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:
where
The Legendre transform from the -variable to the variable is
It is fairly easy to show that the inverse Legendre transformation is simply . |