7.1: Volume Properties
Two volume properties of a closed system are defined as follows: \begin{equation} \textbf{cubic expansion coefficient} \quad \alpha \defn \frac{1}{V}\Pd{V}{T}{p} \tag{7.1.1} \end{equation} \begin{equation} \textbf{isothermal compressibility} \quad \kT \defn -\frac{1}{V}\Pd{V}{p}{T} \tag{7.1.2} \end{equation}
The cubic expansion coefficient is also called the coefficient of thermal expansion and the expansivity coefficient. Other symbols for the isothermal compressibility are \(\beta\) and \(\g_T\).
These definitions show that \(\alpha\) is the fractional volume increase per unit temperature increase at constant pressure, and \(\kT\) is the fractional volume decrease per unit pressure increase at constant temperature. Both quantities are intensive properties. Most substances have positive values of \(\alpha\), and all substances have positive values of \(\kT\), because a pressure increase at constant temperature requires a volume decrease.
The cubic expansion coefficient is not always positive. \(\alpha\) is negative for liquid water below its temperature of maximum density, \(3.98\units{\(\degC\)}\). The crystalline ceramics zirconium tungstate (ZrW\(_2\)O\(_8\)) and hafnium tungstate (HfW\(_2\)O\(_8\)) have the remarkable behavior of contracting uniformly and continuously in all three dimensions when they are heated from \(0.3\K\) to about \(1050\K\); \(\alpha\) is negative throughout this very wide temperature range (T. A. Mary et al, Science , 272 , 90–92, 1996). The intermetallic compound YbGaGe has been found to have a value of \(\alpha\) that is practically zero in the range \(100\)–\(300\K\) (James R. Salvador et al, Nature , 425 , 702–705, 2003).
If an amount \(n\) of a substance is in a single phase, we can divide the numerator and denominator of the right sides of Eqs. 7.1.1 and 7.1.2 by \(n\) to obtain the alternative expressions \begin{gather} \s{ \alpha = \frac{1}{V\m}\Pd{V\m}{T}{\!p} } \tag{7.1.3} \cond{(pure substance, \(P{=}1\))} \end{gather} \begin{gather} \s{ \kT = -\frac{1}{V\m}\Pd{V\m}{p}{T} } \tag{7.1.4} \cond{(pure substance, \(P{=}1\))} \end{gather} where \(V\m\) is the molar volume. \(P\) in the conditions of validity is the number of phases. Note that only intensive properties appear in Eqs. 7.1.3 and 7.1.4; the amount of the substance is irrelevant. Figures 7.1 and 7.2 show the temperature variation of \(\alpha\) and \(\kT\) for several substances.
If we choose \(T\) and \(p\) as the independent variables of the closed system, the total differential of \(V\) is given by \begin{equation} \dif V = \Pd{V}{T}{\!p}\dif T + \Pd{V}{p}{T}\difp \tag{7.1.5} \end{equation} With the substitutions \(\pd{V}{T}{p} = \alpha V\) (from Eq. 7.1.1) and \(\pd{V}{p}{T} = -\kT V\) (from Eq. 7.1.2), the expression for the total differential of \(V\) becomes \begin{gather} \s{ \dif V = \alpha V \dif T - \kT V \difp } \tag{7.1.6} \cond{(closed system,} \nextcond{\(C{=}1\), \(P{=}1\))} \end{gather} To find how \(p\) varies with \(T\) in a closed system kept at constant volume, we set \(\dif V\) equal to zero in Eq. 7.1.6: \(0 = \alpha V\dif T - \kappa _T V\difp\), or \(\difp/\dif T = \alpha /\kappa _T\). Since \(\difp/\dif T\) under the condition of constant volume is the partial derivative \(\pd{p}{T}{V}\), we have the general relation \begin{gather} \s{ \Pd{p}{T}{V} = \frac{\alpha}{\kT} } \tag{7.1.7} \cond{(closed system,} \nextcond{\(C{=}1\), \(P{=}1\))} \end{gather}