
Vol.
32 No. 1
JanuaryFebruary 2010
What is a Mole?: Old Concepts and New continued
A Fixed Avogadro Constant or a Fixed Carbon12 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, JohnstoneStoney 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 wellknown 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 carbon12.” (14th CGPM, 1971)
This present definition of the mole implies a fixed carbon12 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, m_{e} 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 carbon12 isotope and M(12C) molar mass of carbon12; 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 carbon12 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 carbon12 molar mass M(^{12}C) is to be determined.
 A fixed Planck constant h and a fixed carbon12 molar mass M(^{12}C), the Avogadro constant N_{A} is to be determined.
 A fixed Avogadro constant N_{A} and a fixed carbon12 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 carbon12 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 carbon12 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 h N_{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 carbon12 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 carbon12 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 carbon12 atomic weight, so they are called “relative atomic weights.” In the present SI, the atomic weight and molar mass of carbon12 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 carbon12 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, (MarchApril), 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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