9.21: The Entropy Change for A Spontaneous Process at Constant E and V
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For any spontaneous process, we have \(dE={dq}^{spon}\)+\({dw}^{spon}\), which we can rearrange to \({dq}^{spon}=dE-{dw}^{spon}\). Substituting our result from Section 9.15, we have
\[\hat{T}dS>dE-dw^{spon} \nonumber \] (spontaneous process)
If the energy of the system is constant throughout the process, we have \(dE=0\) and
\[\hat{T}{\left(dS\right)}_E>-dw^{spon} \nonumber \] (spontaneous process, constant energy)
The spontaneous work is the sum of the pressure–volume work and the non-pressure–volume work, \(\ dw^{spon}={dw}^{spon}_{PV}+{dw}^{spon}_{NPV}\). If we introduce the further condition that the spontaneous process occurs while the volume of the system remains constant, we have \({dw}^{spon}_{PV}=0\). Making this substitution and repeating our earlier result for a reversible process, we have the parallel relationships
\[{\left(dS\right)}_{EV}>\frac{-dw^{spon}_{NPV}}{\hat{T}} \nonumber \] (spontaneous process, constant \(E\) and \(V\))
\[{\left(dS\right)}_{EV}=\frac{-dw^{spon}_{NPV}}{\hat{T}} \nonumber \] (reversible process, constant \(E\) and \(V\))
(For a reversible process, \(T=\hat{T}\).) If the spontaneous process occurs while \(\hat{T}\) is constant, summing the incremental contributions to a finite change of state produces the parallel relationships
\[{\left(\Delta S\right)}_{EV}>\frac{-w^{spon}_{NPV}}{\hat{T}} \nonumber \] (spontaneous process, constant \(E\), \(V\), and \(\hat{T}\))
\[{\left(\Delta S\right)}_{EV}=\frac{-w^{spon}_{NPV}}{\hat{T}} \nonumber \] (reversible process, constant \(E\), \(V\), and \(\hat{T}\))
Constant \(\hat{T}\) corresponds to the common situation in chemical experimentation in which we place a reaction vessel in a constant-temperature bath. If we introduce the further condition that only pressure–volume work is possible, we have \(dw^{spon}_{NPV}=0\). The parallel relationships become
\[{\left(dS\right)}_{EV}>0 \nonumber \] (spontaneous process, constant \(E\) and \(V\), only \(PV\) work)
\[{\left(dS\right)}_{EV}=0 \nonumber \] (reversible process, constant \(E\) and \(V\), only\(\ PV\) work)
If the energy and volume are constant for a system in which only pressure–volume work is possible, the system is isolated. The conditions we have just derived are entirely equivalent to our earlier conclusions that \(dS=0\) and \(dS>0\) for an isolated system that is at equilibrium or undergoing a spontaneous change, respectively. Summing the incremental contributions to a finite change of state produces the parallel relationships
\[{\left(\Delta S\right)}_{EV}>0 \nonumber \] (spontaneous process, only \(PV\) work)
\[{\left(\Delta S\right)}_{EV}=0 \nonumber \] (reversible process, only \(PV\) work)
The validity of these expressions is independent of any variation in either \(T\) or \(\hat{T}\).