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Home , Amount of substance, Molar mass constant

METROLOGY IN CHIMIA 2009, 63, No. 10 613

doi:10.2533/chimia.2009.613 Chimia 63 (2009) 613–615 © Schweizerische Chemische Gesellschaft Amount of Substance and the

Martin J. T. Milton*a and Ian M. Millsb

Abstract: The unit mole is very familiar amongst both chemists and physicists, but the name of the corresponding quantity ‘amount of substance’ is not so familiar and the concept is still a source of difficulty for many students. This paper reviews and clarifies these concepts and also discusses the definition of the unit mole, and its pos- sible revision. Keywords: Amount of substance . Molar constant . Mole . Redefinition . SI

1. Amount of Substance concept for chemical reactions. Thus the from a specific number of entities and from ratio of volumes of solutions of X and Y the definition of the unit for amount of sub- Amount of substance is a quantity that that react together in a titration are given by stance. measures the size of an ensemble of enti- The quantity amount of substance (n) ties. It appears in thermodynamic relations is thus an alternative to using the quantity (4) such as the law, and in stoichio-  number of entities (N). They are related by metric relations between reacting mol- the equation ecules such as the Law of Multiple Pro- portions.[1,2] Familiar equations involving where the quantity (n /n ) is a simple ra- n = N/N (5) X Y A amount of substance (n) are tional fraction. Hence the of an unknown solution may be determined where N is the . A pV = nRT,(1) from the concentration of a standard solu- One might reasonably ask why we need tion by measuring the volumes in a titration. the quantity amount of substance at all, for an ideal gas, and the equation This is the Law of Multiple Proportions. when the number of entities could be used It is interesting to note that whilst the in its place? We propose three reasons for c = n/V (2) use of amount of substance in the sense preferring to use n rather than N. of referring to a thermodynamic ensemble The first is that equations like Eqn. (3) for the amount of substance concentration can be traced back to Boyle’s research in can be used to determine M, or (usually called simply the concentration) the 17th century, the development of an amounts in terms of moles, without know- of a solution. Here V is the of a understanding of dates to ing the value of theAvogadro constant. The solution containing the amount of solute n. Lavoisier’s work one hundred years later. atomic weights of in the periodic ta- Another important relation is that between Underlying both of the senses in which it ble were known long before the value of amount of substance n and mass m for a is used, is the fact that the quantity amount the Avogadro constant was known with pure sample of substance measures a number of enti- similar accuracy. Even today, the value of ties. This insight can be traced directly to the Avogadro constant is only known to n = m/M (3) two developments made in the early 19th about one part in 107, whereas many - century: ’s explanation of his Law ic weights are known to about one part in where M is the mass per amount of sub- of Multiple Proportions and Avogadro’s 109 or better. stance, usually called the molar mass. hypothesis that “samples of different gas- The reason is practical; the An important application of the quan- es at the same , and number of entities is generally of the or- tity amount of substance in chemistry is volume always contain the same number der 1023, whereas n is generally a number to the way in which react in a of molecules”.[3,4] As discussed below, it is of order 1 when expressed in moles. Thus, titration or more generally in any chemi- now being proposed that the link between for example, in a chemistry laboratory cal reaction. This is the most fundamental the quantity amount of substance and the the concentration of solutions is typically underlying concept of a number of entities quoted in moles per , with numbers in should be strengthened by the introduc- the general order of magnitude 1. It would tion of a definition for the unit of amount be inconvenient to quote in of substance framed directly in terms of a molecules per litre, with numbers of the fixed number of entities. order 1023. Thus we find bottles labelled “Amount of substance is a quantity ‘0.1 M NaOH’, where M is read as ‘molar’ that measures the size of an ensemble of and is an accepted shorthand for the unit . –1 . –3 *Correspondence: Dr. M. J. T. Miltona entities. It is proportional to the number of mol L = mol dm . The quantity amount Tel.: +44 031 943 6826 specified entities and the constant of pro- of substance may be seen as a device to E-mail: [email protected] portionality is the same for all substances. handle the same quantitative information aNational Physical Laboratory NPL Analytical Science Division The entities may be atoms, molecules, with much smaller numbers. Hampton Road, Teddington, , , other particles, or speci- The third reason for introducing the Middlesex, TW11 0LW, UK fied groups of particles.” quantity amount of substance, with the bUniversity of Reading School of Chemistry This definition emphasises the nature mole as a base unit, is that it extends the Reading, Berkshire, RG6 6AD, UK of amount of substance, which is distinct power of to chemis- 614 CHIMIA 2009, 63, No. 10 IN CHEMISTRY

