What is a Mole? Old Concepts and New

A Fixed Avogadro Constant or a Fixed Carbon-12 Molar Mass: Which One to Choose?

by Yves Jeannin

In a recent issue of Chemistry International, Ian Mills and Martin Milton suggested a new definition for the mole, one of the seven base units.^{1} This matter is controversial and needs a careful examination.

The IUPAC Green Book^{2} describes the seven base units and gives their definitions. They are the length unit, the metre; the time unit, the second; the mass unit, the kilogram; the current unit, the ampere; the temperature unit, the Kelvin; the amount of substance unit, the mole; and the luminous intensity unit, the candela. Some of them require the help of another base unit: for instance, the time unit involves the length unit, the current unit involves the length unit, and the amount of substance unit involves the mass unit.

Historically, the first standard for the metre was based upon the earth so that it was accessible to everyone at any time. Later on, Johnstone-Stoney and Planck had a completely different view and recommended the use of fundamental constants of theoretical physics for defining units. In the mean time, and independent of each other, base units and corresponding standards have been defined on a purely experimental basis. Although it provides a set of clearly defined units, this set is not very consistent. Moreover, advances in modern physics led to fundamental constants known with a great accuracy.^{3} This suggests that we should think again about using base unit definitions based upon fundamental constants since it could result in fewer base units.

Presently, discussions are in progress about this subject. As an example, let us take the case of the speed of light c. It has already been decided that c is a fixed value equal to 299 792 458 m s^{-1.} Indeed, the speed of light is a fundamental constant of physics, the value of which is independent of the galilean referential in which it is measured; it allows a clear definition for the unit of length. Now, considering the well-known formula λ=c/v, in which wave length λ is bound to frequency v through c, it appears that it is no longer necessary to define two base units, metre and second, if the speed of light is arbitrarily considered as a constant without a unit. If the length is chosen as a base unit, the time is expressed in m^{-1}. If the second is chosen as a base unit, the length is expressed in s^{-1}. Let us underline that this is a metrology approach. For practical purposes, speed should keep its traditional unit. This view has the great advantage for metrologists of reducing the number of base units by one.

Exploring the development of this idea, Mills, Mohr, Quinn, Taylor, and Williams^{4} presented a choice of four constants to be fixed similar to fixing the speed of light to define the metre: the Planck constant to define the kilogram, the elementary charge to define the ampere, the Boltzmann constant to define the kelvin, and the Avogadro constant to define the mole. Mills and Milton have discussed further the specific example of fixing the Avogadro constant to define the mole, according to the definition:

“The mole is the amount of substance of a system which corresponds to 6.022 141 79 x 10^{23} elementary entities.”

This may be contrasted with the present definition of the mole, which is:

“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.” (14th CGPM, 1971)

This present definition of the mole implies a fixed carbon-12 molar mass M(^{12}C) equal to 12 g/mol exactly and involves the use of another base unit: the kilogram. Therefore, the mass unit has to be defined prior to the mole unit. The definition proposed by Mills et al.4 and further discussed by Mills and Martin^{1} disconnects the mole from the kilogram, which is one of the advantages of this definition.

From the known relation: h = α^{2}cm_{e} / 2R_{∞} (1)

with h Planck constant, α^{ fine structure constant, c speed of light, me mass of one electron, R∞ Rydberg constant, one can deduce:}

h N_{A} = [α^{2}cm_{e} / 2R_{∞}].[M(^{12}C)/m(^{12}C)]

with m(12C) mass of one atom of carbon-12 isotope and M(12C) molar mass of carbon-12; this formula can be shortened as :

h N_{A} = K M(^{12}C)

If fixed values are assigned to the Planck constant, and/or to the carbon-12 molar mass, and/or to the Avogadro constant, there are three possibilities:

- A fixed Planck constant h and a fixed Avogadro constant N
_{A}, the carbon-12 molar mass M(^{12}C) is to be determined. - A fixed Planck constant h and a fixed carbon-12 molar mass M(
^{12}C), the Avogadro constant N_{A}is to be determined. - A fixed Avogadro constant N
_{A}and a fixed carbon-12 molar mass M(^{12}C), the Planck constant h is to be determined.

