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Statistical Thermodynamics

- Dr. Henry Curran
- NUI Galway

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.

The Boltzmann Factor

Boltzmann law for the population of quantised

energy states

Average basis of the behaviour of matter

- Thermodynamic properties are concerned with

average behaviour.

The instantaneous values of the occupation

numbers are never very different from the

averages.

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

Statistical weights

- At any instant the distribution of particles

among energy states involve n0 with energy e0, n1

with energy e1, n2 with energy e2 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.

Statistical weights

- A given configuration can be reached in a number

of different ways. We call the number of ways W

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 e0, n1

with e1 etc, the statistical weight is

Principle of equal a priori probabilities

- Each configuration will be visited exactly

proportional to its statistical weight

- 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

A massive collapse in probability!

Maximisation subject to constraints

- The predominant configuration among N particles

has energy states that are populated according to

where a and b are constants under the conditions

of constant temperature

- The lowest state has energy e0 0 and

occupation number n0

which identifies the constant a, and enables us

to write

- this is a temperature dependent ratio since the

occupation number of the states vary with

temperature

The constant b can be stated as

where k is the Boltzmann constant

Molecular partition function

Any state population (ni) is known if ei, T,

and n0 are known

Molecular partition function

If the total number of particles is N, then

Molecular partition function, q

Determines how particles distribute (or

partition) themselves over accessible quantum

states.

An infinite series that converges more rapidly

the larger both the energy spacing between

quantum states and the value of b is.

Convergence is enhanced at lower temperatures

since b 1/kT. when be gtgt 0, e-be 0

Molecular partition function, q

- If e1-e0 (De) is large (De gtgt kT)
- q 1 (lowest value of q)
- If e1-e0 (De) 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.

Energy states and energy levels

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

The partition function explored

- The total number of particles in our assembly is

N or, expressed intensively, NA per mole

The partition function is a measure of the

extent to which particles are able to escape from

the ground state

The partition function explored

- 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

The partition function explored

- We can characterise the closeness of spacing in

the energy manifold by referring to the density

of states function, D(e), which represents the

number of energy states in unit energy level. - If D(e) is high (translational motion in gas)
- particles find it easy to leave ground state
- q will rise rapidly as T increases
- If D(e) is low (vibrations of light diatomic

molecules) - small value of q ( 1)

The partition function explored

- 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 gtgt 1 for the valid application of the

Boltzmann law in gaseous systems

Canonical partition function

- Molecular ? molar level
- assume value of an extensive function for N

particles is just N times that for a single

particle - true 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

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, , N
- Molecular states a, b, c, d,

Canonical partition function

- Any given molar state can be described by a

suitable combination of individual molecular

states occupied by individual molecules. If we

call the ith state ?i, we can begin to give a

description of this molar state by writing

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)

Canonical partition function (QN)

- State ?i, 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

with energy

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

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.

The Molar energy

- Note that the expressions

are compatible if we assume that the two

different partition functions are related simply

by

where Q is a function of an N-particle assembly

at constant T and V

Distinguishable and indistinguishable particles

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

Distinguishable and indistinguishable particles

If all molecules are of the same type and

indistinguishable by position they do not need

labelling

Distinguishable and indistinguishable particles

- If particles are indistinguishable the number of
- accessible system states is lower than it is for
- distinguishable ones. A system ?i

differs from a state ?j

if particles are distinguishable, because of the

interchange of particles 2 and 3 between states b

and h. However, ?i is identical to ?j if

particles are indistinguishable

Distinguishable and indistinguishable particles

- 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!

Two-level systems

- The 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 De

At temperatures that are comparable to De/k only

the ground state and first excited state are

appreciably populated

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

where the two-level temperature (q2L) is defined

as

Temperature Dependence of the populations

Comparing energy gap with background thermal

energy

Comparing characteristic T (q2L) with Temperature

(T)

T dependence of populations

but total number of particles is constant

T dependence of populations (Fig. 6.2)

Two-level Molecular partition function

- The effect of increasing T
- only two energy states to consider

High T Low T

T 5q2L T

0.5q2L q2L 1 0.82 q2L

1 0.14 q2L 2

q2L 1 Both states equally

only lower state accessible

accessible

The energy of a two-level system

At high T, half the particles occupy the upper

state and the total energy takes the value ½ N De

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

Variation of CV with T is a measure of how

accessible the upper states becomes as T

increases.

