# What is the dimension of molar mass?

I was told that the number of moles is ‘dimensionless’. Does this mean that the dimension of molar mass is simply mass and hence the unit g/mol can be expressed as equal or proportional to the dalton?

• The mole (1 mol ) is 1 of 7 base units of SI system for many years (m, kg, s, A, K, cd, mol). Dec 31, 2021 at 23:04
• The quantity “amount of substance” is not dimensionless and it shall not be called “number of moles” just as the quantity “mass” shall not be called “number of kilograms”. Jan 1 at 11:32
• I think we use 'number of moles' as an informal way of saying it, I've seen chemistry lecturers/students and books use the term any times. Jan 1 at 13:19
• The definition of 'amount of substance' acknowledges 'number of moles' as a deprecated term. It is no longer acceptable, even though it is still very popular in a few reunions of this world. Jan 1 at 15:27
• In some sense, any number alone is dimensionless. Even the Avogadro number is dimensionless, as the opposite to the Avogadro constant. Jan 2 at 7:48

According to ISO ISO/IEC 80000 Quantities and units, BIPM The International System of Units (SI), IUPAC Quantities, Units and Symbols in Physical Chemistry (Green Book), NIST Guide for the Use of the International System of Units (SI) etc., the quantity amount of substance is not dimensionless. Amount of substance and mass are two of the seven ISQ base quantities; the corresponding SI base units are the mole and the kilogram, respectively.

The quantity molar mass (quantity symbol: $$M$$) for a pure sample is defined as $$M = m/n$$ where $$m$$ is mass and $$n$$ is amount of substance.
The dimension of the molar mass is $$\dim M = \mathsf{M}\;\mathsf{N}^{-1}$$ The coherent SI unit for molar mass is ‘kilogram per mole’ (unit symbol: $$\mathrm{kg/mol}$$).
(The usually used unit is gram per mole ($$\mathrm{g/mol}$$) rather than kilogram per mole ($$\mathrm{kg/mol}$$).)

From the IUPAC Gold Book:

The mole, symbol $$\pu{mol},$$ is the SI unit of amount of substance. One mole contains exactly $$\pu{6.02214076E23}$$ elementary entities. This number is the fixed numerical value of the Avogadro constant, $$N_\mathrm{A}$$, when expressed in the unit $$\pu{mol^-1}$$ and is called the Avogadro number.

So a mole is a short notation for a certain amount of molecules/atoms etc. In this respect, it is similar to a e.g. a dozen which is equal to 12 elementary entities.

The unit of the molar mass is $$\pu{g mol^-1}$$. So, one could argue that indeed the unit Dalton, for which you also find a definition in the Gold Book, and $$\pu{g mol^-1}$$ are equal:

$$\frac{\pu{g mol^-1}}{\pu{Da}}=\frac{\pu{g}}{\pu{6.022E23}\times\pu{1.661E-24 g}}=1$$

• would we consider the dimension of g/mol to have equal dimension to mass in that case? I understand the idea of using dozen but a mole is specifically a unit of molecules or atoms, where a dozen is similar to just saying a grouping of 12 of any objects. Jan 1 at 0:19
• @user1007028, you could use the unit $\ce{mol}$ for anything, such as cars, houses, planets... you just usually only have too small amounts to make this reasonable. You could drop the unit altogether also in chemistry, you then would have to work with very small numbers (masses) or large numbers (amounts). The unit is a very handy helper. Between $\ce{Da}$ and $\ce{g mol^{-1}}$, the latter spells out more explicitly how the mass is normalized. Jan 1 at 9:36
• Using the current official definition, you should not substitute a dimensionless number for the unit mole. A dozen is equal to 12, but a mole is not equal to the Avogadro number in the SI unit system. Thus, g/mol and Da are not equal in the SI system, and the given equation is incorrect. However, anyone is free to switch to another unit system; they just have to explicitly state it. (Quantum physicists for example like using the Hartree atomic units, which are all dimensionless.) Jan 3 at 14:08

I was told that the number of moles is ‘dimensionless’.

According to the official definition, mole is a unit that has a dimension. This is counterintutitive perhaps given its definition ("the amount containing $$\pu{6.02214076E23}$$ particles"), but is has a dimension nonetheless.

Does this mean that the dimension of molar mass is simply mass and hence the unit g/mol can be expressed as equal or proportional to the dalton?

The dalton is set as 1/12 of the mass of the 12-carbon isotope, so it has dimensions of a mass. As measurements get more accurate, the value will change a bit over time. The mass of a particle and the molar mass of the pure substance made of those particles is proportional. The ratio of g/mol to Da, using current measurements, is 0.99999999965(30) times the Avogadro constant. So g/mol and dalton have different dimensions and can not be equal.

To say they are proportional does not make sense because both of them are units, not measurements. So while it makes sense to say that the mass and volume of samples of gold are proportional (at certain temperature and pressure), and call the proportionality factor the density of gold, it makes no sense to say that the units mL and g are proportional.

• I should have said 'differ by a factor of a pure number' such as 1kg = 1000kg, they need to have the same dimension on each side for this to be true. Jan 3 at 14:11
• Yes, I acknowledge that your and @Loong's answer follow the current official interpretation of the definition of the mole, and mine does not. Anyway, the paper you linked by Klaus Schmidt-Rohr is a very good treatise on this issue (thank you for sharing) and I fully agree with his reasoning. Jan 4 at 10:49