:
冘 (Y ⫺ Y) ⫽ 冘 (Yˆ ⫺ Y) ⫹ 冘 (Y ⫺ Yˆ )
N
i⫽1
N
i
2
i⫽1
N
i
2
i⫽1
i
i
2
(2.22)
The term on the left-hand side is a constant and depends only on the constituent values provided by
the reference laboratory and does not depend in any way upon the calibration. The two terms on the
right-hand side of the equation show how this constant value is apportioned between the two quantities that are themselves summations, and are referred to as the sum of squares due to regression
and the sum of squares due to error. The latter will be the smallest possible value that it can possibly
be for the given data.
2.3.11
Control Charts
It is often important in practice to know when a process has changed sufficiently so that steps may
be taken to remedy the situation. Such problems arise in quality control where one must, often
quickly, decide whether observed changes are due to simple chance fluctuations or to actual changes
in the amount of a constituent in successive production lots, mistakes of employees, etc. Control
charts provide a useful and simple method for dealing with such problems.
The chart consists of a central line and two pairs of limit lines or simply of a central line and
one pair of control limits. By plotting a sequence of points in order, a continuous record of the
quality characteristic is made available. Trends in data or sudden lack of precision can be made
evident so that the causes may be sought.
The control chart is set up to answer the question of whether the data are in statistical control,
that is, whether the data may be retarded as random samples from a single population of data. Because
of this feature of testing for randomness, the control chart may be useful in searching out systematic
sources of error in laboratory research data as well as in evaluating plant-production or controlanalysis data.1
To set up a control chart, individual observations might be plotted in sequential order and then
compared with control limits established from sufficient past experience. Limits of ⫾1.96␴ corresponding to a confidence level of 95%, might be set for control limits. The probability of a future
observation falling outside these limits, based on chance, is only 1 in 20. A greater proportion of
scatter might indicate a nonrandom distribution (a systematic error). It is common practice with
some users of control charts to set inner control limits, or warning limits, at ⫾1.96␴ and outer
control limits of ⫾3.00␴. The outer control limits correspond to a confidence level of 99.8%, or a
probability of 0.002 that a point will fall outside the limits. One-half of this probability corresponds
to a high result and one-half to a low result. However, other confidence limits can be used as well;
the choice in each case depends on particular circumstances.
Special attention should be paid to one-sided deviation from the control limits, because systematic
errors more often cause deviation in one direction than abnormally wide scatter. Two systematic
errors of opposite sign would of course cause scatter, but it is unlikely that both would have entered
at the same time. It is not necessary that the control chart be plotted in a time sequence. In any
1
G. Wernimont, Ind. Eng. Chem., Anal. Ed. 18:587 (1946); J. A. Mitchell, ibid. 19:961 (1947).
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situation where relatively large numbers of units or small groups are to be compared, the control
chart is a simple means of indicating whether any unit or group is out of line. Thus laboratories,
production machines, test methods, or analysts may be put arbitrarily into a horizontal sequence.
Usually it is better to plot the means of small groups of observations on a control chart, rather
than individual observations. The random scatter of averages of pairs of observations is 1/(2)1/2 ⫽
0.71 as great as that of single observations, and the likelihood of two “wild” observations in the
same direction is vanishing small. The groups of two to five observations should be chosen in such
a way that only change variations operate within the group, whereas assignable causes are sought
for variations between groups. If duplicate analyses are performed each day, the pairs form logical
groups.
Some measure of dispersion of the subgroup data should also be plotted as a parallel control
chart. The most reliable measure of scatter is the standard deviation. For small groups, the range
becomes increasingly significant as a measure of scatter, and it is usually a simple matter to plot the
range as a vertical line and the mean as a point on this line for each group of observations.
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Bergmann, B., B. von Oepen, and P. Zinn, Anal. Chem., 59:2532 (1987).
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Dixon, W. J., and F. J. Massey, Introduction to Statistical Analysis, McGraw-Hill, New York,
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