Compressibility factor

Remarks

For an ideal gas, the ideal gas law $p V_m = RT$ gives $Z = 1$ in every state. For a gas that obeys the van der Waals equation $(p + a/V_m^2)(V_m - b) = RT$ with constants $a, b > 0$, solving for $p$ gives $$Z = \frac{V_m}{V_m - b} - \frac{a}{R T V_m},$$ so $Z \ne 1$ in general, and $Z \to 1$ as $V_m \to \infty$. A non-example: the ratio $pV/(RT)$ with the total volume $V$ in place of $V_m$ is not $Z$. For $n$ moles it equals $nZ$, because it does not use the volume per mole. The virial expansion writes $Z = 1 + B(T)/V_m + C(T)/V_m^2 + \cdots$. By the law of corresponding states, $Z$ is approximately the same function of the reduced temperature and reduced pressure for many gases.