CHM 5414 Lecture Notes
15 October 1996

Eventually, we must leave the rarified air of theoretical chemistry and create something practical. So here is where the rubber meets the road; thermodynamic calculations from first principles and molecular properties. It's not going to be that messy. Promise.

Now we know, for instance, that

E = kT² [ ln(Z)/T]_V,

we want to plow molecular parameters into Z and calculate thermodynamic quantities from first principles. (Don't we? C'mon... a little enthusiasm here!)

Since Z=z^N/N!,

E = RT² [ ln(z)/T]_V

for molar energy, and we can now search for z's factors of translation, rotation, vibration, and electronic modes. Classical thermo makes a big deal out of referencing standard energies of elements to zero at standard conditions (1 atm and 298 K), but here we're referencing them to zero at their zero point levels. Thus, our statistical zero of energy occurs at absolute 0 K.

Translation

The nice thing about that partial derivative with respect to T (at constant V) of ln(z) is that we can extract just the T dependence factor in z_trans and ignore the remaining factors; the ln makes them sums and the derivative kills 'em! So what is z_trans's dependence on T?

_trans = (h²/ 8mL²) (n_x² + n_y² + n_z²)

but we can fixate instead on

_x = (h²/ 8mL_x²) n_x²

and substitute y and z later.

Also, for convenience, let

² = (h²/ 8mkTL_x²) so that _x / kT = ²n_x².

Although z calls for a sum over states, we're going to replace it with an integral since d_trans between two translational energy levels is vanishingly small. Let translational energy be continuous.

z_x = exp(-²n_x²) ~ exp(-²n²)dn, the Gaussian Probability Integral.

Although that integral has no "closed form" solution if the endpoint is different from , with that endpoint, the integral's value is ^½/2.

That makes z_x = (2mkT/h²)^½ L_x, and z_y and z_z likewise.

Unfortunately, Netscape has run out of graphic symbol space (!), so we must continue this on Page 2.

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.)