The SI modern metric system
The SI modern metric system of measurement is the language universally used in
science, the dominant language of international commerce and trade.
The International System of Units (SI)
The SI is founded on seven SI base units for seven base quantities
assumed to be mutually independent.
SI base quantities and their units
Base quantity
length
mass
time
electric current
thermodynamic temperature
amount of substance
luminous intensity
Unit name
meter
kilogram
second
ampere
kelvin
mole
candela
Symbol
m
kg
s
A
K
mol
cd
Definitions of the SI base units
Unit of length: meter
The meter is the length of the path travelled by light in
vacuum during a time interval of 1/299 792 458 of a
second.
Unit of mass: kilogram
The kilogram is the unit of mass; it is equal to the mass
of the international prototype of the kilogram
Unit of time: second
Unit of electric current: ampere
The second is the 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.
The ampere is that constant current which, if
maintained in two straight parallel conductors of infinite
length, of negligible circular cross-section, and placed 1
meter apart in vacuum, would produce between these
conductors a force equal to 2 x 10-7 newton per meter
of length.
The kelvin, unit of thermodynamic temperature, is the
Unit of thermodynamic temperature: kelvin fraction 1/273.16 of the thermodynamic temperature of
the triple point of water.
Unit of amount of substance: mole
The mole is the amount of substance of a system which
contains as many elementary entities as there are
atoms in 0.012 kilogram of carbon 12; its symbol is
"mol.„ (?
NA = 6.02×1023 1/mol
When the mole is used, the elementary entities must be
specified and may be atoms, molecules, ions,
electrons, other particles, or specified groups of such
particles.
SI derived units
Other quantities, called derived quantities, are defined in terms of the seven base
quantities via a system of quantity equations. The SI derived units for these derived
quantities are obtained from these equations and the seven SI base units. Examples
of such SI derived units are given in the table below.
Examples of SI derived units
Derived quantity
area
volume
speed, velocity
acceleration
mass density
current density
magnetic field
Unit name
Symbol
square meter
m2
cubic meter
m3
meter per second
m/s
meter per second squared
m/s2
kilogram per cubic meter
kg/m3
ampere per square meter
A/m2
ampere per meter
A/m
SI derived units with special names and symbols
frequency
force
pressure, stress
energy, work,
power
electric charge
hertz
newton
pascal
joule
watt
coulomb
Hz
N
Pa
J
W
C
s-1
m·kg·s-2
m-1·kg·s-2
m2·kg·s-2
m2·kg·s-3
s·A
Prefixes
The 20 SI prefixes used to form decimal multiples and submultiples of SI units are
given in the table below.
Factor
1024
1021
1018
1015
1012
109
106
103
102
101
Name
yotta
zetta
exa
peta
tera
giga
mega
kilo
hecto
deka
Symbol
Y
Z
E
P
T
G
M
k
h
da
Factor
10-1
10-2
10-3
10-6
10-9
10-12
10-15
10-18
10-21
10-24
Name
deci
centi
milli
micro
nano
pico
femto
atto
zepto
yocto
Symbol
d
c
m
µ
n
p
f
a
z
y
Exercise:
Express thickness of a cell membrane d = 0.000000009m in centi, milli, micro and
nano-meter.
d = 0.000000009 m =
910-9m = 910-710-2m =0.0000009 mm or 910-7cm
d = 0.000000009 m =
910-9m = 910-610-3 =0.000009 mm or 910-6mm
d = 0.000000009 m =
910-9m = 910-310-6 =0.009 μm
d = 0.000000009 m =
d = 0.000000009 =910-9m = 9 nm
Exercise:
Express the density d of the steraic acid in:
a) g/dm3,
b) kg/dm3
c) g/m3,
d) kg/m3,
if in g/cm3 it is 0.847 at 70 °C.
1 dm3 = 1000 cm3 or 1 cm3 = 0.001 dm3 = 10-3 dm3
a) d = 0.847 g/10-3 dm3 = 0.847 × 103 g/dm3 = 847 g/dm3
1 kg = 1000 g = 103 g or 1 g = 0.001 kg = 10-3 kg
b) d = 847 g/dm3 = 847 ×10-3kg/dm3 = 0.847 × kg/dm3 = 0.847 kg/dm3
1 m3 = 1000000 cm3 = 106 cm3 or 1 cm3 = 0.000001 = 10-6 m3
c) d = 0.847 g/cm3 = 0.847g/10-6m3 = 847× 103 g/m3
1 m3 = 1000000 cm3 = 106 cm3 or 1 cm3 = 0.000001 = 10-6 m3
d) d = 0.847 g/cm3 = 0.847×10-3kg/10-6m3 = 847 kg/m3
*Functions and graphs*
1. Linear function:
y(x) = a·x +b
b = 0.05
(y -intercept)
The slope a equals tangent of the angle α:
a = tanα
y = 125×10-5·x + 0.05
y - INTERCEPT
α
Δy
Δx
protractor
2. Quadratic function: y(x) = ax2 + bx + c
E = ½ mv2
parabola
3. Inverse proportionality – simplified hyperbolic
function:
y · x = const.
hyperbola
4. Exponential function: y(x) = ax
Exponential functions occur, for instance, in the study of
the growth of certain populations of bacteria. As an
illustration it might be observed experimentally that the
number of bacteria in a culture doubles every hour.
If 1000 bacteria are present at the start of the
experiment, than the experimenter would obtain the
readings presented below, where t is the time in hours
and y(t) is the bacteria count at time t.
y(t) = 1000·2x
5. Natural exponential function !!!
y(x) = y0ea·x
The number “e” (Euler’s number) is a limit of a certain
arithmetic expression. It appears naturally in the
investigation of many physical phenomena and equals
approximately: 2.718... .
The factor “a” in the
exponent is usually
negative.
y(x) = y0e-a·x
y(x) = y0e-a·x
Examples:
•Attenuation of electromagnetic ionising radiation: I = I0e-µx
µ – linear attenuation coefficient
•Radiation therapy: N = N0e-D/D0
D0 - is called the mean lethal dose
N - number of targets (for instance cells) surviving irradiation.
•Transmittance vs. solution concentration: T = e-ελdc
ελ stands for the absorptivity
•Stress relaxation : S = S0e-t/τ or S = S0e(-1/τ)·t
τ – stress relaxation time
How to determine the factor a? (µ, D0, τ, ελ)
y(x) = y0e-a·x
?
y0=10
if a ·x = 1 then y(x) = y0/e
e = 2.718...
y0/e =10/2.71 ≈ 3.69
x=1.5
a ·x=1, a=1/x = 1/1.5 ≈ 0.67
y(x) = 10·e-?·x
y(x) = 10·e-0.67·x
Laws of exponents and radicals:
a0 = 1
a1 = a
am × an = am+n
102 × 104 = 106
(am)n = am×n
(23)2 = 26 = 64
Exercises. Simplify expressions:
2x 
2 3
1.
2.
3.
4x

