1.14.73: Variables: Gibbsian and Non-Gibbsian

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Michael J Blandamer & Joao Carlos R Reis
University of Leicester & Faculdade de Ciencias

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Experience shows that the thermodynamic state of a closed single phase system can be defined by a minimum set of independent variables where at least one variable is a measure of the ‘hotness’ of the system; e.g. temperature. The volume of an aqueous solution containing \(\mathrm{n}_{1}\) moles of water and \(\mathrm{n}_{j}\) moles of urea is defined by the set of independent variables, \(\mathrm{T}, \mathrm{~p}, \mathrm{~n}_{1} \text { and } \mathrm{n}_{j}\).

\[\mathrm{V}=\mathrm{V}\left[\mathrm{T}, \mathrm{p}, \mathrm{n}_{1}, \mathrm{n}_{\mathrm{j}}\right]\]

Having defined the parameters set out in the brackets [...] the volume of the system, the dependent variable, is uniquely defined. In fact we can replace \(\mathrm{V}\) in this equation by \(\mathrm{G}, \mathrm{~H} \text { and } \mathrm{~S}\) in order to define unique Gibbs energy, enthalpy and entropy respectively.

The set of independent variables in equation (a) is called Gibbsian because the set comprises the intensive variables \(\mathrm{T}\) and \(\mathrm{p}\) together with the extensive composition variables [1]. The general form of equation (a) defining the thermodynamic potential function, Gibbs energy \(\mathrm{G}\) is as follows where \(\xi\) is the extensive composition variable.

\[\mathrm{G}=\mathrm{G}[\mathrm{T}, \mathrm{p}, \xi]\]

Other sets of independent variables are used in conjunction of the thermodynamic potential functions, enthalpy \(\mathrm{H}\), energy \(\mathrm{U}\) and Helmholtz energy \(\mathrm{F}\).

\[\mathrm{F}=\mathrm{F}[\mathrm{T}, \mathrm{V}, \xi]\]

\[\mathrm{U}=\mathrm{U}[\mathrm{S}, \mathrm{V}, \xi]\]

\[\mathrm{H}=\mathrm{H}[\mathrm{S}, \mathrm{p}, \xi]\]

In equations (c) and (d), \(\mathrm{V}\) is an extensive variable and in equations (d) and (e) S is an extensive variable. The sets of independent variables in equations (c), (d) and (e) are called non-Gibbsian [1].

Footnote

[1] J. C. R. Reis, M. J. Blandamer, M. I. Davis and G. Douhéret, Phys. Chem. Chem. Phys., 2001, 3, 1465.