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Derivation of Thermodynamic Derivatives Using Jacobians

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.

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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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