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
\[\kappa =\frac{1}{B}\]
[edit] 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 \(T\) 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.
\[N = \int_V \rho({\mathbf r},t)~{\rm d}{\mathbf r}\]
[edit] Adiabatic compressibility
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.
