Thermodynamics next up previous
Next: Partition function Up: The grand canonical ensemble Previous: The grand canonical ensemble

Thermodynamics

In the canonical ensemble, the Helmholtz free energy A(N,V,T) is a natural function of N, V and T. As usual, we perform a Legendre transformation to eliminate N in favor of tex2html_wrap_inline481 :

eqnarray22

It turns out that the free energy tex2html_wrap_inline483 is the quantity -PV. We shall derive this result below in the context of the partition function. Thus,

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To motivate the fact that PV is the proper free energy of the grand canonical ensemble from thermodynamic considerations, we need to introduce a mathematical theorem, known as Euler's theorem:

Euler's Theorem: Let tex2html_wrap_inline489 be a function such that

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Then f is said to be a homogeneous function of degree n. For example, the function tex2html_wrap_inline495 is a homogeneous function of degree 2, tex2html_wrap_inline497 is a homogeneous function of degree 3, however, tex2html_wrap_inline499 is not a homogeneous function. Euler's Theorem states that, for a homogeneous function f,

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Proof: To prove Euler's theorem, simply differentiate the the homogeneity condition with respect to lambda:

eqnarray41

Then, setting tex2html_wrap_inline503 , we have

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which is exactly Euler's theorem.

Now, in thermodynamics, extensive thermodynamic functions are homogeneous functions of degree 1. Thus, to see how Euler's theorem applies in thermodynamics, consider the familiar example of the Gibbs free energy:

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The extensive dependence of G is on N, so, being a homogeneous function of degree 1, it should satisfy

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Applying Euler's theorem, we thus have

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or, for a multicomponent system,

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But, since

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it can be seen that tex2html_wrap_inline509 is consistent with the first law of thermodynamics.

Now, for the Legendre transformed free energy in the grand canonical ensemble, the thermodynamics are

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But, since

eqnarray67

the thermodynamics will be given by

eqnarray78

Since, tex2html_wrap_inline511 is a homogeneous function of degree 1, and its extensive argument is V, it should satisfy

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Thus, applying Euler's theorem,

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and since

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the assignment tex2html_wrap_inline515 is consistent with the first law of thermodynamics. It is customary to work with PV, rather than -PV, so PV is the natural free energy in the grand canonical ensemble, and, unlike the other ensembles, it is not given a special name or symbol!


next up previous
Next: Partition function Up: The grand canonical ensemble Previous: The grand canonical ensemble

Mark Tuckerman
Tue Feb 1 14:50:00 EST 2000