# Compressibility

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

${\displaystyle B=-V{\frac {\partial p}{\partial V}}}$

The compressibility K or ${\displaystyle \kappa }$, is given by

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

## Isothermal compressibility

The isothermal compressibility, ${\displaystyle \kappa _{T}}$ is given by

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

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

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

where ${\displaystyle N}$ is the total number of particles in the system, i.e.

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

The adiabatic compressibility, ${\displaystyle \kappa _{S}}$ is given by
${\displaystyle \kappa _{S}=-{\frac {1}{V}}\left.{\frac {\partial V}{\partial p}}\right\vert _{S}}$
where ${\displaystyle S}$ is the entropy.