Today, the "molar" quantities are all defined through the isotope
12C. The choice was made because it is useful in mass spectroscopy. It used to be defined through hydrogen (or oxygen). Since the mass of the hydrogen atom is
approximately twelve times smaller than the mass of a carbon atom, we use the factor of 1/12 everywhere.
Ok, here come the definitions:
- An atomic mass unit (a.m.u.) or unified atomic mass (u) is a unit of mass equal to \frac{1}{12} of the mass of the atom of the isotope 12C. This means that the mass of one atom of 12C is exactly 12 u;
- A mole is a quantity of substance that contains as many units as there are atoms in 12 grams of 12C.
- Avogadro's number N_{A} is the number of atoms in one mole.
From 1, 2 and 3 it follows that:
<br />
12 \, u \times N_{A} = 12 \, \mathrm{g} \Rightarrow u = \frac{1 \, \mathrm{g}}{N_{A}}<br />
Until we find the exact value of Avogadro's number, we don't know what the numerical value of the unified atomic mass (or atomic mass unit) in grams is. Nevertheless, the above equality always holds. Also, everything would have worked if we simply substituted the 12 everywhere with 1 (for hydrogen) or 16 (for oxygen), since these factors would always cancel exactly.
We can measure the masses of atoms or molecules in units of
u. Then, we say that they are the
relative atomic or molecular masses:
<br />
A_{r} = \frac{m(\mathrm{atom})}{u}, \; M_{r} = \frac{m(\mathrm{molecule})}{u}<br />
Now comes the business of the unit of mole. It is very inconvenient to always work with this abstract quantity of N_{A} atoms. Also, since the advent of the atomic structure of matter, people saw that the law of conservation of mass in atomic reaction is a consequence of a more stringent law: The law of conservation of the number of atoms of each element in a chemical reaction (since chemical reactions have to do with the interchange of electrons, but do not alter the nuclei). This law is not valid in nuclear reactions. So, instead of quantifying the amount of substance through its mass, we can quantify it through its number of units and use the Avogadro's number as a unit. Then we say:
A mole (unit symbol mol, dimension N) is the quantity of substance (usually dentoted by the symbol
n) that contains Avogadro's number of units.
If there are
N units of something, then the quantity of substance of that something is found by:
<br />
n = \frac{N}{N_{A}}<br />
However,
n is a quantity with dimension of quantity of substance ([n] = \mathrm{N}, N is unitalicized, because it is a symbol for dimension) and
N (italicized because it is a symbol for a physical quantity) is a number and, therefore, a dimensionless quantity. To preserve dimensional consistency, we must have [N_{A}] = \mathrm{N}^{-1}. Because this constant is with dimensions, it is called
Avogadro's constant, in contrast to Avogadro's number. This ambiguity in symbols often leads to confusion:
<br />
N_{A}(\mathrm{constant}) = N_{A}(\mathrm{number}) \, \mathrm{mol}^{-1}<br />
Now, you may define molar masses:
<br />
M = \frac{m}{n}, \; [M] = \mathrm{M} \, \mathrm{N}^{-1}<br />
There is a connection between molar masses and relative masses. We can derive it from the above definitions, by simply expressing the quantity of substance through the total number and the total mass through the mass of one unit (atom, molecule) and the total number:
<br />
M = \frac{m_{a} \, N}{\frac{N}{N_{A}(\mathrm{constant)}}} = m_{a} \, N_{A}(\mathrm{constant}) = A_{r} (M_{r}) \, u \, N_{A}(\mathrm{constant})<br />
But:
<br />
u \, N_{A}(\mathrm{constant}) = \frac{1 \, \mathrm{g}}{N_{A}(\mathrm{number})} \, N_{A}(\mathrm{number}) \, \mathrm{mol}^{-1} = 1 \frac{\mathrm{g}}{\mathrm{mol}}<br />
Therefore:
<br />
M = A_{r}(M_{r}) \, \frac{\mathrm{g}}{\mathrm{mol}}<br />
Notice that we don't need to know the exact value of Avogadro's number, since it cancels exactly in the above relation.
