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Write the partition function in this case, taking into account the degeneracy of each state. Calculate the heat capacity of a system of N molecules in the limit of low and high temperatures and compare them to the classical result. 45 Two-Level System (Princeton) Consider a system composed of a very large number N of distinguishable atoms at rest and mutually noninteracting, each of which has only two (nondegenerate) energy levels: Let E / N be the mean energy per atom in the limit a) What is the maximum possible value of E / N if the system is not necessarily in thermodynamic equilibrium?

12 Hydrogen Rocket (Moscow Phys-Tech) The reaction chamber of a rocket engine is supplied with a mass flow rate m of hydrogen and sufficient oxygen to allow complete burning of the fuel. The cross section of the chamber is A, and the pressure at the cross section is P with temperature T. Calculate the force that this chamber is able to provide. 13 Maxwell-Boltzmann Averages (MIT) a) Write the properly normalized Maxwell–Boltzmann distribution for finding particles of mass with magnitude of velocity in the interval at a temperature b) What is the most likely speed at temperature c) What is the average speed?

Evaluate explicitly the change in entropy of the system. 39 21 Chemical Potential of Ideal Gas (Stony Brook) Derive the expression for the Gibbs free energy and chemical potential of N molecules of an ideal gas at temperature pressure P, and volume V. Assume that all the molecules are in the electronic ground state with degeneracy At what temperature is this approximation valid? 40 Gas in Harmonic Well (Boston) A classical system of N distinguishable noninteracting particles of mass is placed in a three-dimensional harmonic well: a) Find the partition function and the Helmholtz free energy.