(e) In our discussion ofthermodynamics, we will introduce the Helmholtz free energyA = 5 T5. The second lawI of thermodynamics tells us that In equilibrium at temperature T, a system 1will be found in the equilibrium state that has the lowest free energy A. The free energy difference M = E - T113 between two equilibrium states thus tells us about their relative probabilities. To see the connection of this macroscopic view with the simple system considered in this problem, let us treat the ground state as one equilibrium state of the system, and the collection of M excited states as another equilibrium state. Rewrite your answer to part [cl in the form 10(3) : e(ETSNkBT- Mg) Identify AE and .53 in terms of c and M. Comment on the connection to the famous equation on Boltzmann's tombstone, 5 = to in w. [2 pts} [f] Forthe case of a twofold degenerate excited state lei, consider an additional energy level [e'] that lies above the ground state: e' E! e 6| I 9 Compute the proba bility that the system is in any of these three excited states, [ptei +piei) ipig}, again relative to the ground state probability. (2 pts]