Maxwell RelationsJ G Jackson

State Variables• We have seen that internal energy U is constructed from four

distinct state variables P, V, T and S.

• Every state variable depends on every other state variable

and so on for T, S.

• In essence Thermodynamics explains how state variables are changing with respect to each other. This requires the language of calculus.

Functions of many variables• In particular we should be aware of how changes in one

variable can induce changes in many other variables. This requires knowledge of multivariate calculus.

• You have already encountered this concept when partial derivatives were introduced to you.

• For example, the surface of an arbitrary substance.

Calculus of many variables I• Consider arbitrary functions such that

= =

Rule I• Partial derivatives commute

• This means the order in which the operations are applied doesn’t matter.

Calculus of many variables II• Small increments of and are then expressed as

• These will be referred to as the and equations.

Rule II• Consider fixing , so that . Then the equation reads

• And similarly, the equation

• So then we the inverse rule

Loose Ends• Finally, with fixed , the equation reads

• Then can be substituted with (see previous slide)

• Which, along with the inverse rule yields the somewhat counter intuitive relation

Thermodynamic Potentials I• These abstract calculus relations are very useful when applied

to already known energy potentials. You already know two of them, internal energy and enthalpy .

• Recall constant volume heat capacity

• Introducing lets us write

Thermodynamic Potentials II• The fundamental relation for U read

• Lets find an equivalent expression for

• From this it should be clear that

Thermodynamic Potentials III• What about functions of temperature? Lets (re)invent a couple

more potentials to cover all state variables.

• Helmholtz function

• Gibbs function

Maxwell Relations• Now have 4 energy functions

• And 4 state variables

• We can derive useful relations between state variables by applying multivariate calculus rules to our energy potentials.

• These are powerful tools which will allow us to transform easily between variables.