Thermodynamics of Solutions
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Thermodynamics of Solutions
Diffusion has the reputation of being a difficult subject, much harder than . . . . . . solution thermodynamics. In fact it is relatively simple. . . . . . . I can easily explain a diffusion flux. . . . . . . . I suspect that I have never clearly explained chemical potentials to anyone. E.L. Cussler (1984, Diffusion: mass transfer in fluid systems)
We have, so far, considered thermodynamic potentials for phases that are at a fixed composition. However, when the compositions of the phases become variable, we obviously need to take into account the effect of the compositional changes of the phases on their thermodynamic properties. We have seen earlier (Sect. 3.1) that the extensive thermodynamic potentials H, F and G, which are most useful for practical applications of thermodynamics, are, nonetheless, auxiliary functions, and can be derived by systematic Legendre transformations on the fundamental thermodynamic potential U. For a homogeneous system with fixed masses of all species and unaffected by a force field, U is completely determined by specifying the extensive properties S and V. If the mole numbers of the different species in the system change, then we should begin by making the appropriate modification to the expression of U, and derive from that the expressions for the auxiliary thermodynamic potentials. The required modification to the expression of U leads to the introduction of a new intrinsic thermodynamic property known as the chemical potential of a component in a solution.
8.1 Chemical Potential and Chemical Equilibrium Gibbs laid the foundation of chemical thermodynamics in his monumental work entitled “On the Equilibrium of Heterogeneous Substances” that was published in 1875 and 1878 in the Transactions of the Connecticut Academy of Sciences (see Gibbs, 1993). Here Gibbs argued that if n1 , n2 etc. are the number of moles of different species in a homogeneous system that are subject to change by reversible mass J. Ganguly, Thermodynamics in Earth and Planetary Sciences, C Springer-Verlag Berlin Heidelberg 2008 DOI 10.1007/978-3-540-77306-1 8,
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8 Thermodynamics of Solutions
exchange with the surrounding, then U must be a function of these mole numbers in addition to S and V, so that the fundamental relation must now be written as U = U(S, V, n1 , n2 ....),
(8.1.1)
instead of U = U(S,V), as written in Eq. (2.7.6). The total derivative of U is then ⭸U ⭸U ⭸U ⭸U dS+ dV+ dn1 + dn2 +. . . ., dU = ⭸S V,ni ⭸V S,ni ⭸n1 V,S,nk =1 ⭸n2 V,S,nk =2 (8.1.2) where ni stands for the mole numbers of all components (i.e. n1 , n2 ...) and nk stands for the mole numbers of all components except the one appearing in a partial derivative of U with respect to ni . We know from the earlier developments that ⭸U/⭸S = T and ⭸U/⭸V = –P. Thus, the partial derivatives of U with respect to the number of moles or masses of specific components are the new partial derivatives in the representation of the total derivative of U. The partial derivative of U with respect to
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