7. polyatomic ideal gas 3

8/18/2019 7. Polyatomic Ideal Gas 3

1/72

8.2 Energy level and its degeneracy

∑=i

in N ∑=i

iinU ε

0 1 2 3 4 5

1ε 2ε 3ε

Energy levels are said to be degenerate , if the same energy

level is obtained by more than one quantum mechanical state. Theyare then called degenerate energy levels.

The number of quantum states at the same energy level is called

the degree of degeneracy .


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The degree of freedom of movement The degree of freedom of movement

" Translation! x,y,z F #3


4/72


5/72

'ibration'ibration

" A olyatomic molecule containing n atoms has 3 n degrees of freedomtotally. Three of these degrees offreedom can be assigned totranslational motion of the center ofmass, t o or three to rotational motion.

" 3n& for a linear molecule*" 3n&+ for a nonlinear molecule


6/72


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.2.1 Translational article

The e ression for the allo ed translational energy levels of a article of mass m confined ithin a 3&dimensional bo ithsides of length a , b, c is

22 22

t 2 2 2( ) y x z

nn nhm a b cε = + +

here h +.+2+ 14 &3/56s n , ny, n7 are integrals called

quantum numbers . The number of them is 1,2,89 .2

2 2 2t y 73: 2 ( )

hn n n

mV ε = + +;f a#b#c,

equation becomes


8/72

all energy levels e ce t ground energy levelare degenerate.

Example At 344a, 1 mol

of 0 2 as added into a cubic bo . alculate

the energy level t,4 at ground state, and theenergy difference bet een the first e citedstate and ground state.


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Solution Ta=e the 0 2 at the condition as an ideal gas,then the volume of it is

The mass of hydrogen molecule is

31 .31/ 344 4.42/+2 m14132

n!T V

"

× ×= = =

3 23 2?: 2.41 14 : +.422 14 3.3/? 14 =gm # $ − −= = × × = ×


10/72

2 3/ 2/4

t,4 2:3 2? 2:33 (+.+2+ 14 )3 . 11 14 5

3.3/? 14 4.42/+2h

mV ε

−−

−× ×= × = = ×× × ×

2/4

t,1 2:3 + 11.+22 14 5h

mV ε −= × = ×

/4 /4t,1 t,4 (11.+22 . 11) 14 . 11 14 5ε ε ε

− −∆ = − = − × = ×

the energy difference is so small that the translational articlesare e cited easily to o ulate on different e cited states, andthat the energy changes of different energy levels can be thin=of as a continuous change a ro imately.


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8.2.2 Rigid rotator (diatomic)

The equation for rotational energy level of diatomic molecules is2

r 2 ( 1) 4 1 2h

% % % &

ε π

= + = ⋅ ⋅ ⋅

2 21 24 4

1 2

( )m m

& ! !

m m

µ = =+

here % is rotational quantum number, & is the moment ofinertia ( )

' is the reduced mass ( ), The degree of degeneracy is

r, 2 1 % g % = +


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8.2.3 One dimensional !armonic oscillator

v1( ) 4,1, 2,2

hε ν = + = ⋅⋅⋅v v

v,4

12

hε ν =

here v quantum number hen v #4 the

energy is called 7ero oint energy.

v,1

32

hε ν =

v,2 2hε ν = v,3 ?2 hε ν =

-ne dimensional harmonic vibration is non&degenerate.


13/72

8.2." Electron and atomic nucleus

The differences bet een energy levels of electronmotion and nucleus motion are big enough to =ee theelectrons and nuclei stay at their ground states.

@oth degree of degeneracy, g e,4, for electron motion

at ground state and degree of degeneracy, g n,4, for

nucleus motion at ground state are different fordifferent substances, but they are constant for a givensubstance.


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8.# $omputations of t!e partition function8.# $omputations of t!e partition function

" 8.#.% Some features of partition functions" (1) at T #4, the artition function is equal to the

degeneracy of the ground state.