The choice of a fixed Planck constant seems obvious because of its central position in quantum physics. The advantages have been detailed in a note “On the Possible Redefinition of the Kilogram” written by Taylor and Mohr.^{5}

What about N_{A} or M(^{12}C)? Which one to choose? Let us look at the consequences of a new mole definition.

First, the molar mass of carbon-12 is no longer constant if N_{A} is fixed. Increasing the accuracy of experimental methods in the future will consequently yield a better M(^{12}C) value; any improvement will introduce changes on the whole table of element molar masses. Such modifications will indeed remain minor if a fixed M(^{12}C) is chosen as it is today. From a practical point of view, every chemist concerned with synthetic chemistry will not be troubled by these changes. Nevertheless, it is a major modification with respect to the actual situation of stable values for all molar masses; it will raise some feeling of unstability.

Let us consider together the speed of light, the Planck constant, and the Avogadro constant. Physics meets quite a number of such constants which relate to phenomena or to the properties of matter. One might mention the electron charge, the electron mass, the fine structure constant, the permeability of vacuum, and so on. Each of them has a deep physical meaning. Some of them already have fixed values by international agreement.

The nature of N_{A} is completely different. It is nothing but a proportionality constant. When Dalton thought about atomic weights and set up his famous table, which has considerable historical and practical value, he took 1 for the lightest element, hydrogen.^{6} It led to the value 16 for oxygen and 12 for carbon. At that time, nobody really had any idea about the mass of a single atom. Later on, Berzelius proposed to use oxygen’s atomic weight as a starting value because, he noted, oxygen reacts with many more elements than hydrogen to yield compounds, a mandatory step to determine atomic weights. He chose 100.^{7} The chemical community did not follow his proposal. If this value had been retained, the Avogadro constant would have been different. The physics behind the Avogadro constant cannot be compared with the physics of the speed of light or of the Planck constant.

A fixed Avogadro constant leads to a definition of the mole without reference to any other unit. The mole becomes independent of any other unit so that it gets its status of base unit. If M(^{12}C) is kept equal to 12 g/mol exactly as it is today, the mole definition implies another unit, the kilogram. The kilogram definition is presently based upon the standard kept at the Pavillon de Breteuil where the Bureau International des Poids et Mesures is located. Unfortunately, this standard weight slightly changes over the years without any clear explanation: this is not very satisfactory. It seems possible to get a new definition for the mass unit with a fixed Planck constant without unit. By comparing a mechanical power and an electrical power, mass is found to be proportional to frequency.^{8} The kilogram mass unit then would be defined with only the help of the time unit. Then, it would no longer be necessary to consider the mass unit as a base unit. Consequently, the mole would also lose its status of base unit. The choice of a fixed carbon-12 molar mass would decrease the number of base units by one. It is attractive from a metrology point of view.

If N_{A} is fixed, the relation hN_{A} = K M(^{12}C) provides M(^{12}C) by computation. The silicon sphere method compares experimentally the macroscopic volume of a sphere and the microscopical one of a single atom.^{9} A fixed N_{A} yields M(Si) which in turn yields M(^{12}C). There are thus two independent entries to the molar mass table. This is not the most favorable situation. The isotopic abundance determination remains a weak step in the silicon sphere method. One should point out that there are considerable efforts going on to enrich silicon into its most abundant natural isotope so that difficulties with isotopic abundances will be overcome. One may also note that any other isotope could be introduced in place of carbon-12 in relation 1, particularly an element having a single stable isotope that could also be used in place of silicon for the experimental volume comparison. However, is it possible?

The exact number 12 is designated by A_{r}(^{12}C) and called carbon-12 atomic weight; it has no unit. Although it is not strictly speaking a weight, this word is accepted by IUPAC due to its long traditional use and as a tribute to Dalton. By definition, the molar mass M_{r} is this number expressed with a unit that is the kilogram. One can write:

M_{r} = A_{r}M_{u} with M_{u}= 0.001 kg/mol

M_{u} is called molar mass constant. All the other atomic weights are determined relative to the carbon-12 atomic weight, so they are called “relative atomic weights.” In the present SI, the atomic weight and molar mass of carbon-12 have exact values; M_{u} is exactly 0.001 kg/mol.

