# 18.6: Isothermal Compressibilities

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The isothermal compressibility of a fluid is defined as follows:

$c_{f}=-\frac{1}{V}\left(\frac{\partial V}{\partial \rho}\right)_{T} \label{18.15}$

This expression can be also given in term of fluid density, as follows:

$c_{f}=-\frac{1}{\rho}\left(\frac{\partial \rho}{\partial P}\right)_{T} \label{18.16}$

For liquids, the value of isothermal compressibility is very small because a unitary change in pressure causes a very small change in volume for a liquid. In fact, for slightly compressible liquid, the value of compressibility ($$c_o$$) is usually assumed independent of pressure. Therefore, for small ranges of pressure across which $$c_o$$ is nearly constant, Equation \ref{18.16} can be integrated to get:

$c_{o}\left(p-p_{b}\right)=\ln \left(\frac{\rho_{o}}{\rho_{o b}}\right) \label{18.17}$

In such a case, the following expression can be derived to relate two different liquid densities ($$\rho_{o}$$, $$\rho_{ob}$$,ob) at two different pressures (p, pb):

$\rho_{o}=\rho_{o b}\left[1+c_{o}\left(p-p_{b}\right)\right] \label{18.18}$

The Vasquez-Beggs correlation is the most commonly used relationship for $$c_o$$.

For natural gases, isothermal compressibility varies significantly with pressure. By introducing the real gas law into Equation \ref{18.16}, it is easy to prove that, for gases:

$c_{g}=\frac{1}{P}-\frac{1}{Z}\left(\frac{\partial Z}{\partial P}\right)_{r} \label{18.19}$

Note that for an ideal gas, cg is just the reciprocal of the pressure. “cg” can be readily calculated by graphical means (chart of Z versus P) or by introducing an equation of state into Equation \ref{18.19}.