The mole concept

In everyday work, we deal with macroscopic amounts of substances. It is, therefore, useful to have a standard definition of macroscopic quantity of something. Avogadro's number is used to provide such a definition. We define 1 mole of something to be the amount of that something which contains $N_0$ individual entities of that thing. Thus, 1 mole of sodium atoms = $N_0$ sodium atoms. Now, if $m_{X}$ is the mass of one atom of the element X, then the mass of one mole of X is given by

$$m_{X} ({\rm g/atom}) \times N_0 ({\rm atoms/mol}) = m_{X}N_0 ({\rm g/mol})$$

But $m_{X}=A/N_0$. Thus, we see that the relative mass $A$ also measures the mass of 1 mole of X (in g/mol, for example). For this reason, the relative mass $A$ is also called the molar mass of X, however to distinguish relative mass $A$ from molar mass, we will use the symbol $M$ for molar mass. Note, therefore, that 12g of $^{12}$C is exactly 1 mole of $^{12}$C, which explains why Avogadro's number is defined the way it is.