9.25: Summary- Thermodynamic Functions as Criteria for Change
- Page ID
- 152100
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)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:
\[{\left(dS\right)}_{EV}>0 {\left(\Delta S\right)}_{EV}>0 \nonumber \] \[{\left(dS\right)}_{HP}>0 {\left(\Delta S\right)}_{HP}>0 \nonumber \] \[{\left(dE\right)}_{SV}<0 {\left(\Delta E\right)}_{SV}<0 \nonumber \] \[{\left(dH\right)}_{SP}<0 {\left(\Delta H\right)}_{SP}<0 \nonumber \] \[{\left(dA\right)}_{\hat{T}V}<0 {\left(\Delta A\right)}_{\hat{T}V}<0 \nonumber \] \[{\left(dG\right)}_{\hat{T}P}<0 {\left(\Delta G\right)}_{\hat{T}P}<0 \nonumber \]