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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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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.

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Canonical Partition Function Q(N, V, T)

An approximate canonical partition function for a dense gas is:

Q(N, V, T) = 1/N! [2(pi)(mk_nT / h^2]^3N/2 (V - Nb)^Nexp[(alphaN^2/Vk_BT)].

Where m is the mass of the particles and a and b are molecular parameters (which are independent of temperature). Calculate the energy, entropy, pressure, and chemical potential for this system. To what well known thermodynamic approximate equation of state does your answer correspond?

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