1 ideal diatomic gas: internal degrees of freedom polyatomic species can store energy in a variety...

1

Ideal diatomic gas: internal degrees of freedom

• Polyatomic species can store energy in a variety of ways:

– translational motion– rotational motion– vibrational motion– electronic excitation

Each of these modes has its own manifold

of energy states, how do we cope?

2

Internal modes: separability of energies

• Assume molecular modes are separable – treat each mode independent of all others– i.e. translational independent of vibrational,

rotational, electronic, etc, etc

Entirely true for translational modesVibrational modes are independent of:– rotational modes under the rigid rotor

assumption– electronic modes under the Born-

Oppenheimer approximation

3


Thus, a molecule that is moving at high speed is not forced to vibrate rapidly or rotate very fast.

An isolated molecule which has an excess of any one energy mode cannot divest itself of this surplus except at collision with another molecule.

The number of collisions needed to equilibrate modes varies from a few (ten or so) for rotation, to many (hundreds) for vibration.

4


Thus, the total energy of a molecule j:

jel

jvib

jrot

jtrs

jtot

5

Weak coupling: factorising the energy modes

• Admits there is some energy interchange – in order to establish and maintain thermal

equilibrium

• But allows us to assess each energy mode as if it were the only form of energy present in the molecule

• Molecular partition function can be formulated separately for each energy mode (degree of freedom)

• Decide later how individual partition functions should be combined together to form the overall molecular partition function

6


• Imagine an assembly of N particles that can store energy in just two weakly coupled modes and

• Each mode has its own manifold of energy states

and associated quantum numbers

• A given particle can have:- -mode energy associated with quantum number k

- -mode energy associated with quantum number r

rktot

7


The overall partition function, qtot:

statesall

totrieq

)(

expanding we would get:

...

)()()()(

)()()()(

)()()()(

)()()()(

33231303

32221202

31211101

30201000

eeee

eeee

eeee

eeeeqtot

8


but e(a+b) = ea.eb, therefore:

...

......

......

......

221202

211101

201000

eeeeee

eeeeee

eeeeeeqtot

each term in every row has a common factor of

in the first row, in the second, and so on. Extracting these factors row by row:

0e 1e

9


...

...)(

...)(

...)(

...)(

32103

32102

32101

32100

eeeee

eeeee

eeeee

eeeeeqtot

the terms in parentheses in each row are identical and form the summation:

statesall

je

10


statesallstatesall

statesalltot

jj

j

exe

eeeeeq

...3210

If energy modes are separable then we can factorise the partition function and write:

qxqqtot

11

Factorising translational energy modes

jztrs

jytrs

jxtrs

jtottrs ,,,,

which allows us to write:

ztrsytrsxtrstrs

stateszallstatesyallstatesxalltrs

statesallstatesalltrs

qxqxqq

exexeq

eeq

ztrsytrsxtrs

ztrsytrsxtrstottrs

,,,

)(

,,,

,,,,

Total translational energy of molecule j:

12

Factorising internal energy modes

elvibrottrstot qqqqq ...using identical arguments the canonical partition function can be expressed:

elvibrottrstot QQQQQ ...

Total translational energy of molecule j:

but how do we obtain the canonical from the molecular partition function Qtot from qtot? How

does indistinguishability exert its influence?

13


When are particles distinguishable (having distinct configurations, and when are they indistinguishable?

• Localised particles (unique addresses) are always distinguishable

• Particles that are not localised are indistinguishable– Swapping translational energy states between such

particles does not create distinct new configurations

• However, localisation within a molecule can also confer distinguishability

14


When molecules i and j, each in distinct rotational and vibrational states, swap these internal states with each other a new configuration is created and both configurations have to be counted into the final sum of states for the whole system. By being identified specifically with individual molecules, the internal states are recognised as being intrinsically distinguishable.

Translational states are intrinsically indistinguishable.

15

Canonical partition function, Q

This conclusion assumes weak coupling. If particles enjoy strong coupling (e.g. in liquids and solutions) the argument becomes very complicated!

Nelvibrottrstot

Nel

Nvib

Nrot

Ntrs

tot

qqqqN

Q

qqqN

qQ

...!

1!

and thus:

16

Ideal diatomic gas: Rotational partition function

Assume rigid rotor for which we can write successive rotational energy levels, J, in terms of the rotational quantum number, J.

1

12

2

2

)1(

)1(8

)1(8

cmJBJ

cmJJIc

h

hc

E

JJI

hE

JJ

J

joules

where I is the moment of inertia of the molecule, is the reduced mass, and B the rotational constant.

