Compressibility

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

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

Isothermal compressibility

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.

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({\mathbf r},t)~{\rm d}{\mathbf r}}

Adiabatic compressibility

The adiabatic compressibility, 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 \kappa_S} is given by

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 \kappa_S =-\frac{1}{V} \left.\frac{\partial V}{\partial p}\right\vert_{S}}

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 S} is the entropy.

See also

The compressibility equation in statistical mechanics.