1.14.36: Irreversible Thermodynamics

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).

\[\mathrm{dS}=(\mathrm{q} / \mathrm{T})+(\mathrm{A} / \mathrm{T}) \, \mathrm{d} \xi ; \quad \mathrm{A} \, \mathrm{d} \xi \geq 0\]

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.

\[\mathrm{dS}=(\mathrm{q} / \mathrm{T})+\mathrm{d}_{\mathrm{i}} \mathrm{S} ; \quad \mathrm{d}_{\mathrm{i}} \mathrm{S} \geq 0\]

\(\mathrm{d}_{\mathrm{i}\mathrm{S}}\) 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.

\[\mathrm{T} \, \mathrm{d}_{\mathrm{i}} \mathrm{S}=\mathrm{A} \, \mathrm{d} \xi \geq 0\]

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.

\[\mathrm{P}[\mathrm{S}]=\mathrm{d}_{\mathrm{i}} \mathrm{S} / \mathrm{dt}=\int_{\mathrm{V}} \sigma[\mathrm{S}] \, \mathrm{dV} \geq 0\]

We combine equations (c) and (d).

\[\mathrm{P}[\mathrm{S}]=\frac{\mathrm{d}_{\mathrm{i}} \mathrm{S}}{\mathrm{dt}}=\frac{\mathrm{A}}{\mathrm{T}} \, \frac{\mathrm{d} \xi}{\mathrm{dt}} \geq 0\]

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,

\[P[S]=\frac{d_{i} S}{d t}=\frac{A}{T} \, \frac{1}{v_{j}} \, \frac{d n_{j}}{d t} \geq 0\]

We develop this equation into a form which has wider significance. We assume that the system is homogeneous such that for a system volume \(\mathrm{V}\),

\[\sigma[S]=P[S] / V\]

Then

\[\sigma[\mathrm{S}]=\frac{1}{\mathrm{~V}} \, \frac{\mathrm{d}_{\mathrm{i}} \mathrm{S}}{\mathrm{dt}}=\frac{\mathrm{A}}{\mathrm{T}} \, \frac{1}{\mathrm{v}_{\mathrm{j}}} \, \frac{1}{\mathrm{~V}} \, \frac{\mathrm{dn}_{\mathrm{j}}}{\mathrm{dt}} \geq 0\]

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),

\[\sigma[S]=\frac{\mathrm{A}}{\mathrm{T}} \, \frac{1}{\mathrm{v}_{\mathrm{j}}} \, \frac{\mathrm{dc}_{\mathrm{j}}}{\mathrm{dt}} \geq 0\]

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, \(\mathrm{J}_{\mathrm{ch}}\).

\[\mathrm{J}_{\mathrm{ch}}=\left(\mathrm{l} / \mathrm{v}_{\mathrm{j}}\right) \, \mathrm{dc}_{\mathrm{j}} / \mathrm{dt}\]

Then,

\[\sigma[S]=(A / T) \, J_{c h} \geq 0\]

Or,

\[\mathrm{T} \, \sigma[\mathrm{S}]=\mathrm{A} \, \mathrm{J}_{\mathrm{ch}} \geq 0\]

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.

1. Phenomenon--- Electrical Conductivity Law: Ohm’s Law (discovered 1826)
   \(\mathrm{I}=\mathrm{L} \, \mathrm{V}\) where \(\mathrm{L} =\) conductance; resistance \(\mathrm{R} = 1/\mathrm{L}\).
2. Phenomenon--- Diffusion Law: Fick’s Law discovered by Adolf Fick 1855
   \(\mathrm{J}_{\mathrm{j}}=\mathrm{D} \,\left(-\mathrm{d} \mu_{\mathrm{j}} / \mathrm{dx}\right)\)
3. Phenomenon— Thermal conductivity Law : Fourier’s Law (1822)
   \(\mathrm{J}_{\mathrm{q}}=\lambda \,(-\mathrm{dT} / \mathrm{dx})\) \(\lambda =\) thermal conductivity
4. 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.