1.14.37: Irreversible Thermodynamics: Onsager Phenomenological Equations

The major thrust of the account presented in these Topics concerns reversible processes in which system and surroundings are in thermodynamic equilibrium. When attention turns to non-equilibrium processes, the thermodynamic treatment is necessarily more complicated [1-5]. Here we examine several aspects of irreversible thermodynamics for near equilibrium in open systems[1]. In other words there is a strong ‘communication’ between system and surroundings. With increasing displacement of a given system from equilibrium the thermodynamic analysis becomes more complicated and controversial [2-5]. Here we confine attention to processes in near-equilibrium states [6]. The key assumption is that equations describing relationships between thermodynamic properties are valid for small elemental volumes, the concept of local equilibrium. Hence we can in a description of a given system identify the energy per unit volume and the entropy per unit volume in the context respectively of the first and second laws of thermodynamics.

With respect to a small volume \(\mathrm{dv}\) of a given system the change in entropy \(\mathrm{ds}\) is given by the change in entropy \(\mathrm{d}_{\mathrm{i}} \mathrm{s}\) by virtue of processes within a small volume \(\mathrm{dv}\) and by virtue of exchange with the rest of the system, des. The rate of change of \(\mathrm{d}_{\mathrm{i}} \mathrm{s}\), namely \(\mathrm{d}_{\mathrm{i}} \mathrm{s} / \mathrm{dt}\) is the local entropy production and is determined by the following condition.

\[\sigma \equiv \mathrm{d}_{\mathrm{i}} \mathrm{s} / \mathrm{dt} \geq 0\]

Phenomenological Laws

For systems close to thermodynamic equilibrium, the entropy production per unit volume \(\sigma\) can be expressed as the sum of products of forces \(\mathrm{X}_{\mathrm{k}}\) and conjugate flows, \(\mathrm{J}_{\mathrm{k}}\). Thus for \(\mathrm{k}\) flows and forces,

\[\sigma=\sum_{\mathrm{k}} \mathrm{X}_{\mathrm{k}} \, \mathrm{J}_{\mathrm{k}}\]

The condition ‘conjugate’ is important in the sense that for each flow \(\mathrm{J}_{k}\) there is a conjugate force \(\mathrm{X}_{k}\). For near equilibrium systems a given flow is a linear function of the conjugate force, \(\mathrm{X}_{k}\). Then,

\[\mathrm{J}_{\mathrm{k}}=\sum_{\mathrm{j}} \mathrm{L}_{\mathrm{kj}} \, \mathrm{X}_{\mathrm{k}}\]

The property \(\mathrm{L}_{\mathrm{kj}}\) is a phenomenological coefficient describing the dynamic flow and conjugate force.

In simple systems there is only one flow and one force such that the flow is directly proportional to the force. A classic example is Ohm’s law which can be written in the following form.

\[\mathrm{I}=(1 / \mathrm{R}) \, \mathrm{V}\]

Thus \(\mathrm{I}\) is the electric current, the rate of flow of electric charge for a system where the driving force is the electric potential gradient \(\mathrm{V}\). The relevant property of the system under consideration is the resistance \(\mathrm{R}\) or, preferably, its conductance \(\mathrm{L} (= 1/\(\mathrm{R}\)).

A similar phenomenological law is Fick’s Law of diffusion relating the rate of diffusion of chemical substance \(j\), \(\mathrm{J}_{j}\) to the concentration gradient \(\mathrm{dc}_{j}/\mathrm{dx}\) where \(\mathrm{D}_{j}\) describes the property of diffusion. Thus

\[\mathrm{J}_{\mathrm{j}}=\mathrm{D}_{\mathrm{j}} \,\left(\mathrm{dc}_{\mathrm{j}} / \mathrm{dx}\right)\]

The Law of Mass Action is a similar phenomenological law. In other words throughout chemistry (and indeed all sciences) there are phenomenological laws which do not, for example, follow from the first and second laws of thermodynamics.

