http://www.natur.cuni.cz/chemie/fyzchem Faculty of Science, Charles University in Prague Diatomic and polyatomic ideal gas: vibrations, rotations Peter Košovan [email protected]Dept. of Physical and Macromolecular Chemistry Lecture 4, Statistical Thermodynamics, MC260P105, 3.11.2014 If you find a mistake, kindly report it to the author :-)
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• Intermediate temperatures: none of the above, but Euler-MacLaurin:
qrot(T ) =TΘr
(1 +
13
(Θr
T
)+
115
(Θr
T
)2
+4
315
(Θr
T
)3
+ · · ·
)
P. Košovan Lecture 4: Polyatomic ideal gas 9/31
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Euler MacLaurin expansion in more detail
b∑n=a
f (n) =
∫ b
af (n) dn +
12
(f (b) + f (a)
)+∞∑
j=1
(−1)j Bj
(2j)!
(f (2j−1)(a)− f (2j−1)(b)
)f k (a) is the k -th derivative of f (a).{Bj} are Bernoulli numbers:
B1 =16, B2 =
130, B3 =
142, · · ·
Example:
11− eα
=∞∑
j=0
e−αj =1α
+12− α
12+
α3
720+ · · ·
P. Košovan Lecture 4: Polyatomic ideal gas 10/31
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In most cases Θr � boiling temperature
Table from McQuarrie, Statistical Mechanics, University Science Books (2000)
P. Košovan Lecture 4: Polyatomic ideal gas 11/31
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Various approximations to qrot for HCl (Θr = 15.02 K)
0.6
0.7
0.8
0.9
1
1.1
1.2
1.3
0 2 4 6 8 10 12
qro
t(n
)
number of terms
T = 10 K
High T
Euler-MacLaurin
Direct sum
0
1
2
3
4
5
6
7
8
0 2 4 6 8 10 12
qro
t(n
)
number of terms
T = 100 K
High T
Euler-MacLaurin
Direct sum
0
5
10
15
20
0 2 4 6 8 10 12
qro
t(n
)
number of terms
T = 300 K
High T
Euler-MacLaurin
Direct sum
0
5
10
15
20
25
30
0 2 4 6 8 10 12q
rot(n
)
number of terms
T = 400 K
High T
Euler-MacLaurin
Direct sum
P. Košovan Lecture 4: Polyatomic ideal gas 12/31
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Rotational contribution to thermodynamic functions
0
0.1
0.2
0.3
0.4
0.5
0 2 4 6 8 10 12
f J
J
Population of rotational states in HCl at 300 K
Θr = 15.02 K
Euler-MacLaurin
High-T limit
Direct sum (4 terms)
Erot = NkBT 2(∂ ln qrot
∂T
)= NkBT + · · ·
Crotv =
(∂Erot
∂T
)N
= NkB + · · ·
Note that for T →∞:
Erot → NkBT , Cv → NkB
Population of vibrational states:
fJ =(2J + 1)e−ΘrJ(J+1)/T
qrot(T )
Jmax =
(kBT2B
)1/2
− 12≈(
kBT2B
)1/2
=
(T
2Θr
)1/2
P. Košovan Lecture 4: Polyatomic ideal gas 13/31
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Homonuclear diatomics: symmetry of wave functionSymmetry of the total wave function upon interchange of nuclei:• Bosons (integral spin): symmetric• Fermiions (half-integral spin): antisymmetric
Interchange of electrons:1. Inversion of all particles2. Inversion of just the electrons back
ψ′tot = ψtrans ψvib ψrot ψelec exclusive the nuclear part
• ψtrans, ψvib symmetric with respect to inversion• ψelec depends on symmetry of the ground state• The most common Σ+
g ground state is symmetric wrt inversion
• Only ψrot can control the symmetry of ψ′tot.
P. Košovan Lecture 4: Polyatomic ideal gas 14/31
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Implications of inversion for H2
H2 atom nuclei with spins 1/2:• 3 symmetric spin functions:αα, ββ, (αβ + βα)/
√2
• 1 anti-symmetric spin function:(αβ − βα)/
√2
• Nuclei with s = 1/2 arefermions⇒ ψtot anti-symmetricwrt interchange of nuclei
Possible further extensions:• Anharmonic vibrations• Vibration-rotation coupling• Centrifugal distorsion• Molecules with low electronic states: inclusion of more states in qelec
• Molecules with other than Σ ground state: coupling betweenelectronic and rotational angular momenta
P. Košovan Lecture 4: Polyatomic ideal gas 22/31
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Thermodynamic functions of diatomics (T � Θr)
Cv
NkB=
52
+
(hν
kBT
)2 eβhν
(eβhν − 1)2
SNkB
= ln
(2π(m1 + m2)kBT
h2
)3/2Ve5/2
N+ ln
8π2IkBT eσh2
+βhν
eβhν − 1− ln(1− e−βhν) + lnωe1
pV = VkBT(∂ ln Q∂V
)N,T
= NkBT
µ0(T )
kBT=− kBT ln
(2π(m1 + m2)kBT
h2
)3/2
− ln8π2IkBTσh2
+hν
2kBT+ ln(1− e−βhν)− De
kBT− lnωe1
P. Košovan Lecture 4: Polyatomic ideal gas 23/31
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Polyatomic ideal gas
q(V ,T ) = qtrans qrot qvib qelec qnuc
Q(N,V ,T ) =(qtrans qrot qvib qelec qnuc)N
N!
