In statistical mechanics and thermodynamics the compressibility equation refers to an equation which relates the isothermal compressibility (and indirectly the pressure) to the structure of the liquid. It reads:

$${\displaystyle kT\left({\frac {\partial \rho }{\partial p))\right)=1+\rho \int _{V}\mathrm {d} \mathbf {r} [g(r)-1]}$$

${\displaystyle \rho }$

${\displaystyle kT\left({\frac {\partial \rho }{\partial p))\right)}$

Using the Fourier representation of the Ornstein-Zernike equation the compressibility equation can be rewritten in the form:

$${\displaystyle {\frac {1}{kT))\left({\frac {\partial p}{\partial \rho ))\right)={\frac {1}{1+\rho \int h(r)\mathrm {d} \mathbf {r} ))={\frac {1}{1+\rho {\hat {H))(0)))=1-\rho {\hat {C))(0)=1-\rho \int c(r)\mathrm {d} \mathbf {r} }$$

where h(r) and c(r) are the indirect and direct correlation functions respectively. The compressibility equation is one of the many integral equations in statistical mechanics.