rotational partition functions:
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Rotational partition functions:. - PowerPoint PPT PresentationTRANSCRIPT
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.