Up until this point, we have assumed that the vibrational and rotational partition functions are separable. In reality, as a vibrating bond oscillates, so does the moment of inertia of the molecule. So rotations and vibrations are not truly uncoupled. One approach to this more complicated problem is to use first-order perturbation theory to model the energies of rovibrational microstates, EnJ:
EnJ = hw(n + [J(J+1)]) (1)
Two terms in Eq. 1 should look familiar, the coupling term has > 0.
(a) The rovibrational partition function, qrov, can be expanded in powers of X. Derive an expression for qrov that is linear in A, i.e.
qrov = qho * qrr * f(A) (2)
where qho and qrr are the harmonic oscillator and rigid rotor partition functions, respectively, and f(A) is a coupling term.
(b) Derive an expression for the entropy S(T) that is linear in A.
(c) Use your result from (b) to plot the entropy S(T) of a dilute vapor of LiH at P = 0.5 atm from T = 50K to T = 1000K. In addition, on the same figure, plot separately the contributions to the entropy from (i) the harmonic oscillator partition function, (ii) the rigid rotor partition function, (iii) the rovibrational coupling term, and (iv) the translational partition function. Which degrees of freedom contribute most to S(T)? Why do you think this is? Is the rovibrational correction significant in this case? For LiH, use Or = 10.82K, O = 2025K, and a/kp = 0.3070K.
