A new definition for the mole based
on the Avogadro constant;
a journey from physics to chemistry
Martin Milton
NPL
The Royal Society, London
24th January 2012
Outline
•
The development of today’s understanding of:
– the quantity amount of substance,
– the unit mol, and
– the Avogadro constant.
•
How have we come to a definition whereby we
know the mass of a mole, but not the number of
entities in it?
– Is there justification for a change?
– What would the consequences be?
•
How is the mole realised in practice?
– some examples of the best uncertainties currently
achievable
The concept “amount of substance”
Boyle’s Law (1662)
“For a fixed amount of gas kept at a fixed temperature, P and V are inversely
proportional”
Stoichiometry (Lavoisier)
“the relationship between the amounts of substance that react together, and
the products that are formed”
Law of Multiple Proportions– (Dalton 1803)
“when elements combine, they do so in a ratio of small whole numbers”
Law of Definite Proportions (Proust 1806)
“a chemical compound always contains exactly the same proportion of
elements by mass”
Avogadro’s Law (1811)
“Equal volumes of ideal or perfect gases, at the same temperature and
pressure, contain the same number of particles, or molecules.”
The gram-molecule
Kilogrammolekuel and g-Molekuel used by Ostwald
and Nernst in their text books in 1893.
Abbreviation to Mol recorded by Nernst.
gramme-molecule - First used in English in the
Encyclopaedia Britannica (1893).
mole – First used in English in the translation of
Ostwald’s “Principles of Inorganic Chemistry”
(1902).
The gram-molecule in use
“On the Motion of Small Particles Suspended in a Stationary Liquid, as Required by
the Molecular Kinetic Theory of Heat” Einstein, 1905
• Van’t Hoff’s Law for the osmotic pressure Π V = z R T
Where z gram-molecules is dissolved in a a volume V
Let z=n/N where
n suspended particles are present and
N signifies the actual number of molecules contained in a gram-molecule
RT
• The Stokes-Sutherland-Einstein formula
aN A =
6πη D
The gram-molecule in use
“On the Motion of Small Particles Suspended in a Stationary Liquid, as Required by
the Molecular Kinetic Theory of Heat” Einstein, 1905
• Van’t Hoff’s Law for the osmotic pressure Π V = z R T
Where z gram-molecules is dissolved in a a volume V
Let z=n/N where
n suspended particles are present and
N signifies the actual number of molecules contained in a gram-molecule
RT
• The Stokes-Sutherland-Einstein formula
aN A =
6πη D
“A new determination of molecular dimensions” Einstein, 1906
• Calculate the change in viscosity when spheres of radius a are dissolved in
a solvent of viscosity η
*
η
= (1 + 2.5φ )
η
The total volume of dissolved material per unit volume of solvent
4
3
φ = πa 3 ρ
N
M
The gram-molecule defined
Perrin (1909)
“It has become customary to name as the gram-molecule of a
substance, the mass of the substance which in the gaseous
state occupies the same volume as 2 grams of hydrogen
measured at the same temperature and pressure.
Avogadro's proposition is then equivalent to the following:
Any two gram-molecules contain the same number of
molecules.
This invariable number N is a universal constant, which may
appropriately be designated Avogadro's Constant."
J. B. Perrin, “Mouvement brownien et réalité moléculaire”,
Annales de chimie et de physiqe VIII 18, 5-114 (1909).
trans: F. Soddy “Brownian Movement and Molecular Reality”,
Taylor and Francis (London) 1910.
The “Mol” in use
Stille (PTB) explained in 1955 that Mol was being used in two conceptually
different ways.
•
The ”chemical mass unit” for example
1 mol = 22.991 g of sodium, or
1 mol = 58.448 g of sodium chloride
•
The ”number of moles” ( from Molzahl ) given by the equation:
l=ν/L
ν = number of entities
L = {NA}
•
Stille advocated the use of the Molzahl as a dimensionless quantity rather
than the use of the quantity Stoffmenge (literally “amount of substance”)
1 Mol is “the Stoffmenge that contains as many entities as Ar(O) g of
atomic oxygen”.
Stille “Messen und Rechnen in der Physik” 1955
Amount of substance
Guggenheim
•
..”for special problems it may be advantageous to increase
the number of fundamental quantities above the usual
number. It can sometimes be useful in dimensional
analysis to regard the number of atoms as having
dimensions different from a pure number”
– Guggenheim, E. A. 1942 Units and Dimensions
• Phil. Mag. 33 pp479-496.
•
“This quantity was first named “Stoffmenge” in German
and the English translation is amount of substance”
– Guggenheim, E. A. 1961 The Mole and Related Quantities
• J Chem Ed 38 86-87.
The 1971 definition of the mole
– “The mole is the amount of substance of a system that
contains as many elementary entities as there are atoms
in 0.012 kilogramme of carbon 12.
When the mole is used, the elementary entities must be specified
and may be atoms, molecules, ions, electrons, or other particles,
or specified groups of such particles”.