Onsager Equations

Following on a proposal by Lord Rayleigh relating to mechanical properties, in 1931 Onsager [7] extended the ideas discussed above to include all forces and flows. For a system involving two flows and forces we may write the following two equations to describe near –equilibrium systems.

\[\mathrm{J}_{1}=\mathrm{L}_{11} \, \mathrm{X}_{1}+\mathrm{L}_{12} \, \mathrm{X}_{2}\]

\[\mathrm{J}_{2}=\mathrm{L}_{21} \, \mathrm{X}_{1}+\mathrm{L}_{22} \, \mathrm{X}_{2}\]

This formulation recognises that force \(\mathrm{X}_{2}\) may also produce a coupled flow \(\mathrm{J}_{1}\). In each case the products \(\mathrm{L}_{11} \, \mathrm{X}_{1}, \mathrm{~L}_{12} \, \mathrm{X}_{2}, \mathrm{~L}_{21} \, \mathrm{X}_{1}\) and \(\mathrm{L}_{22} \, \mathrm{X}_{2}\) involve conjugate flows and forces such that the product, \(\mathrm{J}_{\mathrm{i}} \, \mathrm{X}_{\mathrm{i}}\) has the dimension of the rate of entropy production. The cross terms \(\mathrm{L}_{12}\) and \(\mathrm{L}_{21}\) are the coupling coefficients such that for example, force X2 produces flow J1.

Onsager’s Law

The key theoretical advance made by Onsager was to show that for near-equilibrium states the matrix of coefficients is symmetric. Then, for example,[8]

\[\mathrm{L}_{12}=\mathrm{L}_{21}\]

The point can be developed by considering a system involving two flows and two forces. According to equation (b)

\[\sigma=\mathrm{J}_{1} \, \mathrm{X}_{1}+\mathrm{J}_{2} \, \mathrm{X}_{2}\]

Hence from equations (f) and (g)

\[\sigma=\mathrm{L}_{11} \, \mathrm{X}_{1}^{2}+\left(\mathrm{L}_{12}+\mathrm{L}_{21}\right) \, \mathrm{X}_{1} \, \mathrm{X}_{2}+\mathrm{L}_{22} \, \mathrm{X}_{2}^{2}>0\]

It also follows that [8]

\[\mathrm{L}_{11} \, \mathrm{X}_{1}^{2} \geq 0 \quad ; \quad \mathrm{L}_{22} \, \mathrm{X}_{2}^{2} \geq 0\]

And,

\[\mathrm{L}_{11} \, \mathrm{L}_{22} \geq \mathrm{L}_{12}^{2}\]

Electrokinetic Phenomena[1]

These phenomena illustrate the application of the equations discussed above. A membrane separates two salt solutions; an electric potential E and a pressure gradient are applied across the membrane. There are two flows;

1. solution flows through the membrane, described as a volume flow;
2. an electric current.

The dynamics of the system are described by the dissipation function \(\phi\) given by equation (m), the sum of products of flows and forces.

\[\phi=\mathrm{J}_{\mathrm{V}} \, \Delta \mathrm{p}+\mathrm{I} \, \mathrm{E}\]

The dynamics of the system are described by two dynamic equations,

\[\mathrm{J}_{\mathrm{V}}=\mathrm{L}_{11} \, \Delta \mathrm{p}+\mathrm{L}_{12} \, \mathrm{E}\]

\[\mathrm{I}=\mathrm{L}_{21} \, \Delta \mathrm{p}+\mathrm{L}_{22} \, \mathrm{E}\]

Onsager’s law requires that,

\[\mathrm{L}_{12}=\mathrm{L}_{21}\]

In an experiment we set \(\mathrm{E}\) at zero. Then

\[\mathrm{L}_{11}=\left(\frac{\mathrm{J}_{\mathrm{v}}}{\Delta \mathrm{p}}\right)_{\mathrm{E}=0}\]

However the electric current is not zero. According to equation (o),

\[\mathrm{I}=\mathrm{L}_{21} \, \Delta \mathrm{p}\]

In other words, there is a coupled flow of ions. Katchalsky and Curran [1] discuss numerous experiments which illustrate this type of coupling of flows and forces.

Footnotes

[1] A. Katchalsky and P. F. Curran, Non-Equilibrium Thermodynamics in Biophysics, Harvard University Press, 1965.

[2] P. Glandsorff and I. Prigogine, Thermodynamics of Structure Stability and Fluctuations, Wiley-Interscience, London,1971.

[3] G. Nicolis and I. Prigogine, Self-Organization in Nonequilibrium Systems, Wiley, New York, 1977.

[4] P Gray and S. K. Scott, Chemical Oscillations and Instabilities, Oxford,1990.

[5] B. Lavenda, Thermodynamics of Irreversible Processes, MacMillan Press, London, 1978.

[6] D. Kondepudi and I. Prigogine, Modern Thermodynamics, Wiley, New York, 1998.

[7] L. Onsager, Phys. Rev.,1931,38,2265.

[8] D. G. Miller, Chem. Rev.,1960,60,15.