# 5.3: Liquid Properties and Liquid Mixtures

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When performing material balances for liquid systems, one must have access to reliable liquid properties. Unlike gases, the densities of liquids are weak functions of pressure and temperature, i.e., *large changes* in pressure and temperature result in *small changes* in the density. The changes in liquid density due to changes in pressure are determined by the** coefficient of compressibility** which is defined by

\[\kappa =\left\{\begin{array}{c} \text{ coefficient of} \\ \text{ compressibility} \end{array}\right\} =\frac{1}{\rho } \left(\frac{\partial \rho }{\partial p} \right)_{T} \label{16}\]

Changes in liquid density due to changes in temperature are determined by the coefficient of thermal expansion which is defined by

\[\beta =\left\{\begin{array}{c} \text{ coefficient of thermal} \\ \text{ expansion} \end{array}\right\} =- \frac{1}{\rho } \left(\frac{\partial \rho }{\partial T} \right)_{p} \label{17}\]

The **coefficient of thermal expansion**, \(\beta\), is defined with a negative sign since the density of most liquids decreases with increasing temperature. Using a negative sign in the definition of the thermal expansion coefficient makes \(\beta\) positive for most liquids. There is an interesting counterexample, however, and it is the density of liquid water at low temperatures. For liquid water between 4 C and the freezing point, 0 C, the coefficient of expansion is negative, i.e., \(\beta <0\). If it were not for this characteristic, water in lakes and rivers would freeze from the bottom during the winter, and this would destroy most aquatic life.

The density of *ideal liquid mixtures* is computed using Amagat’s law. Assuming that the total volume of a mixture is equal to the sum of the volume of the components of the mixture, we obtain

\[V=\sum_{A = 1}^{A = N} m_{A} / \rho_{A}^\text{ o} =\sum_{A = 1}^{A = N}V_{A} \label{18}\]

Here \(V\) is the volume of the mixture, while \(m_{A}\) and \(\rho_{A}^\text{ o}\) represent the masses and densities of the pure components. Equation \ref{18} can be used to compute the density of the mixture according to

\[\rho =\frac{m}{V} =\frac{m}{\sum_{A = 1}^{A = N} m_{A} / \rho_{A}^\text{ o} } =\frac{1}{ \sum_{A = 1}^{A = N} \omega_{A} / \rho_{A}^\text{ o} } \label{19}\]

Non-ideal behavior of liquid mixtures is a very complex topic. At low to moderate pressures, this version of Amagat’s law for liquids is a satisfactory approximation. When some of the components of a liquid mixture are above their boiling points, or if the components are polar, the use of Eqs. \ref{18} or \ref{19} may give significant errors^{2}.