# Derivation of Thermodynamic Derivatives Using Jacobians

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Rules from the book by M. Tribus are explored to deduce specified thermodynamic derivatives of the thermodynamical variables (energy , enthalpy , free energy of Helmholtz , free energy of Gibbs , entropy , temperature , pressure , and volume ) as functions of measurable quantities (temperature , pressure , volume , coefficient of thermal expansion , coefficient of compressibility , heat capacity at constant pressure , and heat capacity at constant volume ). Sometimes the entropy will remain outside the derivatives.

Contributed by: Mikhail Dimitrov Mikhailov (March 2011)

Open content licensed under CC BY-NC-SA

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## Details

The main thermodynamical variables are energy , enthalpy , free energy of Helmholtz , free energy of Gibbs , entropy , temperature , pressure , and volume .

It is possible to form 336 partial derivatives of the type , where , , and are among these eight variables.

It is possible to express one derivative in terms of three outer derivatives. Among the 336 derivatives there are approximately combinations, but only a small fraction of them have practical value.

The Jacobian notation shows thermodynamic derivatives in an elegant manner. The application of Jacobians in thermodynamics appears to have started with Bryan [1].

A collection of Jacobian expressions is presented in [2] and [3].

Based on a suggestion made by E. Jaynes, Tribus developed rules for the deduction of thermodynamic partial derivatives [4]. For efficiency a new rule is added.

[1] G. H. Bryan, (article title unknown), *Encyklopedie der mathematishen Wissenschaften*, Bd V, Teil 1, A. Sommerfeld, ed., Leipzig: G. B. Teubner, 1903 p. 113.

[2] P. W. Bridgman, *A Condensed Collection of Thermodynamic Formulas*, Cambridge: Harvard University Press, 1925.

[3] A. N. Shaw, "The Derivation of Thermodynamical Relations for a Simple System," *Phil. Trans. Roy. Soc. A*, 234(740), 1935 pp. 299–328.

[4] M. Tribus, *Thermostatics and Thermodynamics*, Princton, NJ: Van Nostrand, 1961.

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