With the new proposal of a fixed N_{A}, molar mass M(^{12}C) is known with a standard deviation; it is no longer fixed and will slightly fluctuate at the rythm of the accuracy improvement of experimental methods. Consequently, the molar mass constant M_{u} will also fluctuate so that the value 12 for the carbon-12 atomic weight will remain constant. However, this situation seems rather unfortunate. To use a unit conversion factor that is not really a constant is disturbing. Moreover, Martin and Mills recommend a larger use of M_{u}, especially in teaching.^{1} It will be difficult for pupils and even advanced chemistry students to understand the need for a new constant M_{u} that fluctuates. Finally, if a chemist wants to compute a number of mole, he will use a balance so that the weight of the substance is known with a mass unit: This result will then be divided by the molar mass also expressed with a mass unit, not by the relative atomic weight which has no unit. For this reason, it is important that element molar masses be constant.

A good definition for a base unit is supposed to provide a standard that can be easily used by anyone anywhere in the world. The definition proposed by Mills and Martin means that one has to count atoms. It does not seem possible to get a standard by this method. A weighing balance is the tool used by a chemist to measure an amount of substance with the help of molar masses. While the kilogram is needed in the present definition, it disappears from the new definition, yet it still has to be used to measure an amount of substance. Thus, why not keep a mass unit in the mole definition and maintain the present definition.

For these reasons, the choices of a fixed M(^{12}C) and of the actual definition are favored.

References

- I.M. Mills and M. Milton, Chemistry International, (March-April), 3–7 (2009)
- Quantities, Units, and Symbols in Physical Chemistry , 2nd edn., I.M. Mills, Blackwell Scientific Publications, Oxford (1993); 3rd edn., Royal Society of Chemistry, (2007)
- C.J. Bordé, Phil. Trans. Roy. Soc. A 363, 2177 (2005)
- I.M. Mills, et al, Metrologia, 43, 227 (2006)
- B.N. Taylor and P.J. Mohr, “On the Possible Redefinition of the Kilogram,” document prepared for the 14th CCU meeting (2001)
- A New System of Chemical Philosophy, J. Dalton (1808)
- Théorie des proportions chimiques et table analytique des poids atomiques des corps simples et de leurs combinaisons les plus importantes, J.J. Berzelius (1835)
- B.P. Kibble, J.H. Sanders, and A.H. Wapstra, Atomic Masses and Fundamental Constants, Plenum Press (1975)
- K. Fujii, et al, IEEE Trans. on Instr. and Measur., 54, n°2, 854 (2005)

Yves Jeannin is an emeritus professor at the Pierre and Marie Curie University, Paris, France.

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What is a Mole? Old Concepts and New

Closing Comments from Ian M. Mills

This discussion is really about choosing between two alternative definitions for the unit mole:

- The mole is that amount of substance that contains the same number of entities as 12 grams of carbon 12. This is the current definition. It has the effect of fixing the molar mass of carbon 12 as exactly 12 g/mol.
- The mole is that amount of substance that contains exactly 6.022 141 79 x 10
^{23}entities. This is the proposed new definition. It has the effect of fixing the value of the Avogadro constant to be exactly 6.022 141 79 x 10^{23}mol^{-1}.

The choice 1 is connected with the history of the development of the quantity amount of substance and the unit mole. The choice 2 is thought to be simpler, and removes the dependence on the kilogram; this is thought to be desirable, in order to clarify the distinction between the quantities amount of substance and mass (which are often confused). Although I can see advantages in choice 1, most people prefer choice 2 because of its simplicity.

Jeannin also believes that the Avogadro constant is a “fundamental constant of a lesser breed,” in contrast—for example—to the Planck constant, or the speed of light, which he thinks of as true fundamental constants. He argues that the Avogadro constant is free for us to choose; we could choose to have a different number of entities in a mole; it is at our choice. Many people express that view. However, I believe that is a misunderstanding. It is the numerical value of the Avogadro constant that is free for us to choose, but the value of the Avogadro constant, N_{A}, is a true constant of nature just like c and h.

Consider the effect of choosing 12.044 in place of 6.022. We would then have twice as many entities in a mole, so that we would in effect be defining a new mole that would be twice as large. It should then be given a new name, such as new-mole. We would have 1 new-mole = 2 mole. For the value of the Avogadro constant we would have N_{A} = 12.044 x 10^{23} new-mol^{-1},

but this is equal to 6.022 x 1023 mol^{-1} because we have doubled both the number and the unit, and the value of N_{A} is the number divided by the unit mol. That is perhaps my strongest criticism of Jeannin’s presentation.

