# Compressibility factor

The compressibility factor (Z) is a useful thermodynamic property for modifying the ideal gas law to account for the real gas behaviour. The closer a gas is to a phase change, the larger the deviations from ideal behavior. Values for compressibility are calculated using equations of state (EOS), such as the virial equation and van der Waals equation. The compressibility factor for specific gases can be obtained, with out calculation, from compressibility charts. These charts are created by plotting Z as a function of pressure at constant temperature.

## Derivation

The compressibility factor is defined as:

$Z=\frac{P \tilde{V}}{R T}$

where, $P$ is the pressure, $\tilde{V}$ is the molar volume of the gas, $T$ is the temperature, and $R$ is the gas constant.

The virial equation is especially useful to describe the causes of non-ideality at a molecular level as it is derived directly from statistical mechanics:

$\frac{P \tilde{V}}{RT} = 1 + \frac{B}{\tilde{V}} + \frac{C}{\tilde{V}^2} + \frac{D}{\tilde{V}^3} + \dots$

Where the coefficients in the numerator are known as virial coefficients and are functions of temperature.

The virial coefficients account for interactions between successively larger groups of molecules. For example, B accounts for interactions between pairs, C for interactions between three gas molecules, and so on. Because interactions between large numbers of molecules are rare, the virial equation is usually truncated after the third term.

## Physical Significance

For an ideal gas the compressibility factor is defined as $Z=1$. In real gases this is seldom the case. $Z$ generally increases with pressure and decreases with temperature. At high pressures or low temperatures, molecules are moving less rapidly and are colliding more often. This allows attractive forces between molecules to dominate, making the volume of the real gas ($V_{real}$) less than the volume of an ideal gas ($V_{ideal}$) which causes $Z$ to drop below one. When pressures are lower or temperatures higher, the molecules are more free to move. In this case repulsive forces dominate, making $Z>1$. The closer the gas is to its critical point or its boiling point, the more $Z$ deviates from the ideal case.