Chemistry 21b
Problem set # 3
Out: 19 January 2011
Due: 26 January 2011
Problems are worth: 1a=10, 1b=10, 1c=10, 1d=10; 2a=15, 2b=15; 3a=10, 3b=10, 3c=10.
1. At low temperatures, the rotational part of the partition function
Q
(
T
) =
∞
s
J
=0
(2
J
+ 1)exp(

E
J
/kT
)
dominates the nontranslational degrees of freedom, where
E
J
=
BJ
(
J
+ 1)
cm
−
1
for a simple
diatomic or linear molecule in a
1
Σ (that is, closed shell) electronic state. Roughly, this function tells
you over how many states the population is distributed, and is central to the Boltzmann distribution
(
N
J
/g
J
) = (
N/Q
)
e
−
E
J
/kT
, where
N
J
=population in state
J
,
g
J
=degeneracy of state
J
,
E
J
=energy
of state
J
,
N
=total population, and
Q
=the partition function) and statistical thermodynamics, as
you’ll see later in Ch 21c (or may already know).
(a.) Derive an alternative, approximate form of this partition function by replacing the summation
with an integral and by taking
J
to be a continuous variable. This formula should depend only
on the temperature
T
and the rotation constant
B
.
(b.) Evaluate the values of
Q
(
T
) at
T
=20 and
T
=200 K, both by the direct summation (truncated
at some appropriate values of
J
, use a mathematical package for convenience!) and through
use of your approximate formula for the HCCCN molecule. The rotational constant for this
molecule is
B
=4549.06 MHz.
(c.) Recall that the fractional population in level
J
is given by
p
(
J
) = (2
J
+ 1)exp(

E
J
/kT
)
/Q
(
T
)
.
Derive a formula for the value of
J
=
J
max
at which the maximum population occurs for any
speciFed temperature, based upon your integral form of the partition function.
(d.) ±ind
J
max
for
T
=20 and
T
=200 K for HCCCN.
2.
Prove that the following relationships, asserted in class, are in fact true for a complete,
orthonormal basis set:
(a)
s
f

f >< f

= 1
(the closure relation)
(b)
< m

∂
∂x

k >
=

m
¯
h
2
(
E
m

E
k
)
< m

x

k > .