Rotational partition functions:

1. Rotational partition functions: • We will consider linear molecules only. Usually qRotational» qVibrational. This is because: 1. rotational energy level spacings are very small compared to vibrational spacings and 2. each rotational level has a 2J+1 fold degeneracy. Due to degeneracy the populations of higher J levels are much higher than would be otherwise expected.

2. Rotational Partition functions • For the linear rigid rotor we had earlier: • Erot= J(J+1) = hBJ(J+1) where I and B are, again, the moment of inertia and the rotational constant respectively.

3. Rotational partition functions: • Since each rotational level has a (2J+1) fold degeneracy • qRot= kBT • = (2J+1) • = (2J+1)

4. Rotational Partition functions: • The last formula has no “closed form” expression. If the rotational spacings are small compared to kBT (true for most molecules, except H2, at room T and above) we can replace the summation by an integral and obtain eventually (see text) • qRot = = kBT/hc

5. Rotational Partition Functions: • The last formula is “valid” (i.e. a good approximation) for almost all unsymmetrical linear molecules.Aside: For symmetrical linear molecules rotational levels may not all be populated. Only half are populated for 16O2 (all are populated for 16O18O!). We need a symmetry number, σ, equal to 1 normally, or 2 for symmetric linear molecules.

6. Rotational partition functions: • Our previous formula becomes • qRot = • where σ = 1 (unsymmetrical molecule – eg. HCl) and σ = 2 (symmetrical molecule – eg. C16O2)

7. Typical partition function values:

8. Partition function Comments: • The previous slide shows that, for heavier molecules, many rotational levels are populated (thermally accessible) at 300K. • Populations of individual levels can be calculated using (unsymmetrical molecule) • Pi = (2J+1)/qRot

9. Rotational level populations – CO:

10. Comments on Previous slide: • For 12C16O at 300k the J=0 level does not have the highest population. • The (2J+1) or degeneracy term acts to “push up” Pi values as J increases. • The or “energy term” acts to decrease Pi values as J increases. As always, the = 1. Why?

11. Comments – continued: • Less than 1% of CO molecules are in the J=0 level at 300K.(More than 99.99% of CO molecules are in the v=0 level at 300K) • P0 = 1/qRot The P0 value is small for many linear molecules at room temperature. P0 values can be increased by lowering the temperature of the molecules.

12. HCl and DCl Infrared spectra: • The HCl and DCl spectra obtained in the lab show features consistent with the reults presented here. These spectra are shown on the next slides for consideration/class discussion.

15. The Hydrogen Atom: • Recall, for the 3-dimensional particle in a box problem • E(n1,n2,n3) = • This expression was obtained using the appropriate Hamiltonian (with potential energy V(x,y,z) = 0) after employing separation of variables.

16. The Hydrogen atom: • For the 3-dimensional PIAB we have: • 3 Cartesian coordinates • 3 quantum numbers required to describe E. • With problems involving rotation (especially in 3 dimensions) and energies of electrons in atoms, spherical polar coordinates (r,θ,φ)are a more natural choice than Cartesian coordinates. Why?

17. Atoms and Electronic Energies: • In other chemistry courses electronic energies were discussed using three quantum numbers. • n – principal quantum number (n=1,2,3,4,5 …) • l – orbital angular momentum quantum number l = 0,1,2,3,4…,n-1 • ml– magnetic quantum number – ml= -l, -l+1, ….., l-1, l.

18. Coulombic interactions: • Class discussion of coulombic forces, energies and “work terms” (simple integration). Need for spherical polar coordinates in treating the H atom.