try, and to equations involving chemical Molekel” to “Mol” was first recorded in The mole is that amount of substance of quantities. This follows from the fact that 1898 by Nernst.[8] The term ‘mole’appears a system that contains exactly 6.022 141 79 n is a base quantity with its own dimension, in English for the first in the transla- × 1023 specified elementary entities, which whereas N is dimensionless. tion of Ostwald’s ‘Principles of Inorganic may be atoms, molecules, ions, electrons, Chemistry’ published in 1902, in which other particles or specified groups of such he associated it with a standard number of particles. 2. The Mole molecules. Thus one of X The effect of this new definition would became one mole of X. be to fix the value of the Avogadro constant The mole is the SI unit for the quantity It must be admitted that the name to be 6.022 141 79 × 1023 mol−1 exactly. amount of substance.[5] It is currently de- ‘amount of substance’ is not well chosen, The number would be chosen to be the fined by the statements:[6] because the word ‘amount’ has a common best estimate of the numerical value of the The mole is that amount of substance dictionary meaning, and the additional Avogadro constant at the time the new defi- that contains the same number of elemen- words ‘of substance’ seem inadequate to nition is adopted, thus ensuring continuity tary entities as there are atoms in 12 g of imply the chemist’s specialised use for the in the value of the mole.[9] carbon-12. When the mole is used the enti- name. It was the original intention that the This new definition would be concep- ties must be specified and may be atoms, words ‘of substance’ should be replaced tually simpler than the current definition, molecules, ions, electrons, other particles, by the specification of the entity whenever which is chosen to fix the molar mass of or specified groups of such particles. possible, so that one would say (for exam- carbon 12 rather than the number of entities It follows that the numerical value of ple) ‘amount of benzene, C H ’ or ‘amount in a mole. Also the new definition would 6 6 the Avogadro constant,[7] denoted {N }, of ions, H+’. no longer be dependent on the , so A expressed in the unit mol−1, is simply the Another name for n, which is the name that uncertainties in realising the definition number of atoms in 12 g of carbon 12, so that most chemists use, is simply ‘number of the kilogram would no longer be trans- that the value of the Avogadro constant of moles’. However this is not a good mitted to the mole – as they are at present. is directly related to the definition of the name, because it confuses the name of the mole. quantity with the name of the unit. A clear The effect of this definition is that the understanding requires that we always 5. The Molar Mass Constant M u molar mass of carbon 12, M(12C), is exactly distinguish clearly between quantities and 12 g.mol–1, and the molar mass of any atom units. Thus mass is a quantity, for which Many of the relations between the or molecule X is determined from its atom- kilogram (or gram, or milligram) are units, quantities discussed here can be simplified ic or molecular weight by simply multiply- and similarly we wish to say that amount of by introducing the molar mass constant ing by the unit g.mol–1, without the need to substance is a quantity, for which mole (or M , defined as one twelfth of the molar u know the value of the Avogadro constant. millimole, or micromole) are units. mass of the carbon 12 atom. This is the This is summarised in the relations Setting aside the difficulties with the natural analogue on the macroscopic scale name amount of substance, it is important of the unified constant m on u M(12C) = A (12C) g.mol–1 = 12 g.mol–1 (6) to realise that in the system of quantities the atomic scale, defined as one twelfth of r and units that is now universally adopted in the mass of a carbon 12 atom. The quantity chemistry, amount of substance is regarded m is often used as a unit of atomic mass, u M(X) = A (X) g.mol–1 (7) as a base quantity with its own dimension, denoted either u (for ‘unified’) or Da (for r whereas – by contrast – number of entities Dalton). For example, it is more conven- Here A (X) is the recommended symbol for is regarded as a dimensionless quantity. ient to write Eqns (6) and (7) as r the molecular weight of the entity X. The atomic or molecular weight of an entity is M(12C) = A (12C) M (8) r u actually the relative atomic or molecular 4. A Possible New Definition for the mass, relative to the value for the carbon Mole and 12 atom taken as exactly 12. (The names ‘atomic weight’and ‘molecular weight’are The current definition of the seven m(12C) = A (12C) m (9) r u universally used, and have been officially base units of the SI is given in the SI Bro- sanctioned by IUPAC, although they refer chure {SI brochure}. However there are The molar mass M(X) of any entity X is to dimensionless quantities, which are nei- proposals at present under discussion to then given in terms of the molecular weight ther nor weights.) adopt new definitions for four of the base A (X) by the equation: r units.[9,10] This follows from a desire to 3. The Names ‘Mole’ and ‘Amount define each of the base units in relation M(X) = A (X) M (10) r u of Substance’ to one of the fundamental constants of The name ‘mole’has been – and still is – physics or the properties of a simple atom, just as the atomic mass of the entity X is the cause of some confusion. Its origin has because we believe these to be the most given by been discussed in several publications.[3–5] stable and reliable constants of nature The terms ‘Kilogrammemolekuel’ and ‘g- available. Specifically, new definitions m(X) = A (X) m (11) r u Molekel’ were used by German scientists in are being considered for the kilogram, the 1880s and 1890s. The term “gramme- , and mole. This is the sub- Eqns (8) and (9) can be re-written with ex- molecule” was first used in English in ject known as quantum metrology, and plicit use of N as A 1893 in an article in the Encyclopaedia the proposals are discussed in detail else- Britannica. As the term implies, one gram where.[9,10] However the proposed new M(12C) = N m(12C) (12) A molecular weight of a substance X is that definition of the mole is the subject of the amount in a mass equal to the molecular present discussion,[4] and the suggestion and weight expressed in . is that it should simply specify the number These terms proved too awkward for of entities in a mole. This new definition M = N m (13) u A u everyday use, and the abbreviation of “g- might then read as follows. METROLOGY IN CHEMISTRY CHIMIA 2009, 63, No. 10 615

