Predicting Unmeasurable Quantities

The Mathematics of Partial Derivatives Gives Two More Ways to Predict Unmeasurable Quantities 

In this chapter, wc introduce two more methods of thermodynamics. The first is Maxwell relations. With Maxwell relations you can find the entropic and energetic contributions to the stretching of rubber, the expansion of a surface or a film, or the compression of a bulk material. Second, we use the mathematics of homogeneous functions to develop the Gibbs-Duhem relationship. This is useful for finding the temperature and pressure dependences of chemical equilibria. 

First we generalize our treatment of thermodynamics to allow for forces other than the pressures in pistons. 

Generalize the fundamental energy equation by introducing all the relevant extensive variables X IJ = U(S,V,N,X). To find a fundamental equation, it is always extensive variables that you add, and you add them only to the fundamental energy equation. dU will now be a sum of work terms such as 

With this augmented function dU, you can follow the Legendre transform recipes of Chapter 8 (page 137) to find the fundamental function and extremum principle for the appropriate set of independent variables. Here is an example. 

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We have explored two methods of thermodynamics. The Maxwell relations derive from the Euler expression for state functions. They provide another way to predict unmeasurable quantities from measurable ones. For multicomponent systems, you get another relationship from the fact that many thermodynamic functions are homogeneous. In the following chapters we will develop microscopic statistical mechanical models of atoms and molecules. 

We now aim to relate the two path-dependent quantities, q and lu, to state functions like U and 5, for two reasons. First, such relationships give ways of getting fundamental but unmeasurable quantities U and S from measurable quantities q and w. Second, there is an increase in predictive power w henever q and w depend only on state variables and not on process variables. The First Law of thermodynamics gives such a relationship. 

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