# 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 Summary

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

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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 ...

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