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# The Other Energy Functionals: The Legendre transformations

From Equation 21-1, the internal energy has the natural variables'' . To change to another set of natural variables, a new function is defined by subtracting off a particular conjugate pair:

For example: Enthalpy

 (21-3)

For example: Helmholtz Free Energy

 (21-4)

For example: Gibbs Free Energy

 (21-5)

The Legendre transformation'' is defined as the procedure of subtracting off conjugate pair to change that particular variable.

To see how the new variables appear, consider the Helmholtz free energy:

 (21-6)

implies

 (21-7)

Doing the same for

And for

Trivial Example of a Legendre Transform
Suppose that the internal energy is composed of the thermal part plus a simple spring:

 (21-8)

where

 (21-9)

The Legendre transform from the -variable to the variable is

 (21-10)

It is fairly easy to show that the inverse Legendre transformation is simply .

Next: LeChatelier's Principle Up: Lecture_21_web Previous: Mathematics of Exact Differentials
W. Craig Carter 2002-10-25