– 14th CGPM, 1971
• resolved the confusion arising from the use of both
• g-mol and kg-mol
• 12C and 16O basis
• introduced dimensional analysis to chemistry.
McGlashan, Metrologia, 1995, 31, 447-455.
The atomic mass scale
The N measured atomic masses are related by
the N-1 ratios Ar(X)/Ar(Y).
So we fix the value of the Nth ratio Ar(12C).
m(12C)
mu
Ar(12C)
m(X)
Ar(X)
atomic
level
Fixed
value
me
nit
ss u
ma
of
ca
At o
mic
Ma
ss
Ma
ss
of
th
ee
lec
t
rbo
n-1
2
ron
Atomic masses and fundamental constants
Ar(e)/Ar(12C)
m(12C)
Mass
Ar(12C)
mu
the mole (present definition)
M(12C)
macroscopic
Ar(12C)
Mu
10-3 kg mol-1
atomic
level
Fixed
value
me
Ar(e)/Ar(12C)
m(12C)
Mass
Ar(12C)
mu
the mole (present definition)
M(12C)
macroscopic
Ar(12C)
NA
atomic
level
Fixed
value
me
Ar(e)/Ar(12C)
m(12C)
Mass
Mu
NA
Ar(12C)
mu
Why change
the definition of the mole?
•
There is very little initiative for any change from the
communities of users of the mole.
– there is momentum behind the proposal for a “new SI”
– which could include a fixed value for NA
A possible rationale for change
• The mole has been derived from the gramme-molecule
– the amount of substance of 12g of 12C.
– We know the exact mass of a mole (of 12C),
– do not know the exact number of entities
NA has some uncertainty
Is this sufficient to motivate a change?
the mole (present definition)
M(12C)
macroscopic
Ar(12C)
NA
atomic
level
Fixed
value
me
Ar(e)/Ar(12C)
m(12C)
Mass
Mu
NA
Ar(12C)
mu
the mole (new definition)
M(12C)
macroscopic
Fixing NA means that
another quantity in this
system has to be
determined
experimentally.
atomic
level
Fixed
value
me
Ar(e)/Ar(12C)
Ar(12C)
NA
m(12C)
Mass
Mu
NA
Ar(12C)
mu
A new definition for the mole
•
The proposed new definition would reverse the
present definition
– specify the number of entities in one mole
• equal to NA exactly.
– some uncertainty in the mass of one mole
• one mole of carbon-12 = 12g +/- u(α2).
•
•
•
The molar masses and the atomic masses will have
the same (relative) uncertainties.
A single entity will be an exact amount of substance.
Both approaches will be the same in practice
• to within +/- u(α2)
Possible definition 201X ?
201X
– “The mole is the unit of amount of substance of a
specified elementary entity, which may be an atom,
molecule, ion, electron, any other particle or a specified
group of such particles; its magnitude is set by fixing the
numerical value of the Avogadro constant to be equal to
exactly 6.022 14X 1023 when it is expressed in the unit
mol-1.”
1971
– “The mole is the amount of substance of a system that
contains as many elementary entities as there are atoms
in 0.012 kilogramme of carbon 12.
When the mole is used, the elementary entities must be specified
and may be atoms, molecules, ions, electrons, or other particles,
or specified groups of such particles”.
The debate about a new
definition for the mole
•
Many users are confused about the existing use of the mole.
•
The mole has always been used in conceptually different, but
equivalent ways
“chemical mass unit”
“number of moles”
n= m / Ar(X) Mu
l = ν / {NA}
•
•
“amount of substance”
n= ν / NA
Much of the discussion originates from authors who believe that one
of these is correct to the exclusion of the others.
Would a change in the definition put an end to this discussion?