17


Another expression results from using the characteristic rotational temperature, r,

)()1(8 2

2

jouleskJJE

hcBkk

hcB

Ik

h

rJ

rr

• 1st energy increment = 2kr

• 2nd energy increment = 4kr

18


Rotational energy levels are degenerate and each level has a degeneracy gJ = (2J+1). So:

TJJkTJrot

rJ eJegq /)1(/ )12(

If no atoms in the atom are too light (i.e. if the moment of inertia is not too small) and if the temperature is not too low (close to 0 K), allowing appreciable numbers of rotational states to be occupied, the rotational energy levels lie sufficiently close to one another to write:

19


2

2

0

/)1(

8

)12(

h

IkTTq

eJq

rrot

dJTJJrot

r

• This equation works well for heteronuclear diatomic molecules.

• For homonuclear diatomics this equation overcounts the rotational states by a factor of two.

20

Ideal diatomic gas: Rotational partition function• When a symmetrical linear molecule rotates

through 180o it produces a configuration which is indistinguishable from the one from which it started. – all homonuclear diatomics

– symmetrical linear molecules (e.g. CO2, C2H2)

• Include all molecules using a symmetry factor

rrot

Tq

= 2 for homonuclear diatomics, = 1 for heteronuclear diatomics = 2 for H2O, = 3 for NH3, = 12 for CH4 and C6H6

21

Rotational properties of molecules at 300 K

r/K T/r qrot

H2 88 2 3.4 1.7

CH4 15 12 20 1.7

HCl 9.4 1 32 32

HI 7.5 1 40 40

N2 2.9 2 100 50

CO 2.8 1 110 110

CO2 0.56 2 540 270

I2 0.054 2 5600 2800

22

Rotational canonical partition function

Nrotrot qQ

relates the canonical partition function to the molecular partition function. Consequently, for the rotational canonical partition function we have:

NNN

rrot h

IkT

hcB

TTQ

2

28

23

Rotational Energy

2

28lnlnln

h

IkNTNQrot

this can differentiated wrt temperature, since the second term is a constant with no T dependence

molecules)diatomic(forNkTU

TT

NkTT

QkTU

rot

V

rotrot

ln

ln 22

24

Rotational heat capacity

molecules)(linearRC

RTU

mrot

mrot

,

,

this equation applies equally to all linear molecules which have only two degrees of freedom in rotation. Recast for one mole of substance and taking the T derivative yields the molar rotational heat capacity, Crot, m. Thus, when N = NA, the molar rotational energy is Urot,m

molecules)diatomic(forNkTU rot

25

Rotational entropy

53.106//ln/ 12 KTmkgIRSrot

Srot is dependent on (reduced) mass (I = r2), and there is also a constant in the final term, leading to:

2

2

2

2

8lnln1

8ln

lnlnln

h

kITNkS

h

IkTk

T

NkT

QkT

UQk

T

QkTS

rot

N

rot

Vrot

26

Rotational entropy

282 1010 trsrot qbutq

Typically, qrot at room T is of the order of hundreds for diatomics such as CO and Cl2. Compare this with the almost immeasurably larger value that the translational partition function reaches.

27

Extension to polyatomic molecules

2

1

,,,

zryrxrrot

TTTq

• In the most general case, that of a non-linear polyatomic molecule, there are three independent moments of inertia.

• Qrot must take account of these three moments – Achieved by recognising three independent characteristic

rotational temperatures r, x, r, y, r, z corresponding to the three principal moments of inertia Ix, Iy, Iz

• With resulting partition function:

28

Conclusions

• Rotational energy levels, although more widely spaced than translational energy levels, are still close enough at most temperatures to allow us to use the continuum approximation and to replace the summation of qrot with an integration.

• Providing proper regard is then paid to rotational indistinguishability, by considering symmetry, rotational thermodynamic functions can be calculated.

29

Ideal diatomic gas: Vibrational partition function

Vibrational modes have energy level spacings that are larger by at least an order of magnitude than those in rotational modes, which in turn, are 25—30 orders of magnitude larger than translational modes.– cannot be simplified using the continuum

approximation– do not undergo appreciable excitation at room

Temp.

– at 300 K Qvib ≈ 1 for light molecules

30

The diatomic SHO modelWe start by modelling a diatomic molecule on a simple ball and spring basis with two atoms, mass m1 and m2, joined by a spring which has a force constant k.The classical vibrational frequency, oscis given by:

Hzk

osc

2

1

There is a quantum restriction on the available energies:

...),2,1,0(2

1

voscvib hv

31

The diatomic SHO model

The value is know as the zero point energy

• Vibrational energy levels in diatomic molecules are always non-degenerate.