8
a
2x 3 3 x 2
x 
10
 10
108
6 x 
2x 
3 2
4.

2 3
a a  a
m
1
2
 10 

2 3
 4x 
 2x 
6.
2
a 
m n
4
8
n
am
mn

a
n
a
1 5
a  3a 2 4a 7  
5.
6
4
1
 n
a

2 3
4
n

exponential function
 am n
mn
5. Logarithmic function
The inverse of the exponential function, y = ax , is called the logarythmic
function:
x = logay only if y = ax
(the logarithm of y
with base a)
EXERCISES:
Change the following equations to the logarithmic form:
a)
43 = 64 (log464=3),
b) 27 = 128,
c)
10-3=0.001,
d) m = t e.
Change the following equations to exponential form:
a) log51=0,
b) log327=3,
c) logak=n,
d) log100.01=-2.
Find value of the logarithms:
a) log101000 =
b) log216 =
c) log553 =
d) log100.001 =
e) logee-2 =
f) log1010 =
Laws of logarithms
logex = lnx
Task: change the exponential function: y = y0·e-a·x to the liner form
(y=b+ax).
y = y0·e-a·x /ln
lny =ln( y0·e-a·x )
lny =lny0 + lne-a·x
lny =lny0 + (-ax)lne
y = b - ax
lne = logee = 1
lny = lny0 - ax
y = b - ax
y = y0·e-a·x
lny = lny0 - ax
a = tan α
EXAMPLES
1. Using the Richter scale
formula:
find the magnitude of an
earthquake that has intensity:
a) 100 times that of I0:
b) 10,000 times that of I0,
c) 100,000 times that of I0,
d) 1000,000 times that of I0.
2. Chemists use a number denoted by pH to describe quantitatively the acidity
or basicity of solutions.
By definition:
pH = - log10[H+]
Approximate the pH of each substance:
a) sea water, [H+] ≈ 5.0×10-9
b) carrots, [H+] ≈ 1.0×10-5
c) vinegar,
[H+]
≈
6×10-3
log105 ≈ 0.70
log106 ≈ 0.78
d) soap, [H+] ≈ 1.0×10-9
e) determine the concentration of [H+] ions in the stomach acid if its pH = 1.5
3.16×10-2
3. Calculation of sound intensity level (SIL) in decibels is based on the formula:
where I0 is the intensity of threshold of
hearing at 1000 Hz
Find SIL if:
a) I is 10 times as great as I0,
b) I is 100,000 (this is the intensity level of the average voice) times as great as I0
THE END