At present in the SI, M = 1 g.mol–1 exactly, h is the Planck constant, and A (e) is the ditional uncertainty associated with M as u r u and the Avogadro constant is an experi- relative mass of the on the unified described in this paper. mentally determined quantity (the number atomic mass scale. of atoms in 12 g of carbon 12), whose value The molar mass constant M has not u is currently known with a relative standard been much used in the established litera- 7. Summary and Conclusions uncertainty of about 5 × 10−8.[7] With the ture. It can of course always be replaced new definition proposed above, M will by the expression M(12C)/12, which is how It is something of a paradox that a con- u initially have the same value of 1 g.mol–1, it is defined. We recommend that this con- cept such as the quantity ‘amount of sub- but it will be an experimentally determined stant could be used with advantage more stance’, and its unit ‘mole’, so widely used quantity, with an uncertainty, and its value widely than it is at present, in teaching by practical chemists, are also the subject may change slightly from 1 g.mol–1 due to chemistry for example, to simplify the ex- of widespread misunderstandings. We future adjustments in the values of other pression for calculating the molar mass of have presented a proposal to re-define the constants. However the relative change atoms and molecules. mole on the basis of a fixed number of en- of M from the value 1 g.mol–1 is unlikely tities. The basis for such a proposal is that u ever to be greater than a few parts in 109, the atomic weight of carbon 12 (A (12C)) is r and this is so much smaller than the uncer- 6. Realising the Mole defined to be exactly 12 by IUPAP/IUPAC tainty with which chemical and any revision of the definition of the are likely to be made that for all practical There is an important difference be- mole must retain complete consistency purposes chemists may still treat M as be- tween the way the mole is realised and with this. u ing equal to 1 g.mol–1. the way the other base units of the SI are There are strong arguments in favour of The Figure provides a diagrammatic realised. Whilst the definition of the other the proposal to fix N as part of a revised A representation of how this proposal would base units and their associated mise en pra- definition for the mole. This is most easily work. In the proposed system, it is the two tiques provide specific information about achieved by allowing M to have some very u factors A (12C) and N indicated by the el- how they should be realised, the use of the small uncertainty (equal to the uncertainty r A lipses that would be exact, and the four mole does not depend on a particular meth- in the mass of the electron in the proposed masses at the apexes that would all have od of realisation. For example, the use of system). This approach would have the ad- the same uncertainty. the mole according to its current definition vantage of allowing existing equations to does not depend on the use of an experi- be retained and used without alteration. mental method that determines the number It would also introduce the same (rela- A (12C) of entities in 0.012 kg of carbon-12 or one tive) uncertainties for the parallel systems r that compares the number