The Heat Capacity of a Diatomic Gas

1. The Heat Capacity of a Diatomic Gas Chapter 15

2. 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.

3. 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.

4. 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 EJ = (J + ½) hv , J = 0, 1, 2, ….. • The energies are equally spaced and the ground state has non-zero energy.

5. 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. gj = 1 • The partition function of an oscillator

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

7. The distribution function for B-statistics is 2

8. 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 = NkT2 since

9. For T → 0 For thus

10. 15.3 Vibrational 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!

11. Since Or At high temperatures

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

13. Therefore Cv 0 as T  0 The total energy of a diatomic molecule is made up of four contributions that can be separately treated:

14. 1. The kinetic energy associated with the translational motion • The vibrational motion • 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:

15. J = 0

16. 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