popdistributionsdiatmollect18me501f2011

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Purdue University

School of Mechanical Engineering

ME 501: Statistical ThermodynamicsLecture 18: Population Distributions

for Diatomic Molecules

Prof. Robert P. Lucht

Room 2204, Mechanical Engineering BuildingSchool of Mechanical Engineering

Purdue UniversityWest Lafayette, Indiana

[email protected], 765-494-5623 (Phone)

October 17, 2011


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Lecture Topics

• The nuclear partition function.

• Rotational distributions for heteronuclear molecules.

• Vibrational distributions.

• Electronic distributions.

• Rotational distributions for homonuclear molecules.


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The Nuclear Partition Function

• Let IA and IB be the spins of the ground states of nuclei A and B (don't have to consider excited nuclear states very often):

, ,

, ,

2 1 2 1

2 1 2 1

o A A o B B

nuc o A o B A B

g I g I

Z g g I I

• The nuclear partition function is often grouped with the rotational partition function because of the strong influence of nuclear spin on the relative intensities of even and odd J lines for homonuclear molecules,

1 2 1 2 1rot nuc A Brot

Z Z I I


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Rotational Distribution for a Heteronuclear Molecule

• Boltzmann distribution law is applied separately to each mode: for the translational mode this leads to the Maxwell velocity distribution that will be discussed in detail later.

• Rotational distribution: Assume that T >> rot, heteronuclear molecule, electronic level with electronic degeneracy =1. Then:

v

v

2 1 exp 1 /exp //

rotJ J Bl J

l rot rot

J J J Tg k TNN Z T

• Note that the level population is proportional to (2J+1).

• Determination of the rotational level with the maximum population. Assume that J varies continuously:

vmax

102 2

l J

rot

dN Tfor JdJ


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Rotational Distribution for the OH Molecule• For OH in the ground electronic state, Be = 18.911 cm-1, rot = 27.2 K.

T (K) Jmax

500 2.5

1000 3.5

1500 4.5

2000 5.5

2500 6.5

3000 7.5

Jmax rounded to

nearest half-

integer


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Vibrational Distribution

• Include zero-point energy (ZPE):

Note: ZPE contribution divides out.

vib vibv 1vib2

vib

exp / 1 exp /exp v /

exp / 2B vibl

l vib

g k T TN TN Z T

v exp v / 1 exp /lvib vib

l

N T TN

• Do not include zero-point energy (ZPE):

vib vibvvib

vib

exp /exp v / 1 exp /Bl

vibl

g k TN T TN Z

Note: You get the same answer as long as you are consistent in your treatment of ZPE.


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Electronic Distribution

• Not much simplification because we don’t have an analytical expression for the energy of different electronic levels in a diatomic molecule:

exp /elecl l Bl

elec

g k TNN Z


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Rotational Distributions for Homonuclear Molecules: Hydrogen

• For nuclei of odd mass number (fermions) the total wavefunction tot = rot

vib elec nuc must be antisymmetric with respect to exchange of nuclei:

, , , ,

, , , ,

, , , ,

, , , , v

tot tot

rot rot

rot rot

vib vib

x y z x y z

x y z x y z J odd

x y z x y z J even

x y z x y z all

• Diatomic Hydrogen: Diatomic hydrogen has a symmetric ground electronic level. We construct the following table:

1g

1H2 rot vib elec nuc tot

J even + + + - -J odd - + + + -


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• For H2 molecules with a symmetric nuclear spin wavefunction, only rotational states with odd J are possible. For H2 molecules with an antisymmetric nuclear spin wavefunction, only rotational states with even Jare possible.

• There are a total of (2I + 1)2 = (go)2 nuclear spin states, and of the nuclear spin states are go (go+1)/2 are symmetric, and go (go-1)/2 are antisymmetric.

• The population of the rotational levels is thus given by:

( , )

( , )

1 2 1exp 1 /

2

1 2 1exp 1 /

2

o oJrot

rot nuc

o oJrot

rot nuc

g g JN J J T J oddN Z

g g JN J J T J evenN Z


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• For H2 molecules, I = 1/2 because the proton has a spin of 1/2. The nuclear degeneracy is thus g0 = (2I+1) = 2. For H2:

2( , )

2( , )

2 13 exp 1 / 1,3,5,...

2 11 exp 1 / 0,2,4,...

Jrot

rot nuc

Jrot

rot nuc

JN J J T J ortho HN Z

JN J J T J para HN Z

22( , ) 0

1 1 2 12 2rot nuc

rot rot

T TZ g I


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• CARS spectrum of diatomic hydrogen clearly shows the effect of nuclear spin on the rotational level populations:


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Rotational Distributions for Homonuclear Molecules: Nitrogen

• Diatomic nitrogen also has a symmetric ground electronic level .

• However, the nuclear spin of the 14N nucleus is I = 1, and therefore diatomic nitrogen obeys boson statistics. Nuclei of even mass number are bosons because the nuclear spin will be integral and the total wavefunction tot = rot vib elec nuc must be symmetric with respect to exchange of nuclei :

, , , ,

, , , ,

, , , ,

, , , , v

tot tot

rot rot

rot rot

vib vib

x y z x y z

x y z x y z J odd

x y z x y z J even

x y z x y z all

1g


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• Therefore we develop the following table:

14N2 rot vib elec nuc tot

J even + + + + +J odd - + + - +

• For N2 molecules with a symmetric nuclear spin wavefunction, only rotational states with even J are possible. For N2 molecules with an antisymmetric nuclear spin wavefunction, only rotational states with odd Jare possible.

• For N2 molecules, the nuclear spin I = 1 and the nuclear degeneracy is thus g0 = (2I+1) = 3.


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( , )

( , )

( , )

( , )

1 2 1 1exp

2

2 1 16 exp 0,2, 4,

1 2 1 1exp

2

2 1 13 exp 1,3,5,

o o rotJ

rot nuc

rot

rot nuc

o o rotJ

rot nuc

rot

rot nuc

g g J J JNN Z T

J J Jeven J

Z T

g g J J JNN Z T

J J Jodd J

Z T

• For N2 therefore:


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• The N2 CARS spectrum also shows the effects of nuclear spin coupling with the rotational levels:


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Rotational Distributions for Homonuclear Molecules: Oxygen

• Diatomic oxygen also has an antisymmetric ground electronic level . For diatomic oxygen the nuclei are bosons and the total wavefunction must be symmetric with respect to exchange of nuclei.

• We develop the following table:

3g

16O2 rot vib elec nuc tot

J even + + - - +J odd - + - + +

• For O2 molecules with a symmetric nuclear spin wavefunction, only rotational states with odd J are possible. For O2 molecules with an antisymmetric nuclear spin wavefunction, only rotational states with even Jare possible.


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• For 16O, I = 0, go = 1. The populations of the rotational levels are given by

( , )

( , )

( , )

( , )

1 2 1 1exp

2

2 1 11 exp 1,3,5,

1 2 1 1exp

2

2 1 10 exp 0 0,2, 4,

o o rotJ

rot nuc

rot

rot nuc

o o rotJ

rot nuc

rot

rot nuc

g g J J JNN Z T

J J Jodd J

Z T

g g J J JNN Z T

J J Jeven J

Z T


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• The even rotational levels are missing for the oxygen molecules, at least for the most common isotopic species 16O2. The isotope 16O18O has both even and odd rotational levels, however; this was how the isotope was first discovered.

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