# 18.6: Isothermal Compressibilities

- Page ID
- 591

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, p_{b}):

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

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

## Contributors and Attributions

Prof. Michael Adewumi (The Pennsylvania State University). Some or all of the content of this module was taken from Penn State's College of Earth and Mineral Sciences' OER Initiative.