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

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \kappa ={\frac {1}{B}}}

Isothermal compressibility

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

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\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 Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \chi _{T}} ). where $T$ is the temperature, $\rho$ is the particle number density given by

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \rho ={\frac {N}{V}}}

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

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle N=\int _{V}\rho ({\mathbf {r} },t)~{\rm {d}}{\mathbf {r} }}

Adiabatic compressibility

The adiabatic compressibility, Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \kappa _{S}} is given by

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \kappa _{S}=-{\frac {1}{V}}\left.{\frac {\partial V}{\partial p}}\right\vert _{S}}

where $S$ is the entropy.

See also

The compressibility equation in statistical mechanics.