maxwell relations (2)
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Maxwell RelationsJ G Jackson
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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.
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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.
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Calculus of many variables I• Consider arbitrary functions such that
= =
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Rule I• Partial derivatives commute
• This means the order in which the operations are applied doesn’t matter.
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Calculus of many variables II• Small increments of and are then expressed as
• These will be referred to as the and equations.
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Rule II• Consider fixing , so that . Then the equation reads
• And similarly, the equation
• So then we the inverse rule
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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
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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
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Thermodynamic Potentials II• The fundamental relation for U read
• Lets find an equivalent expression for
• From this it should be clear that
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Thermodynamic Potentials III• What about functions of temperature? Lets (re)invent a couple
more potentials to cover all state variables.
• Helmholtz function
• Gibbs function
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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.