Thermodynamics of Continua

Abstract

The extension of the fundamental equations of Thermodynamics to a description of macroscopic systems in terms of continuous state variables is developed. All the basic relations as mass conservation in the presence of chemical reactions, the equation of motion and the equations for energy and that for entropy which express, in the new formalism, the First and the Second Principles, respectively, must be reformulated. The correct expression for the entropy production and the consequent expressions for the fluxes and the corresponding generalized forces are obtained. In the linear regime, the general relation between the mobility of ionic species and the coefficient of diffusion (Einstein relation) is demonstrated. The thermoelectric phenomena (Seebeck, Peltier, and Thomson effects) are discussed together with the thermodiffusion processes. An appendix concerning the Gibbs–Duhem relation closes the Chapter.

16.9 Appendix—The Gibbs–Duhem Relation

Let us consider a phase in which, for simplicity, one work parameter only is present (say the volume) and let it be composed by n independent chemical components. Recalling the discussion in Sect. 4.3.1 the Fundamental Relation in the Entropy representation will be written as

$$\begin{aligned} S=S\left( U,V, n_{\gamma }\right) ,\quad \gamma =1,2,\dots , n \, \end{aligned}$$

(16.211)

and we see that the number of degrees of freedom is $n+2$. It is clear, however, that if we leave aside the amount of matter, the number of independent state parameters will be $n+1$. If we write Eq. (16.211) for one mole, and we denote by $\mathrm {c_{\gamma }}=n_{\gamma }/n_{tot}$ the molar fraction of component $\gamma$ we obtain:

$$\begin{aligned} S_{m}=S\left( U_{m}, V_{m},\mathrm {c_{\gamma }}\right) , \quad \gamma =1,2,\dots , n \,, \end{aligned}$$

(16.212)

where $n_{tot}$ is the total number of moles and the n mole fractions $n_{\gamma }/n_{tot}$, satisfy the obvious relation:

$$\begin{aligned} \sum _{\gamma =1}^{n}\mathrm {c_{\gamma }}=1\,. \end{aligned}$$

(16.213)

It follows that if we decide to describe the system as a function of T and p plus the k mole fractions (that is, we don’t take into account the size of the system), the number of degrees of freedom is $n+1$. This means that among the $n+2$ quantities T, p and $n_{\gamma }$ there must be one mutual dependence. This relation is called the Gibbs–Duhem Relation. Let us write it explicitly.

It is convenient to refer to the Fundamental Equation in the Gibbs representation Eq. (4.32) and to remember expression Eq. (4.78) of the Gibbs potential integrated at constant temperature and pressure. If we differentiate the latter we have:

$$\begin{aligned} \mathrm {d}G=\sum _{\gamma =1}^{n}\mu _{\gamma }\mathrm {d}n_{\gamma }+\sum _{\gamma =1}^{n}n_{\gamma }\mathrm {d}\mu _{\gamma }\, \end{aligned}$$

(16.214)

and if we substitute Eq. (4.32) we obtain:

$$\begin{aligned} \sum _{\gamma =1}^{n}n_{\gamma }\mathrm {d}\mu _{\gamma }=-S\mathrm {d}T+V\mathrm {d}p\,. \end{aligned}$$

(16.215)

This is the Gibbs–Duhem relation. In particular, for processes at constant T and p, Eq. (16.215) becomes:

$$\begin{aligned} \sum _{\gamma =1}^{n}n_{\gamma }\mathrm {d}\mu _{\gamma }=0\,. \end{aligned}$$

(16.216)

Let us go back to Eq. (16.84) and in particular let us consider the first term of the second member. The first step is to recognize that the summation over the mass concentration gradients are equivalent to the gradients of the specific chemical potentials as shown in the following relation:

$$\begin{aligned} \sum _{\gamma ^{\prime }=1}^{n}\left( \frac{\partial \mu _{\gamma }^{*}}{\partial c_{\gamma ^{\prime }}}\right) \varvec{\nabla }c_{\gamma ^{\prime }}=\varvec{\nabla }\mu _{\gamma }^{*} \,. \end{aligned}$$

(16.217)

Hence we may write

$$\begin{aligned} \varrho \sum _{\gamma =1}^{n}c_{\gamma }\sum _{\gamma ^{\prime }=1}^{n}\left( \frac{\partial \mu _{\gamma }^{*}}{\partial c_{\gamma ^{\prime }}}\right) \varvec{\nabla }c_{\gamma ^{\prime }}=\varrho \sum _{\gamma =1}^{k}c_{\gamma }\varvec{\nabla }\mu _{\gamma }^{*}\, \end{aligned}$$

(16.218)

and since $\mu _{\gamma }^{*}=\left( 1/M_{\gamma }\right) \mu _{\gamma }$ and $c_{\gamma }=m_{\gamma }/m$, we may replace $\left( \varrho c_{\gamma }/M_{\gamma }\right)$ with $n_{\gamma }/V$ and, finally, Eq. (16.218) can be written as

$$\begin{aligned} \varrho \sum _{\gamma =1}^{n}c_{\gamma }\varvec{\nabla }\mu _{\gamma }^{*}=\frac{1}{V}\sum _{\gamma =1}^{n}n_{\gamma }\varvec{\nabla }\mu _{\gamma }\, \end{aligned}$$

(16.219)

and

$$\begin{aligned} \sum _{\gamma =1}^{n}n_{\gamma }\varvec{\nabla }\mu _{\gamma }=0\,, \end{aligned}$$

(16.220)

by the Gibbs–Duhem relation for isothermal and isobaric processes as shown in Eq. (16.216).