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gueDiatomic and polyatomic ideal

gas: vibrations, rotations

Peter Košovanpeter.kosovan@natur.cuni.cz

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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Rigid rotor – harmonic oscillator approximation

Diatomic ideal gas:

µ =m1m2

m1 + m2

Rigorous separation:

H = Htr + Hinternal

ε = εtr + εint

q = qtr qint

Q(N,V ,T ) =1

N!(qtr qint)

N

where

qtr = V(

2π(m1 + m2)kBTh2

)3/2

=VΛ3

Approximate separation

Hint = Hvib + Hrot

εint = εrot + εvib

qint = qrotqvib

Q(N,V ,T ) =1

N!(qtr qrot qvib)N

P. Košovan Lecture 4: Polyatomic ideal gas 1/31

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Rotation and vibrationHarmonic approximation:

u(r) = u(re) + (r − re)

(dudr

)r=re

+12

(r − re)2(

d2udr2

)r=re

+ · · ·

u(r) = u(re) + k(r − re)2 + · · ·

Vibrational energy levels(harmonic oscillator):

εn = hν(n +12

)

ν =1

2π

(kµ

)1/2

ωn = 1 for all n

Image source: wikimedia commons

Rotational energy levels(rigid rotor):

εJ =h2J(J + 1)

2I

ν =h

4π2I(J + 1)

ωJ = 2J + 1P. Košovan Lecture 4: Polyatomic ideal gas 2/31

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Electronic and nuclear degrees of freedom

∗

Further approximate separation ofelectronic and nuclear DOF:

H = Htr + Hrot + Hvib + Hel + Hnucl

ε = εtr + εrot + εvib + εel + εnucl

q = qtr qrot qvib qel qnucl

Q(N,V ,T ) =1

N!

(qtr qrot qvib qel qnucl

)N

Electronic partition function:

qel = ωe,1eDe/kBT + ωe,2e−ε2/kBT + · · · where D0 = De −12

hν

∗Image source: wikimedia commons.P. Košovan Lecture 4: Polyatomic ideal gas 3/31

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Vibrational partition function

Vibrational energy levels and degeneracy:

εn = hν(

n +12

), ωn = 1 for all n

qvib(T ) =∑

n

e−βεn = e−βhν/2∞∑

n=0

e−βhνn =e−βhν/2

1− e−βhν

qvib(T ) is a geometric series and can be summed directly (very rare!).

High temperature limit for later comparison:

qvib(T ) = e−βhν/2∫ ∞

n=0e−βhνdn =

kBThν

=TΘv

for (hν � kBT )

Vibrational temperature: Θv = hν/kB

P. Košovan Lecture 4: Polyatomic ideal gas 4/31

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Vibrational contribution to thermodynamic functionsVibrational temperature: Θv = hν/kB.

Evib = NkBT 2(∂ ln qvib

∂T

)= NkB

(Θv

2+

Θv

eΘv/T − 1

)Cvib

v =

(∂Evib

∂T

)N

= NkB

(Θv

T

)2 Θv

(eΘv/T − 1)2

Note that for T →∞:

Evib → NkBT , Cv → NkB

• Universal form forheteronuclear diatomics

• Values of Θv are tabulated 0

0.2

0.4

0.6

0.8

1

1.2

0 0.5 1 1.5 2

Cvvib

(T

) /

Nk

B

T / ΘvP. Košovan Lecture 4: Polyatomic ideal gas 5/31

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Properties of some common diatomic molecules

Table from McQuarrie, Statistical Mechanics, University Science Books (2000)

P. Košovan Lecture 4: Polyatomic ideal gas 6/31

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Population of vibrational states

fn =e−βhν(n+1/2)

qvib, fn>0 =

∞∑n=1

e−βhν(n+1/2)

qvib= 1− f0 = e−Θv/T

0

0.2

0.4

0.6

0.8

1

0 1 2 3 4 5

f n

n

Population of vibrational states in H2

Θv = 6215 K

H2, 300 KH2, 700 K

0

0.2

0.4

0.6

0.8

1

0 1 2 3 4 5f n

n

Population of vibrational states in Br2

Θv = 463 K

Br2, 300 KBr2, 700 K

Population of states of H2 and Br2 at various temperatures.

