1 statistical thermodynamics dr. henry curran nui galway

1

Statistical Thermodynamics

Dr. Henry CurranNUI Galway

2

Background

Thermodynamic parameters of stable molecules can be found. However, those for radicals and transition state species cannot be readily found. Need a way to calculate these properties readily and accurately.

3

The Boltzmann Factor

Tfxconst

EfQuantityPhysical

T

Eexp

kT

j

i ji

n

n /exp in

jn

i

jji

Boltzmann law for the population of quantised energy states:

4

Average basis of the behaviour of matter

Thermodynamic properties are concerned with average behaviour.

kT

j

i ji

n

n /exp

The instantaneous values of the occupation numbers are never very different from the averages.

5

Distinct, independent particles

• Consider an assembly of particles at constant temperature. These particles are

• distinct and labelled (a, b, c, … etc)• They are independent

– interact with each other minimally– enough to interchange energy at collision

• Weakly coupled – Sum of individual energies of labelled particles

i

idcbaE ...

6

Statistical weightsAt any instant the distribution of particles among energy states involve n0 with energy 0, n1 with energy 1, n2 with energy 2 and so on. We call the instantaneous distribution the configuration of the system.

• At the next moment the distribution will be different, giving a different configuration with the same total energy.

• These configurations identify the way in which the system can share out its energy among the available energy states.

7

Statistical weightsA given configuration can be reached in a number of different ways. We call the number of ways the statistical weight of that configuration. It represents the probability that this configuration can be reached, from among all other configurations, by totally random means.For N particles arriving at a configuration in which there are n0 particles with energy 0, n1 with etc, the statistical weight is:

...!!!!

!

3210 nnnn

N

8

Principle of equal a priori probabilities

• Each configuration will be visited exactly proportional to its statistical weight

i

inN i

iinE

• We must find the most probable configuration– How likely is this to dominate the assembly?– For an Avogadro number of particles with an average

change of configuration of only 1 part in 1010 reduces the probability by:

434max 10 A massive collapse

in probability!

9

Maximisation subject to constraints

• The predominant configuration among N particles has energy states that are populated according to:

ieN

ni

where and are constants under the conditions of constant temperature

10

The lowest state has energy 0 = 0 and occupation number n0

which identifies the constant , and enables us to write:

i

ii

en

n

eN

nee

N

n

i

i

0

0

eN

n0

ieN

ni

11

this is a temperature dependent ratio since the occupation number of the states vary with temperature

The constant can be stated as:

ien

ni 0

kT

1

where k is the Boltzmann constant

12

Molecular partition function

i

i

ji

enn

enn

en

n

i

kTi

kT

j

i

0

/0

/

0

Any state population (ni) is known if: i, T, and n0 are known

13

Molecular partition function

q

Nen

qewritingore

Nen

e

Nn

enenenennN

nnnnnN

i

i

i

i

i

i

i

statesallstatesall

i

statesall

statesall

statesalli

0

00000

3210

...

...

321

If the total number of particles is N, then:

14

Molecular partition function, q

Determines how particles distribute (or partition) themselves over accessible quantum states.

kTeeeq

1,0...1 0

321

An infinite series that converges more rapidly thelarger both the energy spacing between quantum states and the value of is. Convergence is enhanced at lower temperatures since = 1/kT.

when >> 0, e-

15

Molecular partition function, q

• If 1-0 () is large (kT)

– q 1 (lowest value of q)

• If 1-0 () ≈ kT (thermal energy)

– q large number

magnitude of q shows how easily particles spread over the available quantum states and thus reflects the accessibility of the quantum energy states of the particles involved.

