1.14.36: Irreversible Thermodynamics
- Page ID
- 387084
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)According to the Second Law of Thermodynamics, the change in entropy \(\mathrm{dS}\) is related to the affinity for spontaneous change \(\mathrm{A}\) using equation (a).
\[\mathrm{dS}=(\mathrm{q} / \mathrm{T})+(\mathrm{A} / \mathrm{T}) \, \mathrm{d} \xi ; \quad \mathrm{A} \, \mathrm{d} \xi \geq 0 \nonumber \]
In these terms chemists usually have in mind a chemical reaction driven by the affinity \(\mathrm{A}\) for spontaneous chemical reaction producing extent of reaction \(\mathrm{d}\xi\). We generalize the law in the following terms.
\[\mathrm{dS}=(\mathrm{q} / \mathrm{T})+\mathrm{d}_{\mathrm{i}} \mathrm{S} ; \quad \mathrm{d}_{\mathrm{i}} \mathrm{S} \geq 0 \nonumber \]
\(\mathrm{d}_{\mathrm{i}\mathrm{S}}\) is the change in entropy of the system by virtue of spontaneous processes in the system. Comparison of equations (a) and (b) yields the following equation.
\[\mathrm{T} \, \mathrm{d}_{\mathrm{i}} \mathrm{S}=\mathrm{A} \, \mathrm{d} \xi \geq 0 \nonumber \]
We introduce two new terms. A quantity \(\mathrm{P}[\mathrm{S}]\) describes the rate of entropy production within the system; a quantity \(\sigma[\mathrm{S}]\) describes the corresponding rate of entropy production in unit volume of the system.
\[\mathrm{P}[\mathrm{S}]=\mathrm{d}_{\mathrm{i}} \mathrm{S} / \mathrm{dt}=\int_{\mathrm{V}} \sigma[\mathrm{S}] \, \mathrm{dV} \geq 0 \nonumber \]
We combine equations (c) and (d).
\[\mathrm{P}[\mathrm{S}]=\frac{\mathrm{d}_{\mathrm{i}} \mathrm{S}}{\mathrm{dt}}=\frac{\mathrm{A}}{\mathrm{T}} \, \frac{\mathrm{d} \xi}{\mathrm{dt}} \geq 0 \nonumber \]
But if \(\mathrm{dn}_{j}\) is the change in amount of chemical substance \(j\) in the system, \(\mathrm{dn} \mathrm{j}_{\mathrm{j}}=\mathrm{v}_{\mathrm{j}} \, \mathrm{d} \xi\). Then,
\[P[S]=\frac{d_{i} S}{d t}=\frac{A}{T} \, \frac{1}{v_{j}} \, \frac{d n_{j}}{d t} \geq 0 \nonumber \]
We develop this equation into a form which has wider significance. We assume that the system is homogeneous such that for a system volume \(\mathrm{V}\),
\[\sigma[S]=P[S] / V \nonumber \]
Then
\[\sigma[\mathrm{S}]=\frac{1}{\mathrm{~V}} \, \frac{\mathrm{d}_{\mathrm{i}} \mathrm{S}}{\mathrm{dt}}=\frac{\mathrm{A}}{\mathrm{T}} \, \frac{1}{\mathrm{v}_{\mathrm{j}}} \, \frac{1}{\mathrm{~V}} \, \frac{\mathrm{dn}_{\mathrm{j}}}{\mathrm{dt}} \geq 0 \nonumber \]
But the concentration of chemical substance \(j\), \(\mathrm{c}_{\mathrm{j}}=\mathrm{n}_{\mathrm{j}} / \mathrm{V}\). Then, \(\mathrm{dc}_{\mathrm{j}}=\mathrm{dn}_{\mathrm{j}} / \mathrm{V}\). Hence from equation (h),
\[\sigma[S]=\frac{\mathrm{A}}{\mathrm{T}} \, \frac{1}{\mathrm{v}_{\mathrm{j}}} \, \frac{\mathrm{dc}_{\mathrm{j}}}{\mathrm{dt}} \geq 0 \nonumber \]
The quantity \(\left(1 / v_{j}\right) \, d c_{j} / d t\) describes the change in composition of the system, the flow of the system from reactants to products. In these terms we identify a chemical flow, \(\mathrm{J}_{\mathrm{ch}}\).
\[\mathrm{J}_{\mathrm{ch}}=\left(\mathrm{l} / \mathrm{v}_{\mathrm{j}}\right) \, \mathrm{dc}_{\mathrm{j}} / \mathrm{dt} \nonumber \]
Then,
\[\sigma[S]=(A / T) \, J_{c h} \geq 0 \nonumber \]
Or,
\[\mathrm{T} \, \sigma[\mathrm{S}]=\mathrm{A} \, \mathrm{J}_{\mathrm{ch}} \geq 0 \nonumber \]
The latter equation has an interesting feature; \(\mathrm{T} \, \sigma[\mathrm{S}]\) is given by the product of the affinity for spontaneous chemical reaction (the driving force) and the accompanying flow. Indeed \(\mathrm{T} \, \sigma[\mathrm{S}]\) is related to the rate of entropy production in the system. Thermodynamics takes us no further. We make an extrathermodynamic leap and suggest that the flow is proportional to the force; i.e. the stronger the driving force the more rapid the chemical flow from reactants to products.
In general terms phenomenological equations start out from the basis of a linear model described by a phenomenological law of the general form, \(\mathrm{J} = \mathrm{L} \, \(\mathrm{~X}\) where \(\mathrm{J}\) is the flow and \(\mathrm{X}\) is the conjugate force such that the product \(\mathrm{J} \, \mathrm{~X}\) yields the rate of entropy production. These laws are based on experiment. Many such phenomenological laws have been proposed. Some examples are listed below.
- Phenomenon--- Electrical Conductivity Law: Ohm’s Law (discovered 1826)
\(\mathrm{I}=\mathrm{L} \, \mathrm{V}\) where \(\mathrm{L} =\) conductance; resistance \(\mathrm{R} = 1/\mathrm{L}\). - Phenomenon--- Diffusion Law: Fick’s Law discovered by Adolf Fick 1855
\(\mathrm{J}_{\mathrm{j}}=\mathrm{D} \,\left(-\mathrm{d} \mu_{\mathrm{j}} / \mathrm{dx}\right)\) - Phenomenon— Thermal conductivity Law : Fourier’s Law (1822)
\(\mathrm{J}_{\mathrm{q}}=\lambda \,(-\mathrm{dT} / \mathrm{dx})\) \(\lambda =\) thermal conductivity - Phenomenon—Chemical reaction If the patterns described above were followed we might write,
\(J_{c h}=L \, A\)
In 1890 Nernst suggested this approach to chemical kinetics. Unfortunately chemists have no method for measuring the affinity; there is no affinity meter. Instead chemists use the Law of Mass Action.