The Avogadro constant
Invention of new physical
methods: diffusion, Brownian
motion, oil drop
Improvement in X-ray
wavelength measurements
Atomic weight and chemical
purity problems with Silicon
U(MM) contributes 61% of the
published uncertainty of the
2003 natural Si result
Becker, Rep Prog Phys 2001
High accuracy measurements (I)
the composition of the atmosphere
Compound
CIPM 81/91 formula
Giacomo et al (1982)
N2
0.78101
Revised values (2004)
0.78082 ± 0.00012
(Calculated by difference)
Measurement Nitrogen mole fraction in dry
air directly by GC/TCD
To be published 2011
0.780 93 ± 0.00006
Measured
Pure CH4
Pure CO2
Pure Ar
Pure O2
10 % CH4/N2
O2
0.20939
Ar
0.00917
CO2
0.00040
Metrologia 18 (1982) 33-40
0.20945 ± 0.00012
NIST (1970)
0.209 45 ± 0.00012
NIST (1970)
0.009332 ± 0.000006
Kim et al (2004)
0.009 331 ± 0.000006
0.000369 ± 0.000001
0.00040
5 % CO2/N2
0.7 % CH4/N2
0.045 % CH4/N2
0.43 % CO2/N2
0.028 % CH4/N2
0.038 % CO2 + 0.00016 % CH4 + 0.93 % Ar
+ 20.9 % O2 + N2 Balance
Metrologia 41 (2004) 387–395
expanded uncertainties (k=2)
Relative uncertainty of 60 parts-per-million achieved
– with respect to standards prepared gravimetrically
Courtesy of
Dr Jin Seog Kim, KRISS, Korea
Pure N2
High accuracy measurements (I)
the composition of the atmosphere
Compound
CIPM 81/91 formula
Giacomo et al (1982)
N2
0.78101
Revised values (2004)
0.78082 ± 0.00012
(Calculated by difference)
Measurement Nitrogen mole fraction in dry
air directly by GC/TCD
To be published 2011
0.780 93 ± 0.00006
Measured
Pure CH4
Pure CO2
Pure Ar
Pure O2
10 % CH4/N2
O2
0.20939
Ar
0.00917
CO2
0.00040
Metrologia 18 (1982) 33-40
0.20945 ± 0.00012
NIST (1970)
0.209 45 ± 0.00012
NIST (1970)
0.009332 ± 0.000006
Kim et al (2004)
0.009 331 ± 0.000006
0.000369 ± 0.000001
0.00040
5 % CO2/N2
0.7 % CH4/N2
0.045 % CH4/N2
0.43 % CO2/N2
0.028 % CH4/N2
0.038 % CO2 + 0.00016 % CH4 + 0.93 % Ar
+ 20.9 % O2 + N2 Balance
Metrologia 41 (2004) 387–395
expanded uncertainties (k=2)
Relative uncertainty of 60 parts-per-million achieved
– with respect to standards prepared gravimetrically
Courtesy of
Dr Jin Seog Kim, KRISS, Korea
Pure N2
High accuracy measurements (I)
the composition of the atmosphere
Compound
CIPM 81/91 formula
Giacomo et al (1982)
N2
0.78101
Revised values (2004)
0.78082 ± 0.00012
(Calculated by difference)
Measurement Nitrogen mole fraction in dry
air directly by GC/TCD
To be published 2011
0.780 93 ± 0.00006
Measured
Pure CH4
Pure CO2
Pure Ar
Pure O2
10 % CH4/N2
O2
0.20939
Ar
0.00917
CO2
0.00040
Metrologia 18 (1982) 33-40
0.20945 ± 0.00012
NIST (1970)
0.209 45 ± 0.00012
NIST (1970)
0.009332 ± 0.000006
Kim et al (2004)
0.009 331 ± 0.000006
0.000369 ± 0.000001
0.00040
5 % CO2/N2
0.7 % CH4/N2
0.045 % CH4/N2
0.43 % CO2/N2
0.028 % CH4/N2
0.038 % CO2 + 0.00016 % CH4 + 0.93 % Ar
+ 20.9 % O2 + N2 Balance
Metrologia 41 (2004) 387–395
expanded uncertainties (k=2)
Relative uncertainty of 60 parts-per-million achieved
– with respect to standards prepared gravimetrically
Courtesy of
Dr Jin Seog Kim, KRISS, Korea
Pure N2
High accuracy measurements (II)
highly pure metals
“Raw”
Zn 99.995%
Vacuum distilled
Zn 99.99995%
43.0 mg/kg
16 determined
impurities
0.5 mg/kg
38.5 kg/mm2
Vickers
micro hardness
32.6 kg/mm2
BAM-M601
Cd
Fe
Cu
Tl
Pb
w
0.55
2.20
1.89
2.25
15.7
[mg/g]
± 0.06
± 0.09
± 0.11
± 0.09
± 0.3
Courtesy of
Dr Heinrich Kipphardt,
BAM, Germany
High accuracy measurements (II)
highly pure metals
“Raw”
Zn 99.995%
Vacuum distilled
Zn 99.99995%
43.0 mg/kg
16 determined
impurities
0.5 mg/kg
38.5 kg/mm2
Vickers
micro hardness
32.6 kg/mm2
BAM-M601
Cd
Fe
Cu
Tl
Pb
w
0.55
2.20
1.89
2.25
15.7
[µg/g]
± 0.06
± 0.09
± 0.11
± 0.09
± 0.3
Courtesy of
Dr Heinrich Kipphardt,
BAM, Germany
Summary
•
The mole and the Avogadro constant
• Emergence of ideas of stoichiometry and thermodynamic
ensemble (18th and 19th centuries)
• Accurate chemical measurement (21st century)
•
The mole has been used in conceptually different
ways
• chemical mass unit
• number of moles
• amount of substance
•
At present, we know the mass of a mole (of 12C),
but not the number of entities.
– is there sufficient momentum behind proposals to change?
– where should u(α2) lie?
Acknowledgements
•
•
Dr Bernd Güttler (PTB)
Prof Ian Mills
The National Measurement System is the
UK’s national infrastructure of measurement