5.8: Criteria for Spontaneity
In this section we combine the first and second laws in order to derive some general relations for changes during a reversible or irreversible process of a closed system. The temperature and pressure will be assumed to be practically uniform during the process, even if the process is irreversible. For example, the volume might be changing at a finite rate but very slowly, or there might be a spontaneous homogeneous reaction in a mixture of uniform temperature and pressure.
The second law states that \(\dif S\) is equal to \(\dq/T\) if the process is reversible, and is greater than \(\dq/T\) if the process is irreversible: \begin{gather} \s{\dif S \ge \dq/T} \tag{5.8.1} \cond{(\({}\subs{ rev}\sups{ irrev}\), closed system)} \end{gather} or \begin{gather} \s{ \dq \le T \dif S } \tag{5.8.2} \cond{(\({}\subs{ rev}\sups{ irrev}\), closed system)} \end{gather} The inequalities in these relations refer to an irreversible process and the equalities to a reversible process, as indicated by the notation \({}\subs{ rev}\sups{ irrev}\).
When we substitute \(\dq\) from Eq. 5.8.2 into the first law in the form \(\dif U=\dq-p\dif V+\dw'\), where \(\dw'\) is nonexpansion work, we obtain the relation \begin{gather} \s{ \dif U \le T \dif S - p \dif V + \dw' } \tag{5.8.3} \cond{(\({}\subs{ rev}\sups{ irrev}\), closed system)} \end{gather} We substitute this relation for \(\dif U\) into the differentials of enthalpy, Helmholtz energy, and Gibbs energy given by Eqs. 5.3.4–5.3.6 to obtain three more relations:
\begin{gather} \s{ \dif H \le T \dif S + V \difp + \dw' } \tag{5.8.4} \cond{(\({}\subs{ rev}\sups{ irrev}\), closed system)} \end{gather} \begin{gather} \s{ \dif A \le -S \dif T - p \dif V + \dw' } \tag{5.8.5} \cond{(\({}\subs{ rev}\sups{ irrev}\), closed system)} \end{gather} \begin{gather} \s{ \dif G \le -S \dif T + V \difp + \dw' } \tag{5.8.6} \cond{(\({}\subs{ rev}\sups{ irrev}\), closed system)} \end{gather}
The last two of these relations provide valuable criteria for spontaneity under common laboratory conditions. Equation 5.8.5 shows that during a spontaneous irreversible change at constant temperature and volume, \(\dif A\) is less than \(\dw'\). If the only work is expansion work (i.e., \(\dw'\) is zero), the Helmholtz energy decreases during a spontaneous process at constant \(T\) and \(V\) and has its minimum value when the system reaches an equilibrium state.
Equation 5.8.6 is especially useful. From it, we can conclude the following:
-
Ben-Amotz and Honig (
J. Chem. Phys.
,
118
, 5932–5936, 2003;
J. Chem. Educ.
,
83
, 132–137, 2006) developed a “rectification” procedure that simplifies the mathematical manipulation of inequalities. Following this procedure, we can write \begin{equation} \dif S = \dq/T + \dBar\theta \tag{5.8.7} \end{equation} where \(\dBar\theta\) is an
excess entropy function
that is positive for an irreversible change and zero for a reversible change (\(\dBar\theta \geq 0\)). Solving for \(\dq\) gives the expression \(\dq=T\dif S-T\dBar\theta\) that, when substituted in the first law expression \(\dif U=\dq-p\dif V+\dw'\), produces \begin{equation} \dif U = T\dif S-p\dif V+\dw'-T\dBar\theta \tag{5.8.8} \end{equation} The equality of this equation is equivalent to the combined equality and inequality of Eq. 5.8.3. Then by substitution of this expression for \(\dif U\) into Eqs. 5.3.4–5.3.6, we obtain equalities equivalent to Eqs. 5.8.4–5.8.6, for example \begin{equation} \dif G = -S\dif T + V\difp + \dw' - T\dBar\theta \tag{5.8.9} \end{equation} Equation 5.8.9 tells us that during a process at constant \(T\) and \(p\), with expansion work only (\(\dw'{=}0\)), \(\dif G\) has the same sign as \(-T\dBar\theta\): negative for an irreversible change and zero for a reversible change.