Thermodynamic derivatives Jacobian method

The above techniques are used quite commonly to interrelate and manipulate thermodynamic variables. Later in this chapter we introduce two additional and more general methods of manipulating thermodynamic derivatives called Legendre and Jacobian transformations. 

Pinkerton, R.C. A Jacobian method for the rapid evaluation of thermodynamic derivatives, without the use of tables. 1. Phys. Chem. 56, 799-800 (1952)... 

Partial derivatives, as introduced in Section 2.12 are of particular importance in thermodynamics. The various state functions, whose differentials are exact (see Section 3.5), are related via approximately 1010 expressions involving 720 first partial derivatives Although some of these relations are not of practical interest, many are. It is therefore useful to develop a systematic method of deriving them. Hie method of Jacobians is certainly the most widely applied to the solution of this problem. It will be only briefly described here. For a more advanced treatment of the subject and its application to thermodynamics, die reader is referred to specialized texts. 

The global Newton methods, such as the Naphtali-Sandholm method (Sec. 4.2.9), are often used to solve highly nonideal systems. These are frequently prone to failure. Good explanations of the theory of homotopy methods are provided by Seader (86) and Wayburn (83). A homotopy method can greatly expand the global Newton method ability to solve difficult nonideal systems. Homotopy methods have been associated with the Naphtali-Sandholm method, where the derivatives of the if-values and enthalpies with respect to all compositions directly appear within the Jacobian. Using a thermodynamic homotopy for another method such as a Tomich has not been presented in the literature. 

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