Compressibility

$Z=\frac{pV}{N{k}_{B}T}$

The bulk modulus $B$ gives the change in volume of a solid substance as the pressure on it is changed,

$B=-V\frac{\partial P}{\partial V}$

The compressibility $K$ or $\kappa$, is given by

$\kappa =\frac{1}{B}$

The isothermal compressibility, ${\kappa }_T$ is given by

${\kappa }_T=-\frac{1}{V}{\frac{\partial V}{\partial P}|}_T=\frac{1}{\rho }{\frac{\partial \rho }{\partial P}|}_T$

(Note: in Hansen and McDonald the isothermal compressibility is written as ${\chi }_T$). where $\rho$ is the particle number density given by

$\rho =\frac{N}{V}$

where $N$ is the total number of particles in the system, i.e.

$N={\int }_V\rho (r,t)\mathrm{d}r$

Compressibility of an Ideal Gas

From the ideal gas law we see that

$Z=\frac{pV}{N{k}_{B}T}=1$