P. Košovan Lecture 4: Polyatomic ideal gas 7/31

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Rotational partition function (heteronuclear diatomic)Rotational energy levels and degeneracy:

εJ =h2J(J + 1)

2I= BJ(J + 1), ωJ = 2J + 1

where we defined the rotational constant B = h2/(8π2I)

qrot(T ) =∑

J

(2J + 1)e−βBJ(J+1)

• qrot(T ) cannot be summed directly.• Analogous with Θv we define rotational temperature Θr = B/kB

• The high temperature limit (T � Θr):

qrot(T ) =

∫ ∞0

(2J + 1)e−J(J+1)Θr/T dJ

P. Košovan Lecture 4: Polyatomic ideal gas 8/31

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Approximations to qrot(T ) (heteronuclear diatomic)

• The high temperature limit (T � Θr):

qrot(T ) =

∫ ∞0

(2J + 1)e−J(J+1)Θr/T dJ =

∫ ∞0

e−J(J+1)Θr/T d{J(J + 1)}

=TΘr

=8π2IkBT

h2

• Low temperatures (T . 1.4Θr) – first few terms suffice:

qrot(T ) = 1 + 3e−2Θr/T + 5e−6Θr/T + 7e−12Θr/T + · · ·

• 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

• Symmetric spin functionscouple with odd J

• Anti-symmetric spin functionscouple with even J

Rotational eigenstates:• J even: symmetric• J odd: anti-symmetric

Rigid rotor eigenfunctions for J ≤ 2. Imagesource: Wikimedia commons

• Statistical weights 1/3 for even/odd J• Parallel (ortho) and anti-parallel (para) nuclear spins in H2

P. Košovan Lecture 4: Polyatomic ideal gas 15/31

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Generaliztion of the effect of inversion on ψtot

For a nucleus with spin I there are (2I + 1) spin states.• α1, α2, · · ·α(2I+1)

• There are (2I + 1)2 nuclear spin functions in ψtot

• Anti-symmetric spin functions: αi(1)αj(2)− αi(2)αj(1)

• Total (2I + 1)(2I)/2 anti-symmetric functions• Total (2I + 1)2 − (2I + 1)(2I)/2 = (I + 1)(2I + 1) symmetric functions

General rule for symmetric Σ+g electronic states:

• Integral nuclear spin:• I(2I + 1) anti-symmetric spin functions couple with odd J• (I + 1)(2I + 1) symmetric spin functions couple with even J

• Half-integral nuclear spin:• I(2I + 1) anti-symmetric spin functions couple with even J• (I + 1)(2I + 1) symmetric spin functions couple with odd J

P. Košovan Lecture 4: Polyatomic ideal gas 16/31

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Example: acetylene H–C=C–H (linear polyatomic)

Figure from McQuarrie, Statistical Mechanics, University Science Books (2000)P. Košovan Lecture 4: Polyatomic ideal gas 17/31

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Homonuclear diatomic molecule

Integral spins:

qrot,nuc(T ) = (I + 1)(2I + 1)∑

J even

(2J + 1)e−ΘrJ(J+1)/T

+ I(2I + 1)∑

J odd

(2J + 1)e−ΘrJ(J+1)/T

Half-integral spins:

qrot,nuc(T ) = I(2I + 1)∑

J even

(2J + 1)e−ΘrJ(J+1)/T

+ (I + 1)(2I + 1)∑

J odd

(2J + 1)e−ΘrJ(J+1)/T

Rotational and nuclear partition function cannot be separated!

P. Košovan Lecture 4: Polyatomic ideal gas 18/31

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High temperature limit (T � Θr)

Applicable for (T & 5Θr):∑J even

≈∑

J odd

≈ 12

∑J all

≈ 12

∫ ∞0

(2J + 1)e−ΘrJ(J+1)/T dJ =T

2Θr

Equations for integral and half-integral spin both yield

qrot,nuc(T ) =(2I + 1)2T

2Θrcf. heteronucl. diatomic: qrot(T ) =

TΘr

which we can separate to

qrot(T ) =T

2Θr, qnuc = (2I + 1)2

We can combine both homo- and hetero-nuclear into one form:

qrot(T ) ≈ TσΘr

≈ 1σ

∞∑J=0

(2J + 1)e−ΘrJ(J+1)/T symmetry number: σ

P. Košovan Lecture 4: Polyatomic ideal gas 19/31

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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 20/31

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Hydrogen at low T is an exception

qrot,nuc(T ) =∑

J even

(2J + 1)e−ΘrJ(J+1)/T

+ 3∑

J odd

(2J + 1)e−ΘrJ(J+1)/T

Northo

Npara=

3∑

J odd(2J + 1)e−ΘrJ(J+1)/T∑J even(2J + 1)e−ΘrJ(J+1)/T

Cv(300 K) =34

Cv(ortho) +14

Cv(para)

P. Košovan Lecture 4: Polyatomic ideal gas 21/31

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Overall partition function of diatomics

q(V ,T ) = V(

2πmkBTh2

)3/2 (8π2IkBTσh2

) (e−βhν/2

1− e−βhν

)ωe,1eβDe

ENkBT

= kBT 2(∂ ln q∂T

)N,V

=52

+βhν

2+

βhνeβhν−1 − βDe

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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Pra

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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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ence

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Uni

vers

ityin

Pra

gue

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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Facu

ltyof

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ence

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Uni

vers

ityin

Pra

gue

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

P. Košovan Lecture 4: Polyatomic ideal gas 31/31