Problem 5.23 (Schroeder’s Intro to Thermal Physics)

Problem 5.23

By subtracting $\mu_N$ 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),

$\Phi \equiv U - TS - \mu N.$

(a) Derive the thermodynamic identity for $\Phi$ , and the related formulas for the partial derivatives of $\Phi$ with respect to $T$ , $V$ , and $\mu .$
(b) Prove that, for a system in thermal and diffusive equilibrium (with a reservoir that can supply both energy and particles), $\Phi$ tends to decrease.
(c) Prove that $\Phi = -PV.$
(d) As a simple application, let the system be a single proton, which can be “occupied” either by a single electron (making a hydrogen atom, with energy -13.6 eV) or 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 electron concentration of about $2 \times 10^{19}$ per cubic meter. Calculate $\Phi$ 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 Problem 5.20, the prediction for such a small system is only a probabilistic one.)