Two-level heat capacity, CV

- Low T
- kT is small
- small DT has little tendency to excite particles
- overall energy remains constant gt CV low
- Intermediate T
- kT is comparable to De
- small DT has larger effect in exciting particles
- CV is somewhat larger
- High T
- Almost half particles in excited state
- small DT causes very few particles excited to

upper - Overall energy remains constant gt CV low

Two-level heat capacity, CV

0.44Nk

0.42

Variation of CV and Energy of a two-level system

as a function of reduced Temperature

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

The effect of degeneracy

Energy of the degenerate two-level system

and the heat capacity

Toolkit equations

Toolkit equations

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.

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.

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.

Particle in a one-dimensional box

The permitted energy levels, ex, 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

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.

One-dimensional partition function

The one-dimensional partition function, qtrs,x,

is obtained by summing over all the accessible

energy states. Thus

an expression that is exact but cannot be

evaluated except by direct and tedious numerical

summation.

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

resulting in the expression

Extension to three dimensions

We can factorise the translational partition

function

product of three dimensions gives the volume, V

Extension to three dimensions

For the canonical partition function for N

indistinguishable particles

Testing the continuum approximation

- At 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.

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

Ideal monatomic gas thermodynamic functions

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

(for one mole N NA, NAk R)

The equation of state

- This can be derived using

(for one mole N NA, NAk R)

The heat capacity, CV

- This can be derived using

(for one mole N NA, NAk R)

Entropy of an ideal monatomic gas

- All of the simplifying factors that result from

taking partial derivatives of ln Qtrs no longer

hold

lnQtrs is a direct term and so all variables

appear

Entropy of an ideal monatomic gas

- Next, using Stirlings approximation (lnN!

NlnN N)

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

Entropy of an ideal monatomic gas

- gathering together all experimental variables

(M. T, p)

The Sackur-Tetrode Equation

The first term contains all of the experimental

variables, the second consists of constants

Using the Sackur-Tetrode equation

- The second term has a value of 172.29 J K-1

mol-1 or 20.723 R. Thus we can write

Some calculated and measured entropies

Argon

Krypton Tb / K 87.4

120.2 Scalc / R 15.542

17.451 Scalor / R 15.60

17.43

Using the Sackur-Tetrode equation

- The Sachur-Tetrode equation can also be written

as using ln p-1 ln V ln R ln T

where the variables are expressed in SI units

(V/m3, T/K, and M/kg mol-1)

Significance of Sackur-Tetrode equation

volume dependence

temperature dependence

the first two terms are known from classical

thermodynamics

the last two terms (3/2 R lnM and 18.605 R)

could not have been foreseen from classical

thermodynamics

Change in conditions

- Effect of increase in T, V, or M on
- D(e) Q U p CV S
- T nil nil
- V nil nil nil
- M nil nil nil

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?

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 modes
- Vibrational modes are independent of
- rotational modes under the rigid rotor assumption
- electronic modes under the Born-Oppenheimer

approximation

Internal modes separability of energies

- 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.

Internal modes separability of energies

- Thus, the total energy of a molecule j

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

Weak coupling factorising the energy modes

- Imagine an assembly of N particles that can store

energy in just two weakly coupled modes a and w - Each mode has its own manifold of energy states

and associated quantum numbers - A given particle can have
- a-mode energy associated with quantum number k
- w-mode energy associated with quantum number r

Weak coupling factorising the energy modes

- The overall partition function, qtot

expanding we would get

Weak coupling factorising the energy modes

- but e(ab) ea.eb, therefore

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

Weak coupling factorising the energy modes

the terms in parentheses in each row are

identical and form the summation

Weak coupling factorising the energy modes

If energy modes are separable then we can

factorise the partition function and write

Factorising translational energy modes

Total translational energy of molecule j

which allows us to write

Factorising internal energy modes

Total translational energy of molecule j

using identical arguments the canonical

partition function can be expressed

but how do we obtain the canonical from the

molecular partition function Qtot from qtot? How

does indistinguishability exert its influence?

Factorising internal energy modes

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

Factorising internal energy modes

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.

Canonical partition function, Q

and thus

- This conclusion assumes weak coupling. If

particles enjoy strong coupling (e.g. in liquids

and solutions) the argument becomes very

complicated!

Ideal diatomic gas Rotational partition function

- Assume rigid rotor for which we can write

successive rotational energy levels, eJ, in terms

of the rotational quantum number, J.

where I is the moment of inertia of the

molecule, m is the reduced mass, and B the

rotational constant.

Ideal diatomic gas Rotational partition function

- Another expression results from using the

characteristic rotational temperature, qr,

- 1st energy increment 2kqr
- 2nd energy increment 4kqr

Ideal diatomic gas Rotational partition function

- Rotational energy levels are degenerate and each

level has a degeneracy gJ (2J1). So

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

Ideal diatomic gas Rotational partition function

- This equation works well for heteronuclear

diatomic molecules. - For homonuclear diatomics this equation

overcounts the rotational states by a factor of

two.

