15 October 1996

Previous Page

The integral approximation which was spectacularly successful for translation is only appropriate here for non-hydrogen rotations. With an atom as light as H, the moment of inertia, I, becomes quite small, making B large. Indeed, since BJ(J+1)/kT is the dimensionless exponent in the partition function, that means that B/k has units of temperature. In fact, it is called _rot and measures the temperature at which significant rotational excitation can begin to occur. For the vast majority of molecules, such as carbon dioxide, B/k < 1 K, but this rotational onset temperature for hydrogen molecule is 88 K. That means that only a handful of rotational states are available at room temperature for hydrogen, making an integral approximation for z perilous.

But we'll make it anyway! In cases where B is too large, that only means we'll actually have to calculate the sum over states, but few states will contribute. In normal circumstances, the following will suffice instead. Same trick; different mode.

z_rot = (2J+1)exp[-BJ(J+1)/kT]

~ exp[-BJ(J+1)/kT] (2J+1)dJ

= exp[-BJ(J+1)/kT] d[J(J+1)]

= exp[-Bx/kT]dx where x=J(J+1)

= kT/B = T/_rot

If we believe it for diatomic hydrogen, it says that since room temperature is only about 3_rot, z=3 there. Certainly not enough states for a continuum over which to integrate! But for carbon dioxide under the same circumstances, it says z~600; good choice for integration.

Unfortunately, in both these cases, T/_rot overestimates z by a factor of 2. This is because we've assumed that a 360° rotation gives us an indistinguishable configuration of the molecule. But both of these molecules become indistinguishable under only 180° of rotation! We've overcounted.

z_rot = kT/B is the way this correction factor is normally inserted. is only appropriate for molecules whose symmetry is high enough to include rotational symmetries. So normal water has =2 but HOD doesn't. Standard molecular Symmetry Tables give directly as the sum of all real rotations (designated as C operations) plus the identity operation (designated E). Remember to include the number given prior to each C as this represents the number of different rotations of that type present in the molecule!

Not surprisingly, for example, ammonia has =3.

Since z_rot has a single factor of T, as opposed to translation's T^3/2, the derivation of E_rot will yield RT instead of (3/2)RT. However, for non-linear molecule, like ammonia, there will be 3 rotation axes, not 2. Indeed, the general (integral approximation) rotational partition function for non-linear molecules will be given by

z_rot = (1/) (kT)^3/2(/ABC)^½ where A, B, and C are the ²/2I) for each separate rotation axis.

Vibration

The exact partition function for the Harmonic Oscillator (a not-unreasonable model for molecular vibrations near ground state) was given in the solution of the first quiz (fall '95), and was there found to be

z_vib = 1/[1-exp(-_vib/T)]

where _vib = h/k which is generally several times larger than room temperature. In other words, at room temperature, vibrations haven't turned on, and almost all of the molecules will be in their ground vibrational states for all vibration modes.

How many vibration modes can there be? Remember 3N-6 (or 3N-5 if a linear molecule)? That was a count of all the "degrees of freedom" of motion of the molecule (3N from x,y,z for each of the N atoms) less the Center of Mass translation (X,Y,Z of the molecule as a whole) less the 3 (2 if linear) rotational degrees of freedom. An analysis of how each symmetry operation in a Symmetry Table distorts each atomic displacement (dx, dy, dz) can tell you not only how many vibrations there are but also the degeneracy of each and which ones will be infrared or Raman active. See . . . Symmetry Tables are your friend.

If we substitute that partition function in the molar energy expression, we find

E_vib = R_vib / [exp(+_vib/T) - 1] for each vibration mode with its unique _vib.

In the limit of T>>_vib, E reduces to RT for each vibration, but such a T is so high that the molecule will have disintegrated long since!

Chris Parr
University of Texas at Dallas
Programs in Chemistry, Room BE3.506
P.O. Box 830688  M/S BE2.6 (for snailmail)
Richardson, TX  75083-0688

   Voice: (972) 883-2485
     Fax: (972) 883-2925
     BBS: (972) 883-2168 (HST) or -2932 (V.32bis)
Internet: parr@utdallas.edu  (Click on that address to send Chris e-mail.)