1.9.3: Entropy and Spontaneous Reaction
- Page ID
- 375466
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)
( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\id}{\mathrm{id}}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\kernel}{\mathrm{null}\,}\)
\( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\)
\( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\)
\( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)
\( \newcommand{\vectorA}[1]{\vec{#1}} % arrow\)
\( \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow\)
\( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vectorC}[1]{\textbf{#1}} \)
\( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)
\( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)
\( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)It is often stated that the entropy of a system is a maximum at equilibrium. This is not generally true and is certainly not the case for closed systems at either (a) fixed \(\mathrm{T}\) and \(\mathrm{p}\), or (b) fixed \(\mathrm{T}\) and \(\mathrm{V}\).
We rewrite the Master Equation in the following way:
\[\mathrm{dS}=(1 / \mathrm{T}) \, \mathrm{dU}+(\mathrm{p} / \mathrm{T}) \, \mathrm{dV}+(\mathrm{A} / \mathrm{T}) \, \mathrm{d} \xi ; \mathrm{A} \, \mathrm{d} \xi \geq \text { zero } \nonumber \]
Temperature \(\mathrm{T}\) is positive and non-zero. At constant energy and constant volume (i.e. isoenergetic and isochoric), spontaneous processes are accompanied by an increase in entropy. This statement is important in statistical thermodynamics where the condition, ‘constant \(\mathrm{U}\) and constant \(\mathrm{V}\)’ is important.
The following equation defines the enthalpy \(\mathrm{H}\) of a closed system.
\[\mathrm{H}=\mathrm{U}+\mathrm{p} \, \mathrm{V} \nonumber \]
Then
\[\mathrm{dU}=\mathrm{dH}-\mathrm{p} \, \mathrm{dV}-\mathrm{V} \, \mathrm{dp} \nonumber \]
From equation (a),
\[\mathrm{dS}=(1 / \mathrm{T}) \, \mathrm{dH}-(\mathrm{p} / \mathrm{T}) \, \mathrm{dV}-(\mathrm{V} / \mathrm{T}) \, \mathrm{dp}+(\mathrm{p} / \mathrm{T}) \, \mathrm{dV}+(\mathrm{A} / \mathrm{T}) \, \mathrm{d} \xi \text { with } \mathrm{A} \, \mathrm{d} \xi \geq \text { zero } \nonumber \]
Hence,
\[\mathrm{dS}=(1 / \mathrm{T}) \, \mathrm{dH}-(\mathrm{V} / \mathrm{T}) \, \mathrm{dp}+(\mathrm{A} / \mathrm{T}) \, \mathrm{d} \xi ; \mathrm{A} \, \mathrm{d} \xi \geq \text { zero } \nonumber \]
Temperature \(\mathrm{T}\) is always positive. Hence at constant enthalpy and pressure (i.e. iso-enthalpic and isobaric) all spontaneous processes produce an increase in entropy.
We have identified two sets of conditions under which an increase in entropy accompanies a spontaneous process. If we follow through a similar argument with respect to the Gibbs energy, the outcome is not straightforward. By definition,
\[\mathrm{G}=\mathrm{H}-\mathrm{T} \, \mathrm{S} \nonumber \]
Then
\[\mathrm{dG}=\mathrm{dH}-\mathrm{T} \, \mathrm{dS}-\mathrm{S} \, \mathrm{dT} \nonumber \]
Or,
\[\mathrm{S}=-\mathrm{dG} / \mathrm{dT}+\mathrm{dH} / \mathrm{dT}-\mathrm{T} \, \mathrm{dS} / \mathrm{dT} \nonumber \]
But from equation (e)
\[\mathrm{dH} / \mathrm{dT}=\mathrm{T} \, \mathrm{dS} / \mathrm{dT}+(\mathrm{V} / \mathrm{T}) \, \mathrm{dp} / \mathrm{dT}-(\mathrm{A} / \mathrm{T}) \, \mathrm{d} \xi / \mathrm{dT} \nonumber \]
Hence,
\[\mathrm{S}=-(\mathrm{dG} / \mathrm{dT})+\mathrm{V} \,(\mathrm{dp} / \mathrm{dT})-\mathrm{A} \,(\mathrm{d} \xi / \mathrm{dT}) \text { with } \mathrm{A} \, \mathrm{d} \xi \geq \text { zero } \nonumber \]
Clearly no definite conclusions can be drawn about changes in entropy S under isobaric - isothermal conditions. We stress these points because again it is often tempting to link, misguidedly, entropies to the degree of ‘muddled-up-ness’. This is the basis of many explanations of entropy. For example, neither the volume nor energy of a deck of cards change on shuffling. Whether what actually happens on shuffling a new well-ordered deck of cards clarifies the meaning of entropy seems doubtful.