A new definition for the mole based on the Avogadro ... · A new definition for the mole based on...

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”

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.”

Law of Multiple Proportions– (Dalton 1803)

“when elements combine, they do so in a ratio of small whole numbers”

Boyle’s Law (1662)

“For a fixed amount of gas kept at a fixed temperature, P and V are inversely

proportional”

Law of Definite Proportions (Proust 1806)

“a chemical compound always contains exactly the same proportion of

elements by mass”

Stoichiometry (Lavoisier)

“the relationship between the amounts of substance that react together, and

the products that are formed”

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).

Kilogrammolekuel and g-Molekuel used by Ostwald

and Nernst in their text books in 1893.

Abbreviation to Mol recorded by Nernst.

The gram-molecule

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

• The Stokes-Sutherland-Einstein formula

D

RTaN

A

πηπηπηπη6====

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

• The Stokes-Sutherland-Einstein formula

D

RTaN

A

πηπηπηπη6====

“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

The total volume of dissolved material per unit volume of solvent

)5.21(

*

φφφφηηηη

ηηηη++++====

M

Na ρρρρππππφφφφ 3

3

4====

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 gram-molecule defined

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

• 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

ν = number of entities

L = {NA}

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

McGlashan, Metrologia, 1995, 31, 447-455.

• resolved the confusion arising from the use of both

• g-mol and kg-mol

•12C and 16O basis

• introduced dimensional analysis to chemistry.

The atomic mass scale

m(12C)mu m(X)

Ar(12C) Ar(X)

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).

Atomic masses and fundamental constants

atomic

levelm(12C)me

Ar(e)/Ar(12C)

mu

Ar(12C)

MassFixedvalue

Mas

s o

f th

e el

ectr

on

Mas

s o

f ca

rbo

n-1

2

Ato

mic

mas

s u

nit

the mole (present definition)

m(12C)

M(12C)

me mu

Mu

Ar(12C)

Ar(12C)

Fixedvalue

macroscopic

atomic

level

Mass

10-3 kg mol-1

Ar(e)/Ar(12C)


m(12C)

M(12C)

NA

me mu

Mu

Ar(12C)

Ar(12C)

NA

Fixedvalue

macroscopic

atomic

level

Mass

Ar(e)/Ar(12C)

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?


m(12C)

M(12C)

NA

me mu

Mu

Ar(12C)

Ar(12C)

NA

Fixedvalue

macroscopic

atomic

level

Mass

Ar(e)/Ar(12C)

the mole (new definition)

m(12C)

M(12C)

NA

me mu

Mu

NA

Fixedvalue

Fixing NA means that

another quantity in this

system has to be

determined

experimentally.

Ar(12C)

Ar(12C)

macroscopic

atomic

level

Mass

Ar(e)/Ar(12C)

• 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)

A new definition for the mole

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 specifiedand 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

• 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?

n= m / Ar(X) Mu l = ν / {NA} n= ν / NA

“chemical mass unit” “number of moles” “amount of substance”

The Avogadro constant

Becker, Rep Prog Phys 2001

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

High accuracy measurements (I)

the composition of the atmosphereMeasurement Nitrogen mole fraction in dry

air directly by GC/TCD Pure NPure NPure NPure N22220.038 % CO0.038 % CO0.038 % CO0.038 % CO2222 + 0.00016 % CH+ 0.00016 % CH+ 0.00016 % CH+ 0.00016 % CH4444 + 0.93 % + 0.93 % + 0.93 % + 0.93 % ArArArAr+ 20.9 % O+ 20.9 % O+ 20.9 % O+ 20.9 % O2222 + N+ N+ N+ N2222 BalanceBalanceBalanceBalance

Pure OPure OPure OPure O2222Pure Pure Pure Pure ArArArArPure COPure COPure COPure CO22225 % CO5 % CO5 % CO5 % CO2222/N/N/N/N22220.43 % CO0.43 % CO0.43 % CO0.43 % CO2222/N/N/N/N2222Pure CHPure CHPure CHPure CH444410 % CH10 % CH10 % CH10 % CH4444/N/N/N/N22220.7 % CH0.7 % CH0.7 % CH0.7 % CH4444/N/N/N/N22220.045 % CH0.045 % CH0.045 % CH0.045 % CH4444/N/N/N/N22220.028 % CH0.028 % CH0.028 % CH0.028 % CH4444/N/N/N/N2222

Pure NPure NPure NPure N22220.038 % CO0.038 % CO0.038 % CO0.038 % CO2222 + 0.00016 % CH+ 0.00016 % CH+ 0.00016 % CH+ 0.00016 % CH4444 + 0.93 % + 0.93 % + 0.93 % + 0.93 % ArArArAr+ 20.9 % O+ 20.9 % O+ 20.9 % O+ 20.9 % O2222 + N+ N+ N+ N2222 BalanceBalanceBalanceBalance

Pure OPure OPure OPure O2222Pure Pure Pure Pure ArArArArPure COPure COPure COPure CO22225 % CO5 % CO5 % CO5 % CO2222/N/N/N/N22220.43 % CO0.43 % CO0.43 % CO0.43 % CO2222/N/N/N/N2222Pure CHPure CHPure CHPure CH444410 % CH10 % CH10 % CH10 % CH4444/N/N/N/N22220.7 % CH0.7 % CH0.7 % CH0.7 % CH4444/N/N/N/N22220.045 % CH0.045 % CH0.045 % CH0.045 % CH4444/N/N/N/N22220.028 % CH0.028 % CH0.028 % CH0.028 % CH4444/N/N/N/N2222

Relative uncertainty of 60 parts-per-million achieved

– with respect to standards prepared gravimetrically

Compound CIPM 81/91 formula

Giacomo et al (1982)

Revised values (2004)

To be published 2011

N2 0.78101 0.78082 ± 0.00012 (Calculated by difference)

0.780 93 ± 0.00006

Measured

O2 0.20939 0.20945 ± 0.00012

NIST (1970)

0.209 45 ± 0.00012

NIST (1970)

Ar

0.00917

0.009332 ± 0.000006

Kim et al (2004)

0.009 331 ± 0.000006

CO2

0.00040

0.000369 ± 0.000001 0.00040

Metrologia 18 (1982) 33-40 Metrologia 41 (2004) 387–395

expanded uncertainties (k=2)

Courtesy of

Dr Jin Seog Kim, KRISS, Korea

High accuracy measurements (II)

highly pure metals“Raw”

Zn 99.995%Vacuum distilled

Zn 99.99995%

16 determined

impurities

Vickers

micro hardness38.5 kg/mm2

43.0 mg/kg 0.5 mg/kg

32.6 kg/mm2

BAM-M601 w [mg/g]

Cd 0.55 ± 0.06

Fe 2.20 ± 0.09

Cu 1.89 ± 0.11

Tl 2.25 ± 0.09

Pb 15.7 ± 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%

16 determined

impurities

Vickers

micro hardness38.5 kg/mm2

43.0 mg/kg 0.5 mg/kg

32.6 kg/mm2

BAM-M601 w [µg/g]

Cd 0.55 ± 0.06

Fe 2.20 ± 0.09

Cu 1.89 ± 0.11

Tl 2.25 ± 0.09

Pb 15.7 ± 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

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