Suppose we are in the NPT ensemble, and that the entropy S = S(L) depends on the length of a molecule. We also find that S(L) = -3L. Given that dG = -SdT + Vdp + μdN + fdL, what does (∂f/∂T)L equal when L is 2?

Problem 1: Consider a classical ideal gas in three dimensions, with N indistinguishable atoms confined in a box of volume N³. Assume the atoms have zero spin and neglect any internal degrees of freedom. Starting from the energy levels of a single atom in a box, find: (a) The Helmholtz free energy F' Hint: ſ. -ax² d.x e Va (b) The entropy o (c) The pressure p

Consider a classical ideal gas of N diatomic heterogeneous molecules at temperature T. The charac- teristic rotational energy parameter is € = 1 and the natural frequency of vibrations is wo. Consider the temperature region where T≫er/kB, but T is of the order of ħwo/kB. Ignore contributions from all other internal modes. Calculate the canonical partition function, the average energy, and the heat capacity at constant volume, Cv.

Show that the one-particle partition function Z₁ for a 2D ideal gas confined to area A is: A 2²/1 Z₁ = S