Statistical Physics

64 4 Ideal Gases4.7.4 Rotational Part: Z (R)First we consider the system using classical mechanics. If the molecule rotatesabout the z-axis with an angular frequency ω, the velocity is given by v =(d/2)ω, whered is the distance between the centers of the two atoms. Thusthe energy isE (R) =2× m 2( d2 ω ) 2= 1 4 md2 ω 2 = 1 2 Iω2 , (4.51)where I =(1/2)md 2 is the moment of inertia of the molecule. Since ω is a continuousvariable, E (R) is also continuous, and we cannot count the number ofmicroscopic states.The angular momentum associated with this rotation, L = r ×p, has onlya z-component,L z =2× d 2 × md 2 ω = 1 2 md2 ω = Iω. (4.52)This is also a continuous variable in classical mechanics, but is quantized inquantum mechanics:L z = Iω = l , l =0, 1, 2, ··· . (4.53)In terms of this l, the square of the angular momentum isL 2 = l (l +1) 2 , (4.54)and the energy isE (R) = 12I L2 = l (l +1) 22I . (4.55)For each value of l, thereare2l + 1 states owing to the freedom in the choiceof the direction of the axis of rotation. 7 If we accept this result of quantummechanics, the partition function can be calculated as follows:∞∑}Z (R) = (2l +1)exp{−l (l +1) β2 . (4.56)2Il=0This summation cannot be done analytically. However, at high temperatureit can be done approximately. When β 2 /2I ≪ 1, the exponential factorbecomes small only at large l. The factor (2l + 1) makes the contributionsfrom large values of l more important than those from small values. In thiscase it is a good approximation to calculate the summation as an integral. Asthe variable of the integral, we takeε (l) ≡ l (l +1) β22I . (4.57)7 When L 2 = l(l +1) 2 , L z takes 2l + 1 quantized values, which can be expressedas L z = m z, wherem z is an integer that satisfies −l ≤ m z ≤ l.