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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[edit]

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 T is the temperature, \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({\mathbf r},t)~{\rm d}{\mathbf r}

Adiabatic compressibility[edit]

The adiabatic compressibility, \kappa_S is given by

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

where S is the entropy.

See also[edit]

The compressibility equation in statistical mechanics.