Question

2.For a certain system, the energy levels are given by 21 with degeneracy 8(2J+1)2 i) Please express the fraction of molecules in the Jth level (denoted as f, ) in terms of g, and , at a given temperature T. Please plot qualitatively fj as function of J. Please show the math procedure on how to obtain the most probable rotational quantum number J? (here most probable quantum number means that most particles would tend to occupy that particular energy level)

Answer 1

Let 'F_0' be the total number of molecules in lowest vibrational state. And F_J is the number of molecules in 'J' energy level. (a) we have G(J) = E_J/h_c = J(J + 1) strok^2/2 I hc = J(J + 1) strok^2/8 pi^2 k c I G(J) = J(J + 1) h/8 pi^2 I C let B = h/8 pi^2 hc = Rotational constant therefore G(J) = B J(J + 1) or elementof_J = G(J) hc therefore Number of molecules present in given state to Total number of moles F(J) = F_J/Q_r = (2 J + 1) e^-G(J) h c/k_B T/sum over all states (Q_r) we know Rotational partion function Q_r = Sigma^infinity_J = 0 (2 J + 1) e^-f(J)hc_K_B T De- genacy of a given Q_r = integral^infinity_0 (2 J + 1) e^-B J(J + 1) hc/K_B T d J let B J (J + 1) hc/K_B T = x: B(2J + 1) hc/k_Bt dJ = d x.