molecular partition functions

8/12/2019 Molecular Partition Functions

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'igure (.) % schematic diagram showing electronic "bold lines#, vibrational androtational energy levels. The electronic uantum umbers are shown to the extreme right.Vibrational uantum numbers are to shown in the extreme left. The rotational uantumnumbers are shown between the vibrational levels.

3.1 Electronic, Vibrtionl, Rottionl nd Trn!ltionl Prtition

Function!The electronic energy levels are generally very widely separated in energy compared tothe thermal energy -T at room temperature. In each electronic level, there are severalvibrational levels and for each vibrational level, there are several rotational states. This isa simplified and useful model to start with. The total energy is a sum of all these energiesand is given by

Etotal Eelectronic5 Evibrational 5 Erotational5 Etranslational5 Eothers "(.(#

The term Eothersincludes nuclear spin energy levels and may also be used later to includethe interactions between the first four. %ssuming the first three to be independent andneglecting the last term, the molecular partition function "ie, a sum over the molecularenergy states# is given by

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5 E8 "E 5 E 5 E E # - T E E Erot transel ib el rot trans

rot transel ib

ibe e e e e

=

"(.7#2ere, the summation is over the electronic, vibrational and rotational states can be doneseparately since they are assumed to be independent. Therefore,

el vib rot trans "(.9#

The molecular partition function is written as the product of electronic, vibrational,rotational and partition functions.

The partition function is a sum over states "of course with the *olt!mann factor multiplying the energy in the exponent# and is a number. :arger the value of , larger thenumber of states which are available for the molecular system to occupy. $inceEel; Evib; Erot ; Etrans, there are far too many translational states available compared tothe rotational, vibrational and electronic states. elis very nearly unity, viband rotare inthe range of ) to )


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This is usually a very large number ")< 1alculate the translational partition function of an I1molecule at (


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Solution'The value of B for 21is 6


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%t (


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The N in the denominator is due to the indistinguishability of the tiny molecules "orother uantum particles in a collection#.

E$%&le 3.-Rerive the Caxwell *olt!mann distribution of molecular speeds.Solution'If we represent a molecular velocity by v , it has three independent components

vx, vyand v!in the three directions x, y and !. :et us consider monatomic gas of mass m.The probability ' " x , y , !# that given molecule will have velocity components lyingbetween x and x 5 dx, y and y 5 d y and ! and ! 5 d! can be written as' "x, y, !# d! dy d! f "x# f "y# f"!# dx dy d!.' is written as a product of three functions f because x, y and ! are independent and sincenature does not distinguish between x, y and ! "unless directional fields li-e gravitationalor electromagnetic are present#, the form of f is the same in the three directions. %gain,since there is no distinction between positive and negative x, f depends on W x W or x1. &ecan rephrase the above euation as

'" # f " # f " # f" # "(.1?#The only function that satisfies the above euation is an exponential function since

1 1 1 1 1 1x y z x y z

e e e e+ +

= and so we conclude that f " # may be written asf" # > e

1b v

x > e

1b v

x "(.1B#

&e ta-e only the negative exponent "> and b are positive# because a positive exponentimplies that very large velocities have very high probabilities which is highly unli-ely.To evaluate >, &e invo-e the physical argument that the velocity has to lie somewherebetween 8 to 5 and that the total probability is ) i.e.,

11 8 b vxx x xf "v # d v > e d v )

The above integral is a standard integral18 a x

e

d x > " b#

)1 "(.( " b# )1 ) or > " b # )1andf " x # "b #)1 18 b vxe

'rom a probability distribution such as f " x#, average uantities can be determined. Theaverages of x and x1are given by

P x ;

"b #)1x 18 b vxe d x < "(.()#

P x1;

x1"b #)1 18 b vxe d x )1b "(.(1#

The averages have been denoted by P ; . &e have also used another standard integral,11 a x

onsider the molecular partition functions. The rotational energy is given by) ) ) )rot rot

rot

qB kT

q B

= = = =

"(.9)#

The classical expression for the rotational energy is ( )1 1)1 x yI + , where I is the momentof inertia and ]xand ]yare the angular velocities in the x and y directions. The rotationalong the molecular axis "the ! axis here# has no meaning in uantum mechanics becausethe rotations along the molecular axis lead to configurations which are indistinguishablefrom the original configuration. The two degrees of freedom have thus given a value of-T. The translational contribution gives,

(

(

) ( ( (

1 1tr tr

tr

q VkT

q V

= = = = =

"(.91#

Thus, the three translational degrees of freedom in three dimensions satisfy theeuipartition theorem. Turning to the vibrational contribution, we get,

( ) ( )1

) ))

))

h$ h$vib

vib h$h$vib vib

q h $ e eh $ kT i% h $

q q ee

= = = V "8- 01# "Y ^vibY0#

( ){ }

( )

1

1

11

) " # " #

) )

V

V

TVh$ h$ h$

h$T

ee h$ e h$ eTk h $ k

e e

+ =

"(.99#

'or large T, the molar >Vbecomes N%- Q and for small T, >Vgoes to !ero as shown in

the s-etch below. The vibrational temperature V

is defined as V

h$

-. 'or 21, ithas a value of 6(1( G and for I1it is (


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'igure (.( Vibrational heat capacity of a diatomic as a function of V T .

3.3 Rottionl nd Vibrtionl Prtition Function! o) Polto%ic

Molecule!

'or a polyatomic molecule containing n atoms, the total number of coordinate degrees offreedom is (n. 3ut of these, three degrees of freedom are ta-en up for the translationalmotion of the molecule as a whole. The translational partition function is given by thesame formula as E. "(.1


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11 11 1 1) ) )

1 1 1 1 1 1&A B

A A B B & &

A B &

'' 'I I I

I I I = + + = + + = "(.6)#

2ere, ,A B and & are the three angular speeds and :%, :*and :>are the three

angular momenta. 'or a symmetric top molecule such as ammonia, or chloromethane,

two components of the moments of inertia are eual, i.e., I* I>. The rotational energylevels of such a molecule are specified by two uantum numbers and G. The totalangular momentum is determined by and the component of this angular momentumalong the uniue molecular axis is determined by G. The energy levels are given by

1, 1 1

" )# " # X ,? ?J ( B A

h hB J J A B ( B A

$I $I

= + + = = in cm8) "(.61#

&here, ta-es on values


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11

1cos sin )" cos #1 cos cos 1B &

p p p pI I

+ + +

"(.6@#

The rotational partition function is given by1 1

" , #

(< < arry out an analysis similarto that of 21 for R1where the deuterium nucleus has a spin of ).

1# Rerive the thermodynamic functions from the polyatomic rotational partition function.

(# >arry out the integration for the rotational partition function of the symmetric top.

7# >alculate the total partition function and the thermodynamic functions of water at)


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)(# E. "(.9

molecular partition functions

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