Consider a microcanonical ensemble of closed cubical cells with volume V . Let each cell contain precisely N particles of a classical, nonrelativistic, perfect gas and contain a nonrelativistic total energy

 

For the moment (by contrast with the text’s discussion of the microcanonical ensemble), assume that E is precisely fixed instead of being spread over some tiny but finite range.(a) Explain why the region Y_o of phase space accessible to each system is 

where A labels the particles, and L ≡ V^1/3 is the side of the cell.(b) To compute the entropy of the microcanonical ensemble, we compute the volumein phase space that it occupies, multiply by the number density of states in phase space (which is independent of location in phase space), and then take the logarithm. Explain why.

vanishes. This illustrates the “set of measure zero” statement in the text, which we used to assert that we must allow the systems’ energies to be spread over some tiny but finite range.(c) Now permit the energies of our ensemble’s cells to lie in the tiny but finite range E_o − δE_oo. Show that

where V_ν(a) is the volume of a sphere of radius a in a Euclidean space with ν >> 1 dimensions, and where

It can be shown (and you might want to try to show it) that

(d) Show that, so long as 1 >> δE_o/E_o >> 1/N (where N in practice is an exceedingly huge number), 

which is independent of δE_o and thus will produce a value forand thence N_states and S independent of δE_o, as desired. From this and with the aid of Stirling’s approximation (Reif, 2008, Appendix A.6) n! ≈ (2πn)^1/2(n/e)ⁿ for large n, and taking account of the multiplicity M = N!, show that the entropy of the microcanonically distributed cells is given by

This is known as the Sackur-Tetrode equation.

(e) Using Eqs. (3.39), show that this is equivalent to Eq. (4.40) for the entropy, which we derived by a very different method.

Transcribed Image Text:

E = E-Nmc².