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
For example: Helmholtz Free Energy
For example: Gibbs Free Energy
``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:
Doing the same for
|Trivial Example of a Legendre Transform|
Suppose that the internal energy is composed of the thermal part plus
a simple spring:
The Legendre transform from the -variable to the variable is
It is fairly easy to show that the inverse Legendre transformation is simply .