The impact of redefinition of the mole is significant for practical metrology. The following excerpt from section 4.1.4 of ref. 4 (vide supra, Metrologia, 43 (2006) 227–246) summarizes the idea: “One of the most signiﬁcant beneﬁts of redeﬁning the mole so that it is linked to an exactly known value of the Avogadro constant N_{A} (assuming h, e, and k also have exactly known values) is that other constants will become exactly known, namely, the Faraday constant F, molar gas constant R, Stefan–Boltzmann constant σ, and molar volume of an ideal gas V_{m} (at a speciﬁed reference temperature and pressure), all of which have practical importance in a number of ﬁelds of chemistry and physics.”

The actual values are presented and compared in tables 1 and 2 below. Overall, the uncertainties across the new SI will decrease significantly, and this is desirable.

Even though not exactly zero by definition, the molar mass uncertainty in the new SI is sufﬁciently small that it can be considered negligible in calculating molar mass for use in the determination of amount of substance. Consequently, the new deﬁnition of the mole will require no change in current metrological practice in any ﬁeld.

Table 1. Comparison of constant uncertainties in the current SI versus and new SI. |

Table 2: Relative standard uncertainties for a selection of fundamental constants multiplied by 10^{8} (i.e., in parts per hundred million). |

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What is a Mole? Old Concepts and New

The following subarticles follow this article: |

A Fixed Avogadro Constant or a Fixed Carbon-12 Molar Mass: Which One to Choose?, by Yves Jeannin Closing Comments from Ian M. Mills |

In the March-April 2009 issue of Chemistry International, Ian Mills and Martin Milton reviewed concepts familiar to chemists: the quantity “amount of substance” and its unit, the “mole.” They also presented a possible new definition for the mole. The reasoning behind the possible new definition is currently being debated in the community. The IUPAC Interdivisional Committee on Terminology, Nomenclature and Symbols was invited to review the question during its recent meeting in Glasgow in August 2009. CI asked the ICTNS Chair Jack Lorimer to recap the issue.

by Jack Lorimer

The current definition of the base unit for the SI base quantity “amount of substance,” the “mole,” was adopted in 1971 by the CGPM (Conférence Générale des Poids et Mesures). The CGPM is the body in charge of maintaining the International System of Units (SI), in accordance with the requirements of the Metre Convention, which is the legal basis for use of the SI in the many countries that subscribe to the convention. The definition reads^{1}:

- 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”.
- When the mole is used, the elementary entities must be specified and may be atoms, molecules, ions, electrons, other particles, or specified groups of particles.
- (Addendum in 1980) In this definition it is understood that unbound atoms of carbon 12, at rest and in their ground state, are referred to.

Since that time, two associated problems have been recognized. First, the name of the base quantity is awkward, with the word “substance” being a source of confusion. Other names, such as “chemical amount” and the problematic “enplethy” have been suggested,^{2} among others. The second problem is more fundamental for metrology, because the current definition makes use of a second SI base unit, the kilogram.

In 2005, at the ICTNS meeting in Beijing, a proposal from the CCU (Consultative Committee on Units) of the BIPM (Bureau International des Poids et Mesures) to redefine not only the mole but all other SI base units (kg, m, s, A, K, cd) in terms of fundamental physical constants led to a resolution to the IUPAC Bureau in support of this general goal, but specific recommendations were only in a preliminary stage.

Some of the scientists whose breakthroughs contributed to the modern definition of the mole (from left): Lord Kelvin, Johann Josef Loschmidt, Amedeo Avogadro, and Stanislao Cannizzaro. |

In 2009, Mills and Milton^{3} published an article in CI that brought the attention of the ICTNS to a specific proposal for redefinition of the mole in terms of fundamental physical constants. ICTNS also received information from CCU giving specific recommendations for revision of the base units, and in particular, asking for support of the redefinition of the mole by IUPAC, given that the mole has special interest for chemists. At its Glasgow meeting at the IUPAC General Assembly, ICTNS devoted a half-day session to this request, which started with a presentation by Ian Mills, IUPAC’s representative on the CCU and currently president of that body. Vigorous discussion followed, leading to conclusions that the redefinition should be supported, and that redefining the unit should provide an excellent opportunity to redefine the name of the base quantity at the same time. The outcome of the session was a resolution to the Bureau:

