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

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.

References

• McQuarrie, Donald A. (1976). Statistical mechanics. New York: Harper & Row. p. 269. ISBN 978-0-06-044366-5.

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