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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N} is the total number of particles in the system, i.e.

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N = \int_V \rho(r,t)~{\rm d}r}

Compressibility of an Ideal Gas

From the ideal gas law we see that

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Z= \frac{pV}{Nk_BT}=1}