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Grand Canonical Partition Function

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For the grand canonical ensemble we've obtained two expressions for the pressure:

P = (k_B)(T)/Vln(x) or P = (k_B)(T)(dln(x))/dV_Bu,B

where is the grand canonical partition function. Check that the derivative does not give the first expression exactly. Nonetheless, the pressure calculated in the two ways agree - at least to any reasonable accuracy. Why? (HINT: consider what happens for a large system).

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Solution Preview

P = KB*T/V *[ln E] -----(1)

P = KB*T[ d/dV {ln E}] ---- (2)

where I am using "d/dV" to represent the partial derivative of "ln E" w.r.t. V.

For simpler notation, let us ...

Solution Summary

This solution uses two expressions for pressure and continues through step by step to show that they are they same for large systems.