• Degeneracy has to be considered for polyatomic species

– Linear: 3N-5 normal modes of vibration– Non-linear: 3N-6 normal modes of vibration

osch2

1

32

Vibrational partition function, qvib

• Set 0 = 0, the ground vibrational state as the reference zero for vibrational energy.

• Measure all other energies relative to reference ignoring the zero-point energy.

– in calculating values of some vibrational thermodynamic functions (e.g. the vibrational contribution to the internal energy, U) the sum of the individual zero-point energies of all normal modes present must be added

33


The assumption (0 = 0) allows us to write:

Thvib

hhhvib

vib

vib

eeq

eeeeq

hhhh

/

32

4321

1

1

1

1

...1

...,4,3,2,

Under this assumption, qvib may be written as:

a simple geometric series which yields qvib in closed form:

where vib = h/k = characteristic vibrational temperature

34


• Unlike the situation for rotation, vib, can be identified with an actual separation between quantised energy levels.

• To a very good approximation, since the anharmonicity correction can be neglected for low quantum numbers, the characteristic temperature is characteristic of the gap between the lowest and first excited vibrational states, and with exactly twice the zero-point energy, .

osch2

1

35

Ideal diatomic gas: Vibrational partition function

Vibrational energy level spacings are much larger than those for rotation, so typical vibrational temperatures in diatomic molecules are of the order of hundreds to thousands of kelvins rather than the tens of hundreds characteristic of rotation.

Species vib/Kqvib

(@ 300 K)

H2 5987 1.000

HD 5226 1.000

D2 4307 1.000

N2 3352 1.000

CO 3084 1.000

Cl2 798 1.075

I2 307 1.556

36


• Light diatomic molecules have:

– high force constants – low reduced masses

• Thus:– vibrational frequencies (osc) and characteristic vibrational

temperatures (vib) are high

– just one vibrational state (the ground state) accessible at room T

• the vibrational partition function qvib ≈ 1

kosc 2

1

37


• Heavy diatomic molecules have:– rather loose vibrations – Lower characteristic temperature

• Thus:– appreciable vibrational excitation resulting in:

• population of the first (and to a slight extent higher) excited vibrational energy state

• qvib > 1

38


• Situation in polyatomic species is similar complicated only by the existence of 3N-5 or 3N-6 normal modes of vibration.

• Some of these normal modes are degenerate

(1), (2), (3), … denoting individual normal modes 1, 2, 3, …etc.

Species vib/K∏(qvib)

(@ 300 K)

CO2 3360 1.091

1890

954(2)

NH3 4880(2) 1.001

4780

2330(2)

1360

CHCl3 4330 2.650

1745(2)

1090(2)

938

523

374(2)

...)3()2()1()( xqxqxqqq vibvibvibnvib

totvib

39


As with diatomics, only the heavier species show values of qvib appreciably different from unity.

Typically, vib is of the order of ~3000 K in many molecules. Consequently, at 300 K we have:

in contrast with qrot (≈ 10) and qtrs (≈ 1030)

For most molecules only the ground state is accessible for vibration

11

110

eqvib

40

High T limiting behaviour of qvib

At high temperature the equation gives a linear dependence of qvib with temperature.

If we expand , we get:

Tvib vibeq /1

1

Tvibe /1

vibvibvib

T

Tq

...)/(11

1High T limit

41

T dependence of vibrational partition function

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.41.0

1.2

1.4

1.6

1.8

2.0

qvi

b

Reduced Temperature T/

As T increases, the linear dependence of qvib upon T becomes increasingly obvious

42

The canonical partition function, Qvib

Using we can find the first

differential of lnQ with respect to

temperature to give:

VT

QkTU

ln2

N

TNvibvib vibeqQ

/1

1

1

ln/

2

T

vib

Vvib vibe

Nk

T

QkTU

43

The vibrational energy, Uvib

This is not nearly as simple as:

)1( /, T

vibmvib vibe

RU

RTU

kTU

mrot

mtrs

,

, 2

3

linear molecules

44

The vibrational energy, Uvib

This does reduce to the simple form at equipartition (at very high temperatures) to:

)1( /, T

vibmvib vibe

RU

Re

RU

RTU

mvib

mvib

7

1

)1(

300010,

,

Normally, at room T:

(equipartition)

45

The zero-point energy• So far we have chosen the zero-point energy

(1/2h) as the zero reference of our energy scale

• Thus we must add 1/2h to each term in the energy ladder

• For each particle we must add this same amount– Thus, for N particles we must add U(0)vib, m = 1/2Nh

hNe

R

Ue

RU

ATvib

mvibTvib

mvib

vib

vib

2

1

)1(

)0()1(

/

,/,

46

Vibrational heat capacity, Cvib

The vibrational heat capacity can be found using:

