Compressibility

The compressibility, $Z$, is given by

$Z={\frac {pV}{Nk_{B}T}}$

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

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

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 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 \chi_T} ). 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 \rho} is the particle number density 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 \rho = \frac{N}{V}}

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

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

See also

The compressibility equation in statistical mechanics.

Compressibility of an Ideal Gas

From the ideal gas law we see that

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 Z= \frac{pV}{Nk_BT}=1}