1 Helmholtz free energy of a hydrogen atom
(a) For a system that can trade heat but not volume or particles with an infinite reservoir (e.g.
a hydrogen atom), prove that the Second Law of Thermodynamics is equivalent to saying that
the system's Helmholtz free energy F = E - TS tends to decrease. (Hints: start by by writing
dS$_{world}$ = dS$_{system}$ + dS$_{reservoir}$. Then, apply the thermodynamic identity to rewrite dS$_{reservoir}$
and simplify the expression above. You should be able to show that dS$_{world}$ = -$\frac{1}{T}$dF)
(b) The first excited energy level of a hydrogen atom has an energy of 10.2eV, if we take the
ground-state energy to be zero. However, the first excited level is really four independent
states, all with the same energy. What is the entropy of an atom in the ground state? What is
the entropy of an atom in the first excited energy level?
(c) For what temperatures is the Helmholtz free energy of a hydrogen atom in the first excited
level positive, and for what temperatures is it negative? (Comment: When F for the level is
negative, the atom will spontaneously go from the ground state into that level, since F = 0
for the ground state and F always tends to decrease. However, for a system this small, the
conclusion is only a probabilistic statement; random fluctuations will be very significant).
