The Heat Capacity of a Diatomic Gas Chapter 15

15.1 Introduction Statistical thermodynamics provides deep insight into the classical description of a MONATOMIC ideal gas. In classical thermodynamics, the principle of equipartition of energy fails to give the observed value of the specific heat capacity for diatomic gases. The explanation of the above discrepancy was considered to be the most important challenge in statistical theory.

15.1 The quantized linear oscillator A linear oscillator is a particle constrained to move along a straight line and acted on by a restoring force F=-kx F = ma= = -kx If displaced from its equilibrium position and released, the particle oscillates with simple harmonic motion of frequency v, given by Note that the frequency depends on K and m, and is independent of the amplitude X.

Consider an assembly of N one-dimensional harmonic oscillators, in which the oscillators are loosely coupled so that the energy exchange among them is small. In classical mechanics, a particle can oscillate with any amplitude and energy. From quantum mechanics, the single particle energy levels are given by E J = (J + ½) hv, J = 0, 1, 2, ….. The energies are equally spaced and the ground state has non-zero energy.

The internal degrees of freedom include vibrations, rotations, and electronic excitations. For internal degrees of freedom, Boltzmann Statistics applies. The distinguish ability arises from the fact that those diatomic molecules have different translational energy. The states are nondegenerate, i.e. g j = 1 The partition function of an oscillator

Introducing the characteristic temperature θ, where θ = hv/k The solution for the above eq. is (in class derivation)

The distribution function for B-statistics is 2

Note that B statistics and M – B statistics have the same distribution function, the eq derived in chapter 14 for internal energy is also valid here. U = NkT 2 since

For T → 0 For thus

15.3Vibrational Modes of Diatomic Molecules The most important application of the above result is to the molecules of a diatomic gas From classical thermodynamics for a reversible process!

Since Or At high temperatures

At low temperature limit On has So approaching zero faster than the growth of (θ/T) 2 as T → 0

ThereforeC v  0 as T  0 The total energy of a diatomic molecule is made up of four contributions that can be separately treated:

1.The kinetic energy associated with the translational motion 2.The vibrational motion 3.Rotation motion (To be discussed later) Example: 15.1 a) Calculate the fractional number of oscillators in the three lowest quantum states (j=0, 1, 2,) for Sol:

J = 0

15.2) a) For a system of localized distinguishable oscillators, Boltzmann statistics applies. Show that the entropy S is given by Solution: according to Boltzmann statistics So