Compressibility

$Z={\frac {pV}{Nk_{B}T}}$

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

$\kappa _{T}=-{\frac {1}{V}}\left.{\frac {\partial V}{\partial P}}\right\vert _{T}={\frac {1}{\rho }}\left.{\frac {\partial \rho }{\partial P}}\right\vert _{T}$

(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.

$N=\int _{V}\rho (r,t)~{\rm {d}}r$

Compressibility of an Ideal Gas

From the ideal gas law we see that

$Z={\frac {pV}{Nk_{B}T}}=1$