8.08 Problem Set # 2Problems:1. (10 pts.) Cooling by adiabatic demagnetization:(a) Consider N spin-1/2 spins in a magnetic field B. Initially, the system has a temperatureT . If we slowly reduce the magnetic field to B/2, what becomes the temperature of thesystem? If we slowly reduce the magnetic field to zero, what becomes the temperatureof the system? (Hint: the entropy remains unchanged in the above adiabatic process.)(b) Consider N spin-1/2 spins in a magnetic field B. The spin system is in thermal contactwith an ideal gas of N particles in a volume V . Initially, the two systems have atemperature T . Assume gµBB  kBT . If we slowly reduce the magnetic field to zero,what becomes the temperature of the gas?2. (10 pts) Equipartition theorem for a classical oscillator and the violation of equipar-tition theorem for a quantum oscillatorConsider an oscillator of mass m and spring constant K. The total energy of the oscillator isE =p22m+Kx22. The oscillator is in contact with a heat bath of temperature T .(a) View the oscillator as a classical oscillator. Show that the average kinetic energy hp22miand the average potential energy hKx22i are both given by12kBT regardless the value ofm and K. As a result, the average total energy is hEi = kBT .(b) View the oscillator as a quantum oscillator. Calculate the average total energy hEi, andshow that hEi = kBT in high temperature limit. So the equipartition theorem is validfor a quantum oscillator in high temperature limit. Below what temperature we startto see a violation of equipartition theorem for the quantum oscillator? What is hEi inT → 0 limit?3. (10 pts) An adiabatic compression of an O2gasConsider an O2gas at room temperature T . The two rotational degrees of freedom are fullyexcited. We also know that the volume of the O2gas is V and there are N O2molecules inthe volume V . We treat the O2gas as an ideal gas.(a) Find the total average energy of the O2gas.(b) Assume that the O2gas is an isolated system and we slowly change the total volumeby a small amount ∆V . Find the change ∆T in the temperature of the gas. Find thechange ∆P in the pressure of the gas. (Notice how your result differ from that for idealgas of point particles.)4. (20pts) Air pressure at high altitudes:In this problem we treat air as an ideal gas formed by a single kind of molecules (which canbe treated as point particles). Assume the mass of the air molecules is m and the air pressureand temperature at the surface of the earth is P0and T0.(a) From the balance of force, show that the pressure of the air satisfies the following differ-ential equation:dPdz= −mgn(z),where n(z) is the particle density at altitude z and g is the gravitational acceleration.1(b) Calculate the air pressure P at an altitude z assuming the air has a uniform temperatureT0.(c) Calculate the entropy per air molecule assuming the air pressure is P and the air temper-ature is T .(d) The assumption of uniform temperature in (b) is incorrect. The air mass moves up/downin the atmosphere quite freely. Such a convection determines the temperature and the pressuredistribution in the atmosphere. We assume that the air convection is fast enough so thatdiffusion, heat conduction, and other irreversible processes can be ignored. At the same timethe air convection is slow enough so that the process can be regarded as an adiabatic process.As a result, the entropy per molecule is uniform (i.e. independent of the altitude z), whilethe temperature T is not uniform. Calculate the air pressure P at an altitude z in this case.Sketch your result P