# 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

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.