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5.62 Lecture #14: Low and High-T Limits for q
rot
and
q
vib
Reading: Hill, pp. 153-159, Maczek pp. 51-
TEMPERATURE DEPENDENCE OF Erot AND CV
rot
Low T limit of Erot:
T lim Erot = lim (^) (6Nkθr e ) = 0
− 2 θr T→ 0 T→ 0
e
− 2 θr lim CV
rot = lim ⎜
⎛12Nkθ r
2 T
T→ 0 T→ 0 ⎝ T^2 ⎠
Low T Limit High T Limit
rot 2 rot CV 12 θ r e
− 2 θ r T Cv ≅ 1 nR
T
2 nR
Erot ≅ 6 θr e
− 2 θr T Erot ≅ T nR nR
Note maximum in
C
R
v ≅ 1.624 at
T
= 1.0 if we retain the two-term formula for the low-T θrot
CV
rot nR
Erot nR
T/θrot
limit. Actual maximum, derived from the full qV
rot , is CV/nR = 1.098 at T/θrot = 0.8. [Rapid
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
U ( R ) = ( k / 2 )( R − Re)
2
Using harmonic approximation:
ε (^ v) = v + hν = v + hcωe 2 2
zero point energy — when v = 0 ε(v = 0 ) =
hν 2
∞
So q
vib =^
∑ exp^ ⎣^ − ε( ( )^ v^ − ε^ (v^ =^0 ))^ kT
v= 0 ∞ ∞
q
vib =^ ∑^ e
−vθvib T
= ∑ x
v
1 −
x v= 0 v= 0
q
vib
1 − e
−θvib T (^) all values of θvib/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.
High Temp Limit of (^) q
vib θvib^ ^ T or^ εvib^ ^ kT
1 − e
−θvib T
qvib =
T ~ + 1 −
θvib
θvib
2
If θvib T, then e
−θvib
T 2T
2
So q
vib =^ 1 − e
−θvib ⎡ ⎣
2 2 T
T
1 − 1 − θvib T + θvib
θ
T
vib hc
kT
ωe
qvib = high temperature limit
When is high temperature limit form useful? For molecules, not often …
EXACT EXACT
MOLECULE (^) θvib[K] q(T=300K) (^) (300/θvib) q(3000K) (^) (3000/θvib)
H 2 6328 0.0474 1.138 0.
HCl (^4302) 1 + 6 × 10
CO 3124 1 + 3 × 10
Br 2 465 1.269 0.645 6.964 6.
I 2 309 1.556 1.
1 + 7 × 10
Cs 2 60.4 5.481 4.926 50.14 49.
1% error
Only for very heavy molecules at very high T is the high temperature limit form for q
vib
useful.
VIBRATIONAL CONTRIBUTIONS TO THERMODYNAMIC FUNCTIONS
2 T
= e
−θvib qVIB =
e
−θvib
1 − e
−θvib T
2 T q
vib
N = e
−Nθvib 2 T *N QVIB = qvib qvib
ln QVIB = −Nθvib / 2T + N ln q
vib
Evib = kT
2 ⎜
⎛∂ln Q vib
⎟ =^ kT
2 ⎡∂ (− Nθ v / 2T)^
N∂ln q
vib
⎝ ∂T ⎠N,V ⎣
∂T ∂T ⎦
T
NkT (
2 θvib
2
∂ln 1 − e
−θv ⎥ 2T
2 ⎣ (^) ∂T ⎦
∂( 1 − e
−θv
Nk 1 − e
−θv T = ⎢^ ⎥ 2
θvib + NkT
2
⎣ ∂T
T
− 1
T
T ) θvib e
−θv Evib =
Nk θvib + NkT
1 − e
−θv T
2 2 T
2
( 1 − e
−θv
e
θv T − 1
Nkθvib (E − E 0 =
Nkθvibe
−θ
T
v T ) = vib 1 − e
−θv
zero point energy (energy of v = 0 above minimum of potential curve)
E 0 =
Nk
Nhcωe reference all energies with respect to
2
θvib 2 zero point energy
NkTx Define x ≡ θvib/T (^) (E − E 0 ) vib
e
x − 1
Einstein Function (E^ −^ E^0 x
RT
) vib = e
x − 1
plotted vs. x in handout
