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Ideal gas

Recall the canonical partition function expression for the ideal gas:

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Define the thermal wavelength tex2html_wrap_inline609 as

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which has a quantum mechanical meaning as the width of the free particle distribution function. Here it serves as a useful parameter, since the canonical partition can be expressed as

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The grand canonical partition function follows directly from Q(N,V,T):

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Thus, the free energy is

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In order to obtain the equation of state, we first compute the average particle number tex2html_wrap_inline613

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Thus, eliminating tex2html_wrap_inline601 in favor of tex2html_wrap_inline613 in the equation of state gives

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as expected. Similarly, the average energy is given by

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where the fugacity has been eliminated in favor of the average particle number. Finally, the entropy

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which is the Sackur-Tetrode equation derived in the context of the canonical and microcanonical ensembles.



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