4. Recall that one can show that for an ideal monatomic (and nonrelativistic) gas, = 38. However, often one has to assum

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4 Recall That One Can Show That For An Ideal Monatomic And Nonrelativistic Gas 38 However Often One Has To Assum 1 (75.84 KiB) Viewed 35 times

4. Recall that one can show that for an ideal monatomic (and nonrelativistic) gas, = 38. However, often one has to assume that, in equilibrium, the mean pressure on every wall of the rectangular container was the same. Now, you can prove that fact under rather general conditions. Proceed as follows: la, wow that the mean pressure on the wall perpendicular to the r direction is given by 2 (), n2 Pr = n(er, 0,3) mV where the orbital r has energy E, and is specified by the quantum numbers n., Ny, and n. nr(6,0,B) is the mean occupation number in either FD, BE, or "fudged classical" statistics) of orbital r. Corresponding expresions hold for py and pz. (b) Assume that the temperature is high enough that a large number of orbitals are occupied. In that case, nx, Ny, and n, may be treated as continuous variables. Use this fact to rewrite Pe above as an integral over Na, Ny, and nz. (This is the same manipulation we've done plenty of times here.) (e) Change variables in your integral to eliminate any explicit reference to Ly, Ly, and L. from the integrand. Now argue that this shows that Pa = Py = .. (d) The integral expression for prin (c) looks, at first glance, like it says that P, depends only on the temperature, not on the volume or on the total number of particles. Explain where the dependence on N/V (which we know must be there from the classical ideal gas law p = N&T/V) is hiding. (e) In part (b) we had to assume that the temperature is high enough. Is this just a technical assumption (to make it easy to do the problem), or is it really true that at low enough temperature the pressures in the three directions may not necessarily be equal? Explain. (Hint: Consider as a test case a BE gas at T = 0 in an asymmetrical box (L, > L, > L.). There is only one microstate in this case so it's easy to find the pressures in the three directions.