9.25: Summary- Thermodynamic Functions as Criteria for Change

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For a spontaneous process, we conclude that the entropy change of the system must satisfy the inequality \(\Delta S+\Delta \hat{S}>\)\(0\). For any process that occurs reversibly, we conclude that \(\Delta S+\Delta \hat{S}=0\). For every incremental part of a reversible process that occurs in a closed system, we have the following relationships: \[dE=TdS-PdV+dw^{rev}_{NPV} \nonumber \] \[dH=TdS+VdP+dw^{rev}_{NPV} \nonumber \] \[dA=-SdT-PdV+dw^{rev}_{NPV} \nonumber \] \[dG=-SdT+VdP+dw^{rev}_{NPV} \nonumber \]

At constant entropy, the energy relationship becomes:

\[{\left(dE\right)}_S=dw^{rev}_{net} \nonumber \] \[{\left(\Delta E\right)}_S=w^{rev}_{net} \nonumber \]

At constant temperature, the Helmholtz free energy relationship becomes:

\[{\left(dA\right)}_T=dw^{rev}_{net} \nonumber \] \[{\left(\Delta A\right)}_T=w^{rev}_{net} \nonumber \]

For reversible processes in which all work is pressure–volume work:

\[dE=TdS-PdV \nonumber \] \[dH=TdS+VdP \nonumber \] \[dA=-SdT-PdV \nonumber \] \[dG=-SdT+VdP \nonumber \]

From these general equations, we find the following relationships for reversible processes when various pairs of variables are held constant:

\[{\left(dS\right)}_{EV}={-dw^{rev}_{NPV}}/{T} {\left(\Delta S\right)}_{EV}={-w^{rev}_{NPV}}/{T} \nonumber \] \[{\left(dS\right)}_{HP}={-dw^{rev}_{NPV}}/{T} {\left(\Delta S\right)}_{HP}={-w^{rev}_{NPV}}/{T} \nonumber \] \[{\left(dE\right)}_{SV}=dw^{rev}_{NPV} {\left(\Delta E\right)}_{SV}=w^{rev}_{NPV} \nonumber \] \[{\left(dH\right)}_{SP}=dw^{rev}_{NPV} {\left(\Delta H\right)}_{SP}=w^{rev}_{NPV} \nonumber \] \[{\left(dA\right)}_{TV}=dw^{rev}_{NPV} {\left(\Delta A\right)}_{TV}=w^{rev}_{NPV} \nonumber \] \[{\left(dG\right)}_{TP}=dw^{rev}_{NPV} {\left(\Delta G\right)}_{TP}=w^{rev}_{NPV} \nonumber \]

If the only work is pressure–volume work, then \(dw^{rev}_{NPV}=0\), \(w^{rev}_{NPV}=0\), and these relationships become:

\[{\left(dS\right)}_{EV}=0 {\left(\Delta S\right)}_{EV}=0 \nonumber \] \[{\left(dS\right)}_{HP}=0 {\left(\Delta S\right)}_{HP}=0 \nonumber \] \[{\left(dE\right)}_{SV}=0 {\left(\Delta E\right)}_{SV}=0 \nonumber \] \[{\left(dH\right)}_{SP}=0 {\left(\Delta H\right)}_{SP}=0 \nonumber \] \[{\left(dA\right)}_{TV}=0 {\left(\Delta A\right)}_{TV}=0 \nonumber \] \[{\left(dG\right)}_{TP}=0 {\left(\Delta G\right)}_{TP}=0 \nonumber \]

For every incremental part of an irreversible process that occurs in a closed system at constant entropy:

\[{dq}^{spon}<0 \nonumber \]

and

\[{\left(dE\right)}_S<{dw}^{spon}_{net} \nonumber \]

and

\[q^{spon}<0 \nonumber \]

and

\[{\left(\Delta E\right)}_S<w^{spon}_{net} \nonumber \]

For an irreversible process at constant temperature:

\[{dq}^{spon}<\hat{T}dS \nonumber \]

and

\[{\left(dA\right)}_{\hat{T}}<{dw}^{spon}_{net} \nonumber \]

and

\[q^{spon}<\hat{T}\Delta S \nonumber \]

and

\[{\left(\Delta A\right)}_{\hat{T}}<w^{spon}_{net} \nonumber \]

When an irreversible process occurs with various pairs of variables held constant, we find:

\[{\left(dS\right)}_{EV}>{-dw^{spon}_{NPV}}/{\hat{T}} {\left(\Delta S\right)}_{EV}={-w^{spon}_{NPV}}/{\hat{T}} \nonumber \]

\[{\left(dS\right)}_{HP}>{-dw^{spon}_{NPV}}/{\hat{T}} {\left(\Delta S\right)}_{HP}>{-w^{spon}_{NPV}}/{\hat{T}} \nonumber \]

\[{\left(dE\right)}_{SV} \nonumber \]

\[{\left(dH\right)}_{SP} \nonumber \]

\[{\left(dA\right)}_{\hat{T}V} \nonumber \]

\[{\left(dG\right)}_{\hat{T}P} \nonumber \]

For irreversible processes in which the only work is pressure–volume work, these inequalities become:

\[{\left(dS\right)}_{EV}>0 {\left(\Delta S\right)}_{EV}>0 \nonumber \] \[{\left(dS\right)}_{HP}>0 {\left(\Delta S\right)}_{HP}>0 \nonumber \] \[{\left(dE\right)}_{SV}<0 {\left(\Delta E\right)}_{SV}<0 \nonumber \] \[{\left(dH\right)}_{SP}<0 {\left(\Delta H\right)}_{SP}<0 \nonumber \] \[{\left(dA\right)}_{\hat{T}V}<0 {\left(\Delta A\right)}_{\hat{T}V}<0 \nonumber \] \[{\left(dG\right)}_{\hat{T}P}<0 {\left(\Delta G\right)}_{\hat{T}P}<0 \nonumber \]