 # 2. Consider an insulated container of total volume (V) with a solid...

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2. Consider an insulated container of total volume (V) with a solid partition that divides the container into two equal volumes (V/2). On one side of the container there are NA gas atoms of mass MA that can be considered as ideal. On the other side there are NB gas atoms of mass MB that are also ideal. The temperature on both sides of the partition is the same (t) and let Ns = NB = N. a) If the multiplicity of side A is ga and the multiplicity of side B is gB, what is the multiplicity of the whole container. Does the total entropy (a) of the whole container equal TA + OB? Express your answers in terms of ga and gB- b) Derive the Canonical Partition Function (express Z as a function of N, T, and V) of gas A and of gas B separately (i.e., before the partition is removed). Now find the Partition Function of the whole system (i.e., the entire container) and thus calculate its total entropy (assume that the partition is NOT removed). c) Now assume that the partition is removed and the gases mix until they achieve thermodynamic equilibrium. Show that the total entropy of the whole system (i.e., the entire container) is increased by 2N log 2. d) If the atoms are actually identical (A III B), what should be the increase in total entropy after the removal of the partition? e) The results from c) and d) would seem to imply that a paradox exists because we can propose the following thought experiment. Suppose that the particles making up gases A and B are made to be arbitrarily similar to each other in terms of their properties (e.g., they may have virtually the same masses with all other properties being identical). The experiment implies a "discontinuity' in the entropy. Comment on this apparent paradox. You may also want to think about the issue of indistinguishability of free particles in quantum mechanics.

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