16

Energy states and energy levels

levelsall

jjegq

statesall

ieq

However, quantum states can be degenerate with a number (g) of states all sharing the same energy. States with the same energy comprise an energy level and we use the symbol gj to denote the degeneracy of the jth level

17

The partition function explored

The total number of particles in our assembly is N or, expressed intensively, NA per mole

0

0

n

Nq

qnngnN

A

levelsalljj

statesalliA

The partition function is a measure of the extent to which particles are able to escape from the ground state

18


The partition function q is a pure number which can range from a minimum value of 1 at 0 K (when n0 = NA and only the ground state is accessible) to an indefinitely large number as the temperature increases

Fewer and fewer particles are left in the ground state and an indefinitely large number of states become available to the system

19


We can characterise the closeness of spacing in the energy manifold by referring to the density of states function, D(), which represents the number of energy states in unit energy level.

• If D() is high (translational motion in gas): – particles find it easy to leave ground state – q will rise rapidly as T increases

• If D() is low (vibrations of light diatomic molecules)– small value of q ( 1)

20


• If q/NA (the number of accessible states per particle) is small– few particles venture out of the ground state

• If q/NA is large– there are many accessible states and molecules are

well spread over the energy states of the system

• q/NA >> 1 for the valid application of the

Boltzmann law in gaseous systems

21

Canonical partition function

• Molecular molar level– assume value of an extensive function for N

particles is just N times that for a single particletrue for energy of non-interacting particles but not so for other properties (e.g. entropy)

• Particles do interact! Thus we consider:– every system has a set of system energy

states which molecules can populate– these states are not restricted by the need

for additivity but can adjust to any inter-particle interactions that may exist

V

m

qLE

ln

22

Canonical partition function• Molar sum over states

– each possible state of the whole system involves a description of the conditions experienced by all the particles that make up a mole

Suppose N identical particles each with the set of individual molecular states available to them.

Particle labels 1, 2, 3, 4, …, NMolecular states a, b, c, d, …

23

Canonical partition functionAny given molar state can be described by a suitable combination of individual molecular states occupied by individual molecules. If we call the ith state Ψi, we can begin to give a description of this molar state by writing:

...

...8765432187654321tckcfhbai

tckcfhbai

E

with energy:

there is no restriction on the number of particles that can be in the same molecular state (e.g. particles 5 & 7 are both in molecular state c)

24

Canonical partition function (QN)

State Ψi, with energy Ei is just one of many states of the whole system. The predominant molecular configuration is called the canonical distribution – applies to states of an N-particle system– at constant amount, volume, and Temperature

V

Nim

statessystem

N

QEE

eQ i

lnwith energy:

25

The Molar energy

The canonical partition function (QN) is much more general than “the product of N molecular partition functions q” since there is no need to consider only independent molecules

VV

qN

q

q

NE

ln

V

Nim

QEE

ln

26

The Molar energy

If we are able to calculate thermodynamic properties for assemblies of N independent particles using q and for N non-independent particles using QN, then, in the limit of the particles of QN, becoming less and less strongly interdependent, the two methods should eventually converge.

27

The Molar energy

Note that the expressions:

V

N

V

QqN

lnln

and

are compatible if we assume that the two different partition functions are related simply by: NqQwhere Q is a function of an N-particle assembly at constant T and V

28

Distinguishable and indistinguishable particles

itckcfhba

tckcfhbai

Q

E

...exp

...

87654321

87654321

for the canonical partition function we can write:

In every one of the i system states, each particle (1, 2, 3, …) will be in one of its possible j molecular states (a,b, c, d …) just once in each system state. If we factorise out each particle in turn from the summation over the system states and then gather together all the terms that refer to a given particle, we get:

29


...

321

statesmolecular

statesmolecular

statesmolecular

jjj eeeQ

If all molecules are of the same type and indistinguishable by position they do not need labelling

N

N

j

qeQ j

30


If particles are indistinguishable the number ofaccessible system states is lower than it is fordistinguishable ones. A system Ψi

...321

...321

bhaj

hbai

differs from a state Ψj

if particles are distinguishable, because of the interchange of particles 2 and 3 between states b and h. However, Ψi is identical to Ψj if particles are indistinguishable

31


In systems which are not at too high a density and are also well above 0 K, the correction factor for this over-counting of configurations is 1/N!

