Solutions to Practice Exam 2 Problems

Heat Engines (Chapter 4)

Problem 1:

A heat engine employing a Carnot cycle with an efficiency of $\eta =10\mathrm{\%}$ is used as a refrigerating machine, with the thermal reservoirs being the same. Find its refrigerating Coefficient of performance COP.

Problem 2:

Find the efficiency of a cycle consisting of two isobaric (constant pressure) and two isothermal (constant temperature) lines if the pressure varies n-fold and the absolute temperature $\tau$ -fold within the cycle. The working substance is an ideal gas with the adiabatic exponent $\gamma =C_P/C_V$.

a) Assume there are two reservoirs at $T_1$ and $T_2=\tau T_1$. Assume that the isobaric processes occur while the system is in contact with the high temperature reservoir. What is the change in entropy of the environment?
b) If this cycle is taken reversibly (by appropriately designing the reservoirs), what would be the efficiency?

Free Energy and Phases (Chapter 5)

Problem 1:

The ground state of a molecule has zero energy, and degeneracy equal to one. The first excited level has degeneracy four, and energy $ϵ$.
a) Calculate the difference in Helmholtz Free energy between the ground state and the first excited state at temperature $T$. At what temperature does the free energy become zero?
b) Using the Boltzmann distribution, at what temperature is the excited state equally probable as the ground state. Compare to (a).

Statistical Mechanics (Chapter 6)

Problem 1:

Consider a three state molecule with energies $E_i=\{-E,0,+E\}$. Compute the Helmholtz free energy of a collection of $N$ molecules at temperature $T$. Use this to find the entropy in two limits: $T\to 0$ and $T\to \mathrm{\infty }$.