Transcript Slide 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.
ROTATIONAL PARTITION FUNCTIONS
For the linear rigid rotor we had earlier:
h2
Erot = 2 J(J+1) = hBJ(J+1) where I and B
8𝜋 I
are, again, the moment of inertia and the
rotational constant respectively.
ROTATIONAL PARTITION FUNCTIONS:
Since each rotational level has a (2J+1) fold
degeneracy
−ϵ
qRot = 𝑛
𝑖=0 gi e i/ kBT
=
∞
𝐽=0
2/8𝜋2Ik T
−J(J+1)h
B
(2J+1)e
=
∞
𝐽=0
(2J+1)e−hBJ(J+1)/kBT
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)
kBT
qRot =
= kBT/hcB
hB
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
16O (all are populated for 16O18O!). We need
2
a symmetry number, σ, equal to 1 normally,
or 2 for symmetric linear molecules.
ROTATIONAL PARTITION FUNCTIONS:
Our previous formula becomes
qRot
kBT
=
σhB
where σ = 1 (unsymmetrical molecule – eg.
HCl) and σ = 2 (symmetrical molecule – eg.
C16O2)
TYPICAL PARTITION FUNCTION VALUES:
Molecule
H2
H35Cl
D35Cl
16O
2
CsI
H-C≡C-F
B(MHz)
1,824,300
312,991
161,656
43,101
708.3
9706
σ
2
1
1
2
1
1
qRot (300K)
1.71
20.0
38.7
72.5
8830
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)
P = (2J+1)e −hBJ(J+1)/kBT /q
i
Rot
ROTATIONAL LEVEL POPULATIONS – CO:
(2J+1)e −hBJ(J+1)/kT
Pi
1
1
0.00927
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
J
2J+1
0
1
1
e−hBJ(J+1)/kT
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 e −hBJ(J+1)/kBT or “energy term” acts
to decrease Pi values as J increases. As
always, the ∞
𝑖=𝑜 Pi = 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
n12
n22
n32
E(n1,n2,n3) =
+ 2 + 2
2
a
b
c
This expression was obtained using the
appropriate Hamiltonian (with potential
energy V(x,y,z) = 0) after employing
separation of variables.
ℎ
8𝑚
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.