particles) ishableindistingufor

particles)habledistinguisfor

(

(

!N

qQ

qQN

N

32

Two-level systemsThe simplest type of system is one which comprises particles with only two accessible states in the form of two non-degenerate levels separated by a narrow energy gap :

un

ln

0

0

At temperatures that are comparable to /k only the ground state and first excited state are appreciably populated

33

Effect of increasing Temperature

The average population ratio of the two levels is an assembly of such two-level (or two-state) particles is given by:

k

een

n

L

T

l

u L

2

/2

where the two-level temperature (2L) is defined as:

34

Temperature Dependence of the populations

)0(14.0

)(5.0

)1(82.0

)(5

5.0/1

2

5/1/

2

2

2

en

nthen

TlowTif

eeen

nthen

ThighTif

TkT

l

u

L

T

l

u

L

L

L

Comparing energy gap with background thermal energy

Comparing characteristic T (2L) with Temperature (T)

35

T dependence of populations

Ne

Ne

en

Ne

eN

en

Nne

Nnn

nen

l

u

u

ul

ul

1

1

1

11

1

1

but total number of particles is constant

36

T dependence of populations (Fig. 6.2)

T00.1 1 10

0.0

0.5

1.0

nu

nl

nu

mb

er o

f p

arti

cles

/ N

Reduced temperature T/2L

T

37

Two-level Molecular partition function

• The effect of increasing T– only two energy

states to consider)1(0

)1(

20

)(2

0

00

eqif

ee

eeeq

L

statesboth

Li

High T Low T T = 52L T = 0.52L

q2L = 1 + 0.82 q2L = 1 + 0.14

q2L 2 q2L 1

Both states equally only lower state accessible accessible

38

The energy of a two-level system

1

2

11

1

1

0

eNe

N

xNe

nxnxnenE uuli

iiL

At high T, half the particles occupy the upper state and the total energy takes the value ½ N

39

Two-level heat capacity, CV

The spacing of energy levels in discussing two-level systems is not affected by changes in volume, so the relevant heat capacity is CV not CP

T

U

T

UC

VV

Variation of CV with T is a measure of how accessible the upper states becomes as T increases.

40


• Low T– kT is small – small T has little tendency to excite particles– overall energy remains constant => CV low

• Intermediate T– kT is comparable to – small T has larger effect in exciting particles– CV is somewhat larger

• High T– Almost half particles in excited state– small T causes very few particles excited to

upper– Overall energy remains constant => CV low

41


22

2

2

1

2

1)(

1

1

e

eNkC

kTeeN

dT

dE

eNE

V

L

42

Variation of CV and Energy of a two-level system as a function of reduced Temperature

0.1 1 100.0

0.1

0.2

0.3

0.4

0.5

E/E

max

CV/N

k

Reduced temperature (T/)

0.0

0.2

0.4

0.6

0.8

1.0

0.44Nk

0.42

43

The effect of degeneracy

Consider a two-level system with degenerate energy levels. If the degeneracy of the lower level is g0 and that of the upper level is g1 then the two-level molecular partition function is:

)( 102 eggq L

44

The effect of degeneracy

egg

Ngg

E

0

1

0

1

Energy of the degenerate two-level system:

and the heat capacity:

2

0

1

0

1

2

egg

egg

Nk

CV

45

Toolkit equations

V

VVV

T

V

T

QkTQkSEntropy

T

QkT

T

QkTCcapacityHeat

V

QkTpstateofEquation

T

QkTUUenergyInternal

QkT

AAbridgeMassieu

lnln:.5

lnln2:.4

ln:.3

ln)0(:.2

ln)0(

:.1

2

22

2

46

Toolkit equations

T

TV

V

QkTVQkTGGenergyfreeGibbs

V

QkTV

T

QkTHHEnthalpy

lnln)0(:.7

lnln)0(:.6 2

In order to relate the partition function to classical thermodynamic quantities, the equations for internal energy and entropy are needed. Once these have been expressed in terms of the canonical partition function, Q, the Massieu bridge can be derived. This in turn provides the most compact link to classical thermodynamics.

