Statistical Physics

4Ideal GasesHere, we shall apply statistical physics to an ideal gas. We calculate the temperatureand pressure as defined by statistical physics, and show that thesestatistical-mechanical definitions give the same temperature and pressure asdo the thermodynamic definitions. We also discuss the thermal properties ofan ideal gas consisting of diatomic molecules.4.1 Quantum Mechanics of a Gas MoleculeSuppose that we enclose N moleculesinaboxofvolumeV with adiabaticwalls, and let them move with arbitrary velocities. If the influence of gravitycan be neglected, the molecules continue to move and they become distributeduniformly, as we have seen in Chap. 1. 1 If there is no interaction between themolecules, the speed of every molecule is conserved. In this case it is impossiblefor the system to iterate through all possible microscopic states. Therefore weallow a weak interaction between the molecules so that every microscopic statecan be realized. A model with this property describes an ideal gas, and alsodescribes a real gas in the dilute limit.We shall investigate this model by the use of statistical physics in this chapter.For this purpose, we must distinguish between microscopic states. Thistask cannot be accomplished if we are discussing the motion of a moleculeusing classical mechanics. In classical mechanics, the state of a molecule isdetermined by its position r and its momentum p. These variables vary continuously,and so we cannot count the number of states that have energiesbetween E and E + δE. The way out of this difficulty is to use quantummechanics, which is the correct set of laws governing the microscopic world.The essence of quantum mechanics is that a molecule behaves both asa particle and as a wave. A wave function ψ(r), which is a continuous1 Note that since there is no energy transfer between the adiabatic walls and themolecules, the energy of a molecule must be conserved in a collision with a wall.The reflection must be perfectly elastic.