1.14.57: Reversible Change
- Page ID
- 390920
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)In thermodynamics the term 'reversible' means that in such a system the affinity for spontaneous change \(\mathrm{A}\) is zero; we can in fact characterize the composition of the system by the symbol \(\xi^{\mathrm{eq}}\), indicating a time independent extent of chemical reaction. The composition of the system does not change because the affinity for spontaneous change is zero.
For a reversible change the affinity for spontaneous change is zero at all stages. The composition is represented by \(\xi^{\mathrm{eq}}\), and the rate of change \(\mathrm{d} \xi^{\mathrm{eq}} / \mathrm{dt}\) is zero, at defined \(\mathrm{T}\) and \(\mathrm{p}\). We represent the volume of the system using following equation.
\[\mathrm{V}=\mathrm{V}\left[\mathrm{T}, \mathrm{p}, \xi^{\mathrm{eq}}, \mathrm{A}=0\right] \nonumber \]
This equation means that the volume, a dependent variable, is unambiguously defined by the set of variables in the square brackets, [... ]. The pressure is changed from \(\mathrm{p}\) to \(\mathrm{p} + \Delta \mathrm{p}\), such that the new equilibrium composition is \(\xi + \Delta \xi\) where the affinity for spontaneous change is zero.
\[\mathrm{V}=\mathrm{V}\left[\mathrm{T},(\mathrm{p}+\Delta \mathrm{p}), \xi^{\mathrm{eq}}(\mathrm{p}+\Delta \mathrm{p}), \mathrm{A}=0\right] \nonumber \]
Under these circumstances the change from \(\mathrm{V}(\mathrm{p})\) to \(\mathrm{V}(\mathrm{p} + \Delta \mathrm{p})\) is from one equilibrium state where \(\mathrm{A} = 0\) to another equilibrium state where \(\mathrm{A}\) is also zero. Such an equilibrium transformation is, in thermodynamic terms, reversible. All changes under the constraint that \(\mathrm{A}\) remains at zero are reversible.