of entities in an of atomic and molar masses. (i.e. The mass 12 M( C) Mu unknown sample with the number of enti- of one mole of X and one molecule of X ties in 0.012 kg of carbon-12. Clearly both would have the same relative uncertainty of these would be impractical. If a new def- u (M ) = u (m )). r u r u inition based on a fixed number of entities, The proposed new definition for the such as the one discussed here, is adopted, mole would simplify the link between the NA NA it will still not require the use of such hypo- mole and the Avogadro constant, which thetical methods that count the number of has its own long and rich history. entities in a sample directly. The challenge of realising the mole would be unchanged Received: July 15, 2009 12 Ar( C) under the new definition. [1] ‘Quantities, Units and Symbols in Physical 12 m m( C) u The challenge of realising the mole has Chemistry’ (IUPAC Green Book), 3rd ed., RSC been discussed elsewhere.[4,11] The recog- Publishing, Cambridge, 2007. [2] M. L. McGlashan, ‘Physico-chemical quantities Fig. Diagrammatic representation of Eqns nised solution is that it is realised by the and units’, 3rd ed., Royal Institute of Chemistry, (8, 9 10 and 11) subject to the proposed re- valid use of a primary method of measure- [12] London, 1971. definition of the mole such that A (12C) and N ment, which has been defined by the r A [3] L. Cerruti, Metrologia 1994, 31, 159. are fi xed. The quantities at the apexes of the CCQM as [4] M. J. T. Milton, I. M. Mills, Metrologia 2009, four-sided fi gure all have the same uncertainty. “a method having the highest metro- 46, 332. The quantities within ellipses are all fi xed. logical qualities, whose operation can be [5] M. L. McGlashan, Metrologia 1995, 31, 447. completely described and understood, for [6] BIPM, ‘The International System of Units’, 8th ed., Sèvres, France; Bureau International des which a complete uncertainty statement Poids et Mesures, 2006. can be written down in terms of SI units.” [7] P. J. Mohr, B. N. Taylor, Rev. Mod. Phys. 2008, By far the most widely used exam- 80, 633. The relation between the Avogadro ple of a primary method is by the weigh- [8] W. Nernst, ‘Theoretische Chemie von Stand- constant and the Planck constant, which ing a sample of material of known purity punkte der Avogadro’schen Regel und der Thermodynamik’, Stuttgart, Enke, 2nd ed., would be used to determine M from the and hence known relative u 1898, p. 392. theoretical expression for the Rydberg (A (X)) and then use of the formula for the r [9] I. M. Mills, P. J. Mohr, T. J. Quinn, B. N. Taylor, constant R , remains true under both the amount of substance (Eqn. (15)): E. R. Williams, Metrologia 2006, 43, 227. ∞ current and the new definition of the mole [10] M. J. T. Milton, J. M. Williams, S. J. Bennett, (Eqn. (14)): Metrologia 2007, 44, 356.  (15) [11] R. J. C. Brown, M. J. T. Milton, Chem. Soc. Rev. 2007, 36, 904. (14) [12] M. J. T. Milton, T. J. Quinn, Metrologia 2001,  where m is the mass of the sample in kg 38, 289. (after correction for the mass of impuri- ties). The use of this formula will continue In this equation α is the fine structure con- to be extremely wide and will only differ stant, c is the in vacuum, by the introduction of the very small ad-