• Separation of individual degrees of freedom: rigid rotor, harmonicoscillator.
• Analogy with diatomics• Three degrees of freedom per rotation• (3n − 5) or (3n − 6) vibrational degrees of freedom
P. Košovan Lecture 4: Polyatomic ideal gas 24/31
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Vibrations in polyatomics
• Normal coordinates, normal modes• (3n − 5) or (3n − 6) independent harmonic oscillators
ε =α∑
j=1
(nj +
12
)hνJ
νj =1
2π
(kj
µj
)1/2
Θv,j =hνj
kB
qvib =α∏
j=1
e−Θv,j/2T
(1− e−Θv,j/T )
Evib = NkB
α∑j=1
(Θv,j
2+
Θv,je−Θv,j/T
1− e−Θv,j/T )
)
CvibV = NkB
α∑j=1
((Θv,j
2
)2
+e−Θv,j/T
1− e−Θv,j/T
)P. Košovan Lecture 4: Polyatomic ideal gas 25/31
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Rotations in polyatomicsLinear polyatomics (analogy with diatomics):
qrot =8π2IkBTσh2 =
TσΘr
Non-linear polyatomics – principal moments of inertia: IA, IB, IC .
A =h2
8π2IA, B =
h2
8π2IB, C =
h2
8π2IC
ΘA =8π2IAkB
h2 , ΘB =8π2IBkB
h2 , ΘC =8π2ICkB
h2 ,
Special cases:• Spherical top: IA = IB = IC• Symmetric top: IA = IB 6= IC• Asymmetric top: IA 6= IB 6= IC
P. Košovan Lecture 4: Polyatomic ideal gas 26/31
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Special casesSpherical top (ΘA = ΘB = ΘC):
εJ =J(J + 1)~2
2I
ωJ = (2J + 1)2
High T limit:
qrot =1σ
∫ ∞0
(2J + 1)2e−J(J+1)~2/2IkBT dJ
≈ 1σ
∫ ∞0
4J2e−J2~2/2IkBT dJ =π1/2
σ
(8π2IkBT
h2
)3/2
(1)
qrot =π1/2
σ
(T 3
ΘAΘBΘC
)1/2
P. Košovan Lecture 4: Polyatomic ideal gas 27/31
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Special cases
Symmetric top (ΘA = ΘB 6= ΘC):
εJK =~2
2
(J(J + 1)
IA+ K 2
(1IC− 1
IA
))J = 0,1,2, · · · ; K = J, J − 1, · · · ,−J
ωJK = (2J + 1)
High T limit:
qrot =1σ
∞∑J=0
(2J + 1)2e−αAJ(J+1)+J∑
K =−J
e−αCK 2, αj =
~2
2IjkBT, j = A,C;
qrot =π1/2
σ
(8π2IAkBT
h2
)(8π2ICkBT
h2
)1/2
=π1/2
σ
(T 3
ΘAΘBΘC
)1/2
P. Košovan Lecture 4: Polyatomic ideal gas 28/31
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Special casesAsymmetric top (ΘA 6= ΘB 6= ΘC):• Very involved at quantum level.• Can be solved numerically.• Analytically solvable in the classical limit.High T limit:
qrot =π1/2
σ
(8π2IAkBT
h2
)1/2(8π2IBkBTh2
)1/2(8π2ICkBTh2
)1/2
Common formulation for all cases using rotational temperatures:
qrot =π1/2
σ
(T 3
ΘAΘBΘC
)1/2
Erot =32
NkBT , CrotV =
32
NkB, Srot = NkB ln
(π1/2
σ
(T 3e3
ΘAΘBΘC
)1/2)
P. Košovan Lecture 4: Polyatomic ideal gas 29/31
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Thermodynamic functions of a non-linear polyatomic
q = V(
2πMkBTh2
)3/2 π1/2
σ
(T 3
ΘAΘBΘC
)1/2(
3n−6∏j=1
e−Θv,j/2T
1− e−Θv,j/T
)ωe,1eβDe
ENkBT
=32
+32
+3n−6∑j=1
(Θv,je−Θv,j/T
1− e−Θv,j/T
)− De
kBT
− ANkBT
= ln(
2πMkBTh2
)3/2 VeN
+ lnπ1/2
σ
(T 3
ΘAΘBΘC
)1/2
−3n−6∑j=1
(Θv,j
2T+ ln(1− e−Θv,j/T )
)+
De
kBT+ lnωe,1
P. Košovan Lecture 4: Polyatomic ideal gas 30/31
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Thermodynamic functions of a non-linear polyatomic
− ANkBT
= ln(
2πMkBTh2
)3/2 VeN
+ lnπ1/2
σ
(T 3
ΘAΘBΘC
)1/2
−3n−6∑j=1
(Θv,j
2T+ ln(1− e−Θv,j/T )
)+
De
kBT+ lnωe,1
SNkB
= ln(
2πMkBTh2
)3/2 Ve5/2
N+ ln
π1/2e3/2
σ
(T 3
ΘAΘBΘC
)1/2
−3n−6∑j=1
(Θv,j/T
eΘv,j/T − 1− ln(1− e−Θv,j/T )
)+ lnωe,1
Next lecture(s):• Statistical thermodynamics in the classical limit• Chemical equilibrium in dilute gases