compressibility equation

{{short description|Equation which relates the isothermal compressibility to the structure of the liquid}}

{{one source |date=March 2024}}

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:$$kT\backslash left(\backslash frac\{\backslash partial\; \backslash rho\}\{\backslash partial\; p\}\backslash right)=1+\backslash rho\; \backslash int\_V\; \backslash mathrm\{d\}\; \backslash mathbf\{r\}\; [g(r)-1]$$where $\backslash rho$ is the number density, g(r) is the radial distribution function and $kT\backslash left(\backslash frac\{\backslash partial\; \backslash rho\}\{\backslash partial\; p\}\backslash right)$ is the isothermal compressibility.

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

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

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.