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5.62 Physical Chemistry II
Spring 2008
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5.62 Lecture #14: Low and High-T Limits for qrot and
qvib
Reading: Hill, pp. 153-159, Maczek pp. 51-53
TEMPERATURE DEPENDENCE OF Erot AND CVrot
Low T limit of Erot:
lim E rot = lim ( 6Nkθr e−2θr
T→0
T→0
T
)=0
⎛ 12Nkθ2r −2θ T ⎞
lim CVrot = lim ⎜
e r ⎟=0
T→0
T→0 ⎝
⎠
T2
1.0
E rot nR
Crot
V nR
1.0
Low T Limit
rot
V
2
r
C
12θ
≅ 2 e−2θr
nR
T
E rot
≅ 6θr e−2θr
nR
Note maximum in
T
T/θrot
High T Limit
T
Crot
v
≅1
nR
E rot
≅T
nR
Cv
T
≅ 1.624 at
= 1.0 if we retain the two-term formula for the low-T
R
θrot
limit. Actual maximum, derived from the full q Vrot , is CV/nR = 1.098 at T/θrot = 0.8. [Rapid
5.62 Spring 2008
Lecture 14, Page 2
change in CV is a signal of a gap in the level spacing measured in units of kT. What gap would
be relevant here? At what value of T/θrot would you expect the most rapid change in CV?]
QUANTUM BEHAVIOR
VIBRATIONAL MOLECULAR PARTITION FUNCTION qvib
2
U (R ) = (k / 2)(R − Re )
Using harmonic approximation:
⎛ 1
⎞
⎛ 1
⎞
ε (v
) = ⎜ v + ⎟ hν = ⎜ v + ⎟ hcωe
⎝
2
⎠
⎝
2
⎠
zero point energy — when v = 0
ε (v = 0 ) =
1
hν
2
revised 3/10/08 2:17 PM
5.62 Spring 2008
Lecture 14, Page 3
Calculate qvib
∞
q vib = ∑ e
−ε ( v) kT
∞
= ∑ e−hcωe
v=0
θvib =
Define
( v+1/2 ) kT
v=0
hcωe
k
“vibrational temperature” [K]
∞
q vib = ∑ e− v+1/2 θvib
(
)
T
v=0
For vibration, θvib > T almost always. Must sum over each vibrational level.
−θvib 2 T
q vib = e
∞
∑
e−vθvib
T
pull zero point energy out of sum
v=0
Let x = e−θvib
T
∞
∑
x
q vib = x
1/2
v
v=0
∞
Now
∑
x v = 1+ x + x 2 +… =
v=0
1
1− x
converges for |x| < 1, but 0 ≤ e−θvib T <1 for
all T, thus we have qvib valid for all T.
q vib
x1/2
e−θvib 2 T
=
=
1− x 1− e−θvib T
Molecular Vibrational Partition Function
Zero of Evib is set at minimum of potential energy curve
Define q*vib
q vib = e
−θvib 2 T
∞
∑
e−θvib T
v=0



q*vib
revised 3/10/08 2:17 PM
5.62 Spring 2008
Lecture 14, Page 4
∞
So
q
*
vib
= ∑ exp ⎡⎣− (ε (v) − ε (v = 0 )) kT⎤⎦
v=0
∞
q
*
vib
=∑ e
−vθvib T
= ∑ xv =
v=0
q*vib =
∞
v=0
1
1 − e−θvib
1
1− x
all values of θvib/T
T
Put this result aside. We will see how it is useful later in redefining our zeros of energy.
q*vib effectively shifts the zero of Evib to the energy of the v = 0 level.
θvib  T or εvib  kT
High Temp Limit of q*vib
q*vib =
If
1
1 − e−θvib
T
then e−θvib
θvib  T,
q*vib =
So
1
1 − e−θvib
q*vib 
T
T

~ +1 −
θvib θ2vib
+
−…
T 2T2
1
⎡
1 − ⎣1 − θvib T + θ2vib 2T2 ⎤⎦
T
kT
=
high temperature limit
θvib hcωe
When is high temperature limit form useful? For molecules, not often …
θvib[K]
6328
EXACT
q*(T=300K)
1 + 7 × 10–10
(300/θvib)
0.0474
EXACT
q*(3000K)
1.138
(3000/θvib)
0.474
HCl
4302
1 + 6 × 10–7
0.0697
1.313
0.697
CO
3124
1 + 3 × 10–5
0.0961
1.546
0.961
Br2
465
1.269
0.645
6.964
6.45
I2
309
1.556
1.029
10.22
10.29
Cs2
60.4
5.481
4.926
50.14
49.64
1% error
MOLECULE
H2
revised 3/10/08 2:17 PM
5.62 Spring 2008
Lecture 14, Page 5
Only for very heavy molecules at very high T is the high temperature limit form for q*vib
useful.
VIBRATIONAL CONTRIBUTIONS TO THERMODYNAMIC FUNCTIONS
e−θvib 2 T
= e−θvib 2 Tq*vib
−θvib T
1− e
Q VIB = q Nvib = e−Nθvib 2 Tq*N
vib
q VIB =
ln Q VIB = −Nθvib / 2T + N ln q*vib
⎡ ∂ (−Nθv / 2T) N∂ln q*vib ⎤
⎛ ∂ln Q vib ⎞
E vib = kT2 ⎜
+
⎟ = kT2 ⎢
⎥
⎣
⎝ ∂T ⎠N,V
∂T
∂T ⎦
⎡
−θv T −1 ⎤
(
)
NkT2θvib
2 ∂ln 1 − e
⎥
=
+ NkT ⎢
2
⎣
⎦
2T
∂T
−1 ⎤
⎡ (
Nk
∂ 1 − e−θv T ) ⎥
2
−θv T
(
)
⎢
=
θvib + NkT 1 − e
⎣
⎦
2
∂T
E vib
(E − E 0 )vib
Nk
e−θv T
2
−θv T θ vib
(
)
=
θvib + NkT 1 − e
2
T2 (1 − e−θv T )2
Nkθvib e−θv T Nkθvib
=
= θv T
1 − e−θv T
e
−1
↑
zero point energy (energy of v = 0 above minimum of potential curve)
E0 =
Nk
Nhcωe
θvib =
2
2
(E − E 0 )vib =
Define x ≡ θvib/T
(E − E 0 )vib
RT
reference all energies with respect to
zero point energy
=
x
x
e −1
NkTx
e x −1
Einstein Function
plotted vs. x in handout
revised 3/10/08 2:17 PM