47

Ideal monatomic gas

Consider an assembly of particles constrained to move in a fixed volume. This system consists of many, non-interacting, monatomic gas particles in ceaseless translational motion. The only energy that these particles can possess is translational kinetic energy.

48

Translational partition function, qtrs

In classical mechanics, all kinetic energies are allowed in a system of monatomic gas particles at a fixed volume V and temperature T. Quantum restrictions place limits on the actual kinetic energies that are found. To determine the partition function for such a system to need to establish values for the allowed kinetic energies. We consider a particle constrained to move in a cubic box with dimensional box with dimensions lx, ly, and lz.

49

Particle in a one-dimensional boxThe permitted energy levels, x, for a particle of mass m that is constrained by infinite boundary potentials at x = 0, and x = lx to exist in a one-dimensional box of length lx are given by:

2

22

8 x

xx m

hn

l

and similarly for the y- and z-directions. The translational quantum number, nx, is a positive integer and the quantum numbers in the y- and z-directions are ny and nz, respectively.

50

One-dimensional partition function

The one-dimensional partition function, qtrs,x, is obtained by summing over all the accessible energy states. Thus:

222 8/,

xx mhn

nallxtrs eq l

an expression that is exact but cannot be evaluated except by direct and tedious numerical summation.

51

One-dimensional partition function

For all values of lx in any normal vessel, these energy levels are very densely packed and lie extremely close to each other. They form a virtual continuum, so the summation can be replaced by an integration with the running variable nx

h

mq

dneq

xtrs

xmhn

xtrsxx

x

l

l2

1

,

0

8/,

2

222

resulting in the expression:

52

Extension to three dimensions

3

2

3

,,,

2

h

mqxqxqq ztrsytrsxtrstrs

zyx lll

We can factorise the translational partition function:

product of three dimensions gives the volume, V:

Vh

mkTV

h

mqtrs

2

3

2

2

3

2

22

53

Extension to three dimensions

For the canonical partition function for N indistinguishable particles

NN

Ntrs

trs Vh

mkT

NV

h

m

NN

qQ

2

3

2

2

3

2

2

!

12

!

1

!

54

Testing the continuum approximationAt its boiling temperature of 4.22 K, one mole of He occupies 3.46 x 10-4 m3. How many translational energy states are accessible at this very low temperature, and determine whether the virtual continuum approximation is valid.

)(105.4

)1046.3()22.410003.4)(10942.5(

)()/10942.5(

)(2

24

42

3330

2

3

2

5

2

332

530

2

32

3

2

unitlessx

xxxx

VMTKkgmmolx

VMTNh

kq

Atrs

55

Ideal monatomic gas: thermodynamic functions

Since all classical thermodynamic functions are related to the logarithm of the canonical partition function, we start by taking the logarithm of Qtrs:

!lnln3lnln2

3)2ln(

2

3ln NhNVNTNmNQtrs

56

Ideal monatomic gas: thermodynamic functions

V

N

V

Q

T

N

T

Q

T

N

T

Q

T

trs

V

trs

V

trs

ln

2

3ln

2

3ln

22

2

57

The translational energy, Etrs

For a monatomic gas, the translational kinetic energy, Etrs, is the only form of energy that the monatomic particles possess, so we can equate it directly to the internal energy, U, and substitute the value of the derivative:

RTU

NkTT

QkTUE

m

V

trstrs

2

3

2

3ln2

(for one mole N = NA, NAk = R)

58

The equation of state

This can be derived using:

RTpVorV

RTp

V

NkT

V

QkTp

mm

T

trs

ln


59

The heat capacity, CV

This can be derived using:

RC

NkNkNk

T

QkT

T

QkTC

mV

V

trs

V

trsV

2

3

2

3

2

33

lnln2

,

2

22


60

Entropy of an ideal monatomic gasAll of the simplifying factors that result from taking partial derivatives of ln Qtrs no longer hold

!lnln3lnln2

3)2ln(

2

3ln

lnln

NhNVNTNmNQ

T

QkTQkS

trs

V

trstrs

lnQtrs is a direct term and so all variables appear

)!lnln(ln!

