91Ó°ÊÓ

Imagine a world in which space is two-dimensional, but the laws of physics are otherwise the same. Derive the speed distribution formula for an ideal gas of non-relativistic particles in this fictitious world, and sketch this distribution. Carefully explain the similarities and differences between the two-dimensional and three-dimensional cases. What is the most likely velocity vector? What is the most likely speed?

Some advanced textbooks define entropy by the formula $$S=-k \sum_{s} \mathcal{P}(s) \ln \mathcal{P}(s),$$ where the sum runs over all microstates accessible to the system and \(\mathcal{P}(s)\) is the probability of the system being in microstate \(s.\) (a) For an isolated system, \(\mathcal{P}(s)=1 / \Omega\) for all accessible states \(s\). Show that in this case the preceding formula reduces to our familiar definition of entropy. (b) For a system in thermal equilibrium with a reservoir at temperature \(T\) \(\mathcal{P}(s)=e^{-E(s) / k T} / Z .\) Show that in this case as well, the preceding formula agrees with what we already know about entropy.

Consider a large system of \(N\) indistinguishable, noninteracting molecules (perhaps in an ideal gas or a dilute solution). Find an expression for the Helmholtz free energy of this system, in terms of \(Z_{1}\), the partition function for a single molecule. (Use Stirling's approximation to eliminate the \(N !\).) Then use your result to find the chemical potential, again in terms of \(Z_{1}\).