Compressibility equation

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

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

where $ρ$ is the number density, g(r) is the radial distribution function and $kT\left(\frac{\partial \rho}{\partial p}\right)$ is the isothermal compressibility.

Using the Fourier representation of the Ornstein-Zernike equation the compressibility equation (1) can be rewritten in the form:

$\frac{1}{kT}\left(\frac{\partial p}{\partial \rho}\right) = \frac{1}{1+\rho \int h(r) d \rm{r}}=\frac{1}{1+\rho \hat{H}(0)}=1-\rho\hat{C}(0)=1-\rho \int c(r) d \rm{r}$ (2)

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.