The mass of a macroscopic system is directly related to the total number of atoms or molecules in the system, denoted $N$. Because $N$ is determined simply by counting, it is a number with no units. A typical macroscopic system has $N\sim {10}^{25}$ atoms, an incredibly large number.

The symbol $\sim$, if you are not familiar with it, stands for “has the order of magnitude.” It means that the number is known only to within a factor of 10 or so. The statement $N\sim {10}^{25}$, which is read “$N$ is of order ${10}^{25}$,” implies that $N$ is somewhere in the range ${10}^{24}$ to ${10}^{26}$. It is far less precise than the “approximately equal” symbol $\approx$. Saying $N\sim {10}^{25}$ gives us a rough idea of how large $N$ is and allows us to know that it differs significantly from ${10}^5$ or even ${10}^{15}$.

It is often useful to know the number of atoms or molecules per cubic meter in a system. We call this quantity the number density. It characterizes how densely the atoms are packed together within the system. In an $N$-atom system that fills volume $V$, the number density is

The SI units of number density are $\text{m}{}^{-3}$. The number density of atoms in a solid is $(N\text{/}V){}_{\text{solid}}\sim {10}^{29}\text{m}{}^{-3}$. The number density of a gas depends on the pressure, but is usually less than ${10}^{27}{\text{m}}^{-3}$. As FIGURE 18.1 shows, the value of N/V in a uniform system is independent of the volume V. That is, the number density is the same whether you look at the whole system or just a portion of it.

FIGURE 18.1 The number density of a uniform system is independent of the volume.