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.

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.

ROTATIONAL PARTITION FUNCTIONS: Since each rotational level has a (2J+1) fold

degeneracy qRot = kBT

= (2J+1)

= (2J+1)

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

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.

ROTATIONAL PARTITION FUNCTIONS:

Our previous formula becomes qRot =

where σ = 1 (unsymmetrical molecule – eg. HCl) and σ = 2 (symmetrical molecule – eg. C16O2)

TYPICAL PARTITION FUNCTION VALUES: Molecule B(MHz) σ qRot (300K)H2 1,824,300 2 1.71H35Cl 312,991 1 20.0D35Cl 161,656 1 38.716O2 43,101 2 72.5CsI 708.3 1 8830H-C≡C-F 9706 1 644

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

ROTATIONAL LEVEL POPULATIONS – CO:J 2J+1 (2J+1) Pi

0 1 1 1 0.00927

1 3 0.9816 2.945 0.02729

2 5 0.9459 4.730 0.04383

5 11 0.7573 8.330 0.07720

8 17 0.5131 8.723 0.08085

10 21 0.3608 7.578 0.07023

15 31 0.1082 3.353 0.03108

20 41 0.00204 0.8366 0.00775

25 51 0.00242 0.1235 0.00114

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?

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.

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.

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.

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?

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.

COULOMBIC INTERACTIONS:

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