In Section 6.5, I derived the useful relation F = ? kT ln

Chapter 7, Problem 7P

(choose chapter or problem)

Problem 7P

In Section 6.5, I derived the useful relation F = − kT ln Z between the Helmholtz free energy and the ordinary partition function. Use an analogous argument to prove that

Ф = − kTlnZ

where z is the grand partition function and Ф is The grand free energy introduced in Problem 1.

Problem 1:

By subtracting μN from U, H, F, or G, one can obtain four new thermodynamic potentials. Of the four, the most useful is the grand free energy (or grand potential),

Φ ≡ U − TS − μN.

(a) Derive the thermodynamic identity for Φ, and the related formulas for the partial derivatives of Φ with respect to T, V, and μ.

(b) Prove that, for a system in thermal and diffusive equilibrium (with a reservoir that can supply both energy and particles), Φ tends to decrease.

(c) Prove that Φ = – PV.

(d) As a simple application, let the system he a single proton, which can be “occupied” either by a single electron (making a hydrogen atom, with energy –13.6 eV) of by none (with energy zero). Neglect the excited states of the atom and the two spin states of the electron, so that both the occupied and unoccupied states of the proton have zero entropy. Suppose that this proton is in the atmosphere of the sun, a reservoir with a temperature of 5800 K and an election concentration of about 2 × 1019 per cubic meter. Calculate Φ for both the occupied and unoccupied states, to determine which is more stable under these conditions. To compute the chemical potential of the electrons, treat them as an ideal gas. At about what temperature would the occupied and unoccupied states be equally stable, for this value of the electron concentration? (As in below Problem 2, the prediction for such a small system is only a probabilistic one.)

Problem 2:

The first excited energy level of a hydrogen atom has an energy of 10.2 eV, 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. We can therefore assign it an entropy of S = k ln 4, since for this given value of the energy, the multiplicity is 4. Question: 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.)