5.6: Expressions for Heat Capacity

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As explained in Sec. 3.1.5, the heat capacity of a closed system is defined as the ratio of an infinitesimal quantity of heat transferred across the boundary under specified conditions and the resulting infinitesimal temperature change: \(\tx{heat capacity} \defn \dq/\dif T\). The heat capacities of isochoric (constant volume) and isobaric (constant pressure) processes are of particular interest.

The heat capacity at constant volume, \(C_V\), is the ratio \(\dq/\dif T\) for a process in a closed constant-volume system with no nonexpansion work—that is, no work at all. The first law shows that under these conditions the internal energy change equals the heat: \(\dif U=\dq\) (Eq. 5.3.9). We can replace \(\dq\) by \(\dif U\) and write \(C_V\) as a partial derivative: \begin{gather} \s{ C_V = \Pd{U}{T}{V} } \tag{5.6.1} \cond{(closed system)} \end{gather}

If the closed system has more than two independent variables, additional conditions are needed to define \(C_V\) unambiguously. For instance, if the system is a gas mixture in which reaction can occur, we might specify that the system remains in reaction equilibrium as \(T\) changes at constant \(V\).

Equation 5.6.1 does not require the condition \(\dw'{=}0\), because all quantities appearing in the equation are state functions whose relations to one another are fixed by the nature of the system and not by the path. Thus, if heat transfer into the system at constant \(V\) causes \(U\) to increase at a certain rate with respect to \(T\), and this rate is defined as \(C_V\), the performance of electrical work on the system at constant \(V\) will cause the same rate of increase of \(U\) with respect to \(T\) and can equally well be used to evaluate \(C_V\).

Note that \(C_V\) is a state function whose value depends on the state of the system—that is, on \(T\), \(V\), and any additional independent variables. \(C_V\) is an extensive property: the combination of two identical phases has twice the value of \(C_V\) that one of the phases has by itself.

For a phase containing a pure substance, the molar heat capacity at constant volume is defined by \(\CVm \defn C_V/n\). \(\CVm\) is an intensive property.

If the system is an ideal gas, its internal energy depends only on \(T\), regardless of whether \(V\) is constant, and Eq. 5.6.1 can be simplified to \begin{gather} \s{ C_V = \frac{\dif U}{\dif T}} \tag{5.6.2} \cond{(closed system, ideal gas)} \end{gather} Thus the internal energy change of an ideal gas is given by \(\dif U=C_V\dif T\), as mentioned earlier in Sec. 3.5.3.

The heat capacity at constant pressure, \(C_p\), is the ratio \(\dq/\dif T\) for a process in a closed system with a constant, uniform pressure and with expansion work only. Under these conditions, the heat \(\dq\) is equal to the enthalpy change \(\dif H\) (Eq. 5.3.7), and we obtain a relation analogous to Eq. 5.6.1: \begin{gather} \s{ C_p = \Pd{H}{T}{\!p} } \tag{5.6.3} \cond{(closed system)} \end{gather} \(C_p\) is an extensive state function. For a phase containing a pure substance, the molar heat capacity at constant pressure is \(\Cpm = C_p/n\), an intensive property.

Since the enthalpy of a fixed amount of an ideal gas depends only on \(T\) (Prob. 5.1), we can write a relation analogous to Eq. 5.6.2: \begin{gather} \s{ C_p = \frac{\dif H}{\dif T}} \tag{5.6.4} \cond{(closed system, ideal gas)} \end{gather}