1ln

!

1lnln

NqNkqN

k

qN

kQk

N

Ntrs

61

Entropy of an ideal monatomic gas

Next, using Stirling’s approximation (lnN! ≈ NlnN – N)

pN

RT

Nh

MkTRS

N

V

h

mkTNkS

N

qNkNk

N

qNkS

N

qNkNNNqNkQk

AAm

trs

2

3

2

2

3

2

2ln

2

5

2ln

2

5

ln2

5

2

3ln1

ln1)lnln(ln

One mole: (N = NA, NAk = R, NAm = M and V = RT/p

62

Entropy of an ideal monatomic gas

gathering together all experimental variables (M. T, p)

2

52

3

2

2

5

2

3

)(2

lnln ekhN

Rp

TMRS

Am

The Sackur-Tetrode Equation

The first term contains all of the experimental variables, the second consists of constants

63

Using the Sackur-Tetrode equationThe second term has a value of 172.29 J K-1 mol-1 or 20.723 R. Thus we can write:

72320)/()/(/ln/ 12

3

2

31 .PapKTmolkgMRSm

Argon Krypton

Tb / K 87.4 120.2

Scalc / R 15.542 17.451

Scalor / R 15.60 17.43

Some calculated and measured entropies

64

Using the Sackur-Tetrode equationThe Sachur-Tetrode equation can also be written as using ln p-1 = ln V – ln R – ln T

RMRTRVRSm 605.18ln2

3ln

2

3ln

where the variables are expressed in SI units (V/m3, T/K, and M/kg mol-1)

65

Significance of Sackur-Tetrode equation

1

2

1

2

1

2

ln

ln2

3ln

T

TC

T

TRS

V

VRS

V

VT

the last two terms (3/2 R lnM and 18.605 R) could not have been foreseen from classical thermodynamics

volume

dependencetemperaturedependence

RMRTRVRSm 605.18ln2

3ln

2

3ln

the first two terms are known from classical thermodynamics

66

Change in conditions

Effect of increase in T, V, or M on:

D() Q U p CV S

T nil nil

V nil nil nil

M nil nil nil

67

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?

68

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

69


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.

70


Thus, the total energy of a molecule j:

jel

jvib

jrot

jtrs

jtot

71

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

72


• 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

73


The overall partition function, qtot:

statesall

totrieq

)(

expanding we would get:

...

)()()()(

)()()()(

)()()()(

)()()()(

33231303

32221202

31211101

30201000

eeee

eeee

eeee

eeeeqtot

74


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

75


...

...)(

...)(

...)(

...)(

32103

32102

32101

32100

eeeee

eeeee

eeeee

eeeeeqtot

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

statesall

je

76


statesallstatesall

statesalltot

jj

j

exe

eeeeeq

...3210

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

qxqqtot

77

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:

78

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?

79


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

80


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.

81

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:

82

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.

83


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

84


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:

85


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.

86

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

87

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

88

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

89

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

90

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

91

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

92

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.

93

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:

94

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.

95

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

96

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

97

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

98

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

99


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

100


• 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

101

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

102


• 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

k

osc 2

1

103


• 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

104


• 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

105


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

e

qvib

106

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

107

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

108

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

1ln

/2

T

vib

Vvib vibe

Nk

T

QkTU

109

The vibrational energy, Uvib

This is not nearly as simple as:

)1( /, T

vibmvib vibe

RU

RTU

kTU

mrot

mtrs

,

, 2

3

linear molecules

110

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)

111

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(

/

,/,

112

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,

113

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

114

The Einstein heat capacity

0.1 1 100.0

0.5

1.0

FE

Reduced temperature T/

High Tlow T

115

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

116

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

117

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

118

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.

119

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.

120


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.

121

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

122

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

123

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.

124

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)

125


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

126

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

127


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 statistical thermodynamics dr. henry curran nui galway

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