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 s

s 2 for homonuclear diatomics, s 1 for

heteronuclear diatomics s 2 for H2O, s 3 for

NH3, s 12 for CH4 and C6H6

Rotational properties of molecules at 300 K

- qr/K s T/qr

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

Rotational canonical partition function

relates the canonical partition function to the

molecular partition function. Consequently, for

the rotational canonical partition function we

have

Rotational Energy

this can differentiated wrt temperature, since

the second term is a constant with no T dependence

Rotational heat capacity

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

Rotational entropy

Srot is dependent on (reduced) mass (I mr2),

and there is also a constant in the final term,

leading to

Rotational entropy

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.

Extension to polyatomic molecules

- 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 qr, x, qr,

y, qr, z corresponding to the three principal

moments of inertia Ix, Iy, Iz - With resulting partition function

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.

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 2530 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

The diatomic SHO model

- We 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, wosc, is

given by

There is a quantum restriction on the available

energies

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

Vibrational partition function, qvib

- Set e0 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

Vibrational partition function, qvib

The assumption (e0 0) allows us to write

Under this assumption, qvib may be written as

a simple geometric series which yields qvib in

closed form

where qvib hw/k characteristic vibrational

temperature

Vibrational partition function, qvib

- Unlike the situation for rotation, qvib, 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, .

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.

Vibrational partition function, qvib

- Light diatomic molecules have
- high force constants
- low reduced masses
- Thus
- vibrational frequencies (wosc) and characteristic

vibrational temperatures (qvib) are high - just one vibrational state (the ground state)

accessible at room T - the vibrational partition function qvib 1

Vibrational partition function, qvib

- 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 gt 1

Vibrational partition function, qvib

- 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.

Vibrational partition function, qvib

- As with diatomics, only the heavier species show

values of qvib appreciably different from unity. - Typically, qvib 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

High T limiting behaviour of qvib

- At high temperature the equation

gives a linear dependence of qvib with

temperature. - If we expand , we get

High T limit

T dependence of vibrational partition function

As T increases, the linear dependence of qvib

upon T becomes increasingly obvious

The canonical partition function, Qvib

- Using we can find the

first differential of lnQ with respect to

temperature to give

The vibrational energy, Uvib

- This is not nearly as simple as

linear molecules

The vibrational energy, Uvib

- This does reduce to the simple form at

equipartition (at very high temperatures) to

(equipartition)

Normally, at room T

The zero-point energy

- So far we have chosen the zero-point energy

(1/2hw) as the zero reference of our energy scale - Thus we must add 1/2hw 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

1/2Nhw

Vibrational heat capacity, Cvib

- The vibrational heat capacity can be found using

The Einstein Equation

This equation can be written in a more compact

form as

Vibrational heat capacity, Cvib

- FE with the argument qvib/T is the Einstein

function

The Einstein function

The Einstein heat capacity

High T

low T

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

The vibrational entropy, Svib

- We know and N NA for one mole,

thus

Variation of vibrational entropy with reduced

temperature

Electronic partition function

- Characteristic electronic temperatures, qel, 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.

Electronic partition function

- It is tempting to decide that qel will not be a

significant factor. Once we assign e0 0, we

might conclude that

To do so would be unwise! One must consider

degeneracy of the ground electronic state.

Electronic partition function

- The correct expression to use in place of the

previous expression is of course

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.

Electronic partition function

- Atoms 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 G as the general term in the

RussellSaunders spin-orbit coupling

approximation, we denote the spectroscopic state

of the ground state of an atom as - spectroscopic atom ground state (2S1)GJ

Electronic partition function

- spectroscopic atom ground state (2S1)GJ
- where S is the total spin angular momentum

quantum number which gives rise to the term

multiplicity (2S1). The degeneracy, g0, of the

electronic ground states in atoms is related to J

through - g0 2J1 (atoms)

Electronic partition function

- For 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

(S, P, D, etc. corresponding to S, P, D, etc. in

atoms). - As with atoms, the term multiplicity (2S1) is

added as a superscript to denote the multiplicity

of the molecular term.

Electronic partition function

- In 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 (2S1)G
- for which the ground-state degeneracy is
- g0 2S 1 (molecules)

Electronic partition function

Electronic partition function

- Where 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 (qel/T is not

very much less than unity) it is necessary to

consider the first excited state. - The electronic partition function becomes

Electronic partition function

- For F atom at 1000 K we have

For NO molecule at 1000 K we have

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