“Given that: (a) definition of the mole in a way that is independent of mass is desirable; (b) the mole is often thought of by chemists as an Avogadro number of entities; and (c) the name of the ISQ (International System of Quantities) base quantity “amount of substance” has been a source of much confusion, ICTNS recommends to the Bureau that:

The recommendation of the CCU (Consultative Committee on Units) of the BIPM, that the mole be defined as follows:

“The mole, 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, is such that the Avogadro constant is equal to exactly 6.022 141 79 x 10^{23} per mole.

Thus, we have the exact relation N_{A} = 6.022 141 79 x 10^{23} mol^{-1}. The effect of this definition is that the mole is the amount of substance of a system that contains 6.022 141 79 x 10^{23} specified elementary entities.”

be supported by the IUPAC, with the following suggestions:

- The greatest effort should be made to change the name of the ISQ base quantity “amount of substance” at the same time that a new definition of the mole is approved.
- A note should accompany the new definition to explain that the molar mass of
^{12}C will be an experimental quantity, with a relative measurement uncertainty of about 1.4 x 10^{-9}.”

The ICTNS had at its disposal, prior to the meeting, a number of relevant documents. These included a dissenting view to the recommendation by former IUPAC President Yves Jeannin on behalf of the Chemistry Section, French Academy of Sciences, and a supporting view from the U.S. National Institute of Standards and Technology (NIST). One of the co-authors of the NIST paper is Peter Mohr, who is also the current chair of SUNAMCO (Symbols, Units, Nomenclature and Atomic Masses Committee), the counterpart of ICTNS in the International Union of Pure and Applied Physics. The definition in the ICTNS resolution was taken directly from this latter document. The document by Jeannin is reproduced below, and followed by closing comments by Ian Mills, in which the arguments are summarized and the relation between the old and new definitions is discussed.

It may be of interest to readers to know the sequence of resolutions that must accompany any approved change in the SI, and also to be aware of the alphabet soup of acronyms that describes the various committees involved. The BIPM was set up in 1875 by the Metre Convention to ensure worldwide unification of measurements,^{1} and has it headquarters and laboratories in Sèvres, just outside Paris, on international territory ceded by the French government. It operates under supervision of the CIPM (International Committee on Weights and Measures), which in turn is under the authority of the CGPM (General Conference on Weights and Measures). Delegates from Member States of the Metre Convention attend the General Conference every four years, and ratify recommendations that arise, in this case, through (in succession) the CCU, CIPM, and CGPM, with the CGPM having responsibility for final decisions. Any changes in the SI are thus subjected to extensive scrutiny over a number of years. The process of redefining the SI base units is currently at the CIPM stage. Input from IUPAC is possible at either the CCU or CIPM stages through ICTNS, which has responsibility for interactions with international organizations outside IUPAC, but in important cases, ICTNS makes recommendations to the Bureau. As noted, IUPAC has a representative on the CCU, and the director of BIPM is a member of ICTNS.

The ICTNS hoped that presentation of these articles would provide IUPAC members with a broad picture of the problems associated with definition of the mole and with the cogent arguments that led to the support of ICTNS for redefinition. Those interested in the redefinitions of the other SI base units will also find relevant information.

References

- Le système international d’unités/The International System of Units, SI. (the SI Brochure) 8th ed., BIPM (Bureau international des poids et mesures), Sèvres (2006); pp. 95, 115.
- Quantities, Units and Symbols in Physical Chemistry. 3rd ed. (the IUPAC Green Book). RSC Publishing, Cambridge, UK (2007); p. 4.
- I.M. Mills and M. Milton, Chemistry International 31 (March-April), 3–7 (2009).

J.W. Lorimer <lorimer@uwo.ca> is an emeritus professor of chemistry at the University of Western Ontario, in London, ON, Canada. He was chair of ICTNS from January 2004 to December 2009.

Post Scriptum: On behalf of the Bureau, the IUPAC Executive Committee at its 2 October 2009 meeting reviewed and endorsed the ICTNS recommendations to support the redefinition of the mole as proposed by the CCU.

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