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A special case of the grand canonical ensemble
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[QUOTE="Ted Ali, post: 6399231, member: 664600"] Temperature ##T## is given by equation ##(2)##, in the microcanonical ensemble and calculated in Wikipedia ([URL]https://en.wikipedia.org/wiki/Einstein_solid[/URL]). The final result is: $$\frac{q}{N} = \frac{1}{e^{hf/kT} - 1}\hspace{1cm} (9)$$ The chemical potential ##\mu## is given by ##(1)##, when ##U, V## are held constant. The calculation of ##(1)## can be found in the solutions manual of D. Schroeder's book: "An Introduction to Thermal Physics". The final result is $$\mu = -kT\ln(1+ \frac{q}{N})\hspace{1cm} (10)$$ (exercise 3.36). In the case we examine it is assumed that ##N \gg q## so ##(10)## becomes: $$\mu = -kT\frac{q}{N}\hspace{1cm} (11)$$ From ##(9)## and ##(11)## we have equation ##(8)##, in its final form. Finally let's comment that in ##(6)##, we have neglected the ground state energy of each one quantum harmonic oscillator of the Einstein solid, in the calculation of the canonical partition function ([URL]https://en.wikipedia.org/wiki/Einstein_solid[/URL]). And that in ##(7)## we have assumed that the oscillators are distinguishable. Equations ##(3), (4), (5)## come from M. Bellac's book: "Equilibrium and non-Equilibrium Statistical Thermodynamics", pg. 148. So the "sum" of the questions still remains open: Can we use ##(8)## in the grand partition function? [/QUOTE]
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A special case of the grand canonical ensemble
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