Jacobian Transformations and Thermodynamic Partial Derivatives

If we consider the ten most common thermodynamic variables P, V, T, U, S, G, H, A, q and w, there exists a very large number of partial derivatives and relations between their derivatives (see Margenau and Murphy, 1956, p. 15). For example, there are 720 (= 10 X 9 X 8) ways of choosing any 3 different variables from a set of 10 hence there must be 720 partial derivatives of the form dx/dy)z relating these variables. Now we have shown above that any one such partial derivative may generally be related to three other mutually independent derivatives by the following kind of manipulation. Given a function 

For an alternative method of obtaining the same relationships, see Denbigh (1966, p. 92). The total number of equations such as (2.19) relating any combination of 

For example, to find (dV/dS)p use the second and third rows of the Table to obtain 

This relates an almost unmeasurable quantity to three experimentally accessible variables T, Cp, and (dV/dT)p. 

Show that dV for an ideal gas is an exact differential by integrating equa- 

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