9.22: The Entropy Change for A Spontaneous Process at Constant H and P

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For any spontaneous process, we have

\[dH=dE+PdV+VdP=dq^{spon}-P_{applied}dV+dw^{spon}_{NPV}+PdV+VdP \nonumber \]

If the pressure is constant (\(P=P_{applied}=\mathrm{constant}\)), this becomes \(dq^{spon}=dH-dw^{spon}_{NPV}\). Substituting our result from Section 9.15, we have

\[\hat{T}{\left(dS\right)}_P>dH-dw^{spon}_{NPV} \nonumber \] (spontaneous process, constant \(P\))

If the enthalpy of the system is also constant throughout the process, we have

\[\hat{T}{\left(dS\right)}_{HP}>-dw^{spon}_{NPV} \nonumber \] (spontaneous process, constant \(H\) and \(P\))

Dividing by \(\hat{T}\) and repeating our earlier result for a reversible process, we have the parallel relationships

\[{\left(dS\right)}_{HP}>\frac{-dw^{spon}_{NPV}}{\hat{T}} \nonumber \] (spontaneous process, constant \(H\) and \(P\)) \[{\left(dS\right)}_{HP}=\frac{-dw^{rev}_{NPV}}{\hat{T}} \nonumber \] (reversible process, constant \(H\) and \(P\))

If it is also true that the temperature of the surroundings is constant, summing the incremental contributions to a finite change of state produces the parallel relationships

\[{\left(\Delta S\right)}_{HP}>\frac{-w^{spon}_{NPV}}{\hat{T}} \nonumber \] (spontaneous process, constant \(H\), \(P\), and \(\hat{T}\))

\[{\left(\Delta S\right)}_{HP}>\frac{-w^{rev}_{NPV}}{\hat{T}} \nonumber \] (reversible process, constant \(H\), \(P\), and \(\hat{T}=T\))

If only pressure–volume work is possible, we have \(dw^{spon}_{NPV}=0\), and

\[{\left(dS\right)}_{HP}>0 \nonumber \] (spontaneous process, constant \(H\), \(P\), only \(PV\) work)

\[{\left(dS\right)}_{HP}=0 \nonumber \] (reversible process, constant \(H\) and \(P\), only \(PV\) work)

and for a finite change of state,

\[{\left(\Delta S\right)}_{HP}>0 \nonumber \] (spontaneous process, only \(PV\) work)

\[{\left(\Delta S\right)}_{HP}=0 \nonumber \] (reversible process, only \(PV\) work)

In this and earlier sections, we develop criteria for spontaneous change that are based on \(dE\) and \(dH\). We are now able to develop similar criteria for a spontaneous change in a system that is in thermal contact with constant-temperature surroundings. These criteria are based on \(dA\) and \(dG\). However, before doing so, we develop a general relationship between the isothermal work in a spontaneous process and the isothermal work in a reversible process, when these processes take a system from a common initial state to a common final state.