2/

/2,

, )1(

T

Tvib

V

mvibmvib vib

vib

e

e

TR

T

UC

The Einstein Equation

This equation can be written in a more compact form as:

T

RC vibEmvib

F,

47

Vibrational heat capacity, Cvib

FE with the argument vib/T is the Einstein function

The Einstein function

Tu

e

eu vibu

u

E

2

2

)1(F

48

The Einstein heat capacity

0.1 1 100.0

0.5

1.0

FE

Reduced temperature T/

High Tlow T

49

The Einstein function

• The Einstein function has applications beyond normal modes of vibration in gas molecules.

• It has an important place in the understanding of lattice vibrations on the thermal behaviour of solids

• It is central to one of the earliest models for the heat capacity of solids

50

The vibrational entropy, Svib

vibvibvib

vibvibvibvibvib

QkT

UUT

AA

T

UUS

ln)0(

)0()0(

Nvibvib qQ • We know and N = NA for one mole, thus:

TT

vibmvib

vibvibAvib

vib

vibe

e

T

R

S

qRqkNQ

//

, 1ln)1(

/

lnlnln

51

Variation of vibrational entropy with reduced temperature

0.1 1 100.0

0.5

1.0

1.5

2.0

2.5

3.0

Svi

b/R

Reduced temperature T/

TT0

52

Electronic partition function• Characteristic electronic temperatures, el,

are of the order of several tens of thousands of kelvins.

• Excited electronic states remain unpopulated unless the temperature reaches several thousands of kelvins.

• Only the first (ground state) term of the electronic partition function need ever be considered at temperatures in the range from ambient to moderately high.

53

Electronic partition function

It is tempting to decide that qel will not be a significant factor. Once we assign 0 = 0, we might conclude that:

1)(00/,

i

kTel termshighereeq iel

To do so would be unwise!One must consider degeneracy of the

ground electronic state.

54


The correct expression to use in place of the previous expression is of course:

00

0/

)(0, gtermshigheregegqi

kTiel

iel

Most molecules and stable ions have non-degenerate ground states. A notable exception is molecular oxygen, O2, which has a ground state degeneracy of 3.

55

Electronic partition functionAtoms frequently have ground states that are degenerate. Degeneracy of electronic states determined by the value of the total angular momentum quantum number, J.Taking the symbol as the general term in the Russell—Saunders spin-orbit coupling approximation, we denote the spectroscopic state of the ground state of an atom as:

spectroscopic atom ground state = (2S+1)J

56

Electronic partition functionspectroscopic atom ground state = (2S+1)J

where S is the total spin angular momentum quantum number which gives rise to the term multiplicity (2S+1). The degeneracy, g0, of the electronic ground states in atoms is related to J through:

g0 = 2J+1 (atoms)

57

Electronic partition functionFor diatomic molecules the term symbols are made up in much the same way as for atoms.

• Total orbital angular momentum about the inter-nuclear axis. Determines the term symbol used for the molecule ( etc. corresponding to S, P, D, etc. in atoms).As with atoms, the term multiplicity (2S+1) is added as a superscript to denote the multiplicity of the molecular term.

58

Electronic partition functionIn the case of molecules it is this term multiplicity that represents the degeneracy of the electronic state.For diatomic molecules we have:

spectroscopic molecular ground state = (2S+1)

for which the ground-state degeneracy is:g0 = 2S + 1 (molecules)

59


SpeciesTerm

Symbol gn el/K

Li 2S1/2 g0 = 2

C 3P0 g0 = 1

N 4S3/2 g0 = 4

O 3P2 g0 = 5

F 2P3/2 g0 = 42P1/2 g1 = 2 590

NO 21/2 g0 = 223/2 g1 = 2 178

O23-

g g0 = 31g g1 = 1 11650

60

Electronic partition functionWhere the energy gap between the ground and the first excited electronic state is large the electronic partition function simply takes the value g0.

When the ground-state to first excited state gap is not negligible compared with kT (el/T is not very much less than unity) it is necessary to consider the first excited state.The electronic partition function becomes:T

eleleggq /

10

61


For F atom at 1000 K we have:

109.524 1000/590/10 eeggq T

elel

674.322 1000/178/10 eeggq T

elel

For NO molecule at 1000 K we have:

1 ideal diatomic gas: internal degrees of freedom polyatomic species can store energy in a variety...

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