" (2) hen T is so high that for each term i:(T #4,

" (3) factori&ation property ;f the energy is a sumof those from inde endent modes of motion, then

44limT ) g → =i

(T i

i) g e

ε −

=∑

limT )→∞ = ∞


15/72

t r e n) ) ) ) ) )= v , , , , ,i t i r i i e i n iε ε ε ε ε ε = + + + +v

, , , , ,i t i r i i e i n i g g g g g g = v The artition functions for mode motions are e ressed as

, , ,

, ,

, , ,

, ,

* *

*

t i r i v i

e i e i

(T (T (T t t i r r i v v ii i i

(T (T e e i e e i

i i

) g e ) g e ) g e

) g e ) g e

ε ε ε

ε ε

− − −

− −

= = =

= =

∑ ∑ ∑∑ ∑


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, r,it,i r,i

v,i e,iv,i e,i

n,in.i

A e ( )B A e ( )B

A e ( )B A e ( )B

A e ( )B

t i

i i

i i

i

) g g (T (T

g g (T (T

g (T

ε ε

ε ε

ε

= − × − ×

− × − ×

−

∑ ∑∑ ∑∑

t r v e n) ) ) ) )= ⋅ ⋅ ⋅ ⋅


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8.#.2 'ero point energy8.#.2 'ero point energy

" &ero point energy is the energy at ground state orthe energy as the tem erature is lo ered toabsolute 7ero.

" Cu ose some energy level of ground state is 4,and the value of energy at level i is i, the energyvalue of level i relative to ground state is

" Ta=ing the energy value at ground state as 7ero,e can denote the artition function as ) 4.

44i iε ε ε = −


18/72

4

4i

(T i

i

) g eε

−=∑ 44 (T ) e )ε =,4 ,4 ,4

,4 ,4

: : :4 4 4

: :4 4* **

t r v

e n

(T (T (T

t t r r v v(T (T

e e n n

) e ) ) e ) ) e )) e ) ) e )

ε ε ε

ε ε = = == =Cince t,4D4, r,4#4, at ordinary tem eratures.

4 4,t t r r ) ) ) )≈ =


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" The vibrational energy at ground state is

" therefore

" the number of distribution in any levels does notde end on the selection of 7ero& oint energy.

1,4 2 hε ν =v

4 :2h (T ) e )ν =v v

4 44

4

: ( ): ::4 4

i i i(T (T (T i i i i(T

N N N n g e g e g e

) ) e )ε ε ε ε

ε − − + −

−= = =×


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8.#.3 ranslational partition function

222 2

,t 2 2 2( ) y x z

i

nnh n

m a b c

ε = + +

,it ,i e ( )t t i

) g (T

ε = −∑

Energy level for translation

The artition function


21/72

22 22

2 2 21 1 1

2 2 22 2 2

2 2 21 1 1

, , ,

e :

e e e

x y z

x y z

y x z t

n n n

x y z n n n

t x t y t z

nn nh) (T m a b c

h h hn n n

m(Ta m(Tb m(Tc

) ) )

∞ ∞ ∞

= = =

∞ ∞ ∞

= = =

= − + + ÷ ÷ = − × − × − ÷ ÷ ÷

=

∑∑∑

∑ ∑ ∑

22

, 21

e x

t x x

n

h) n

m(Ta

∞

=

= − ÷

∑2

2, 2

1

e y

t y yn

h) n

m(Tb

∞

=

= − ÷ ∑

22

, 21

e z

t z z n

h) n

m(Tc

∞

=

= − ÷ ∑


22/72

Ta=e ) t, as an e am le

22

t , 21

e ( ) x

x x

n

nh)m(T a

∞

== − ⋅∑

2 2t, 4

e ( )d x x x) n nα ∞

= −∫

%or a gas at ordinary tem erature *2 1, the summation

converts into an integral.

2α

2

2 2 221

e ( ) ( x

xn

hn m(Taα α

∞

== − =∑


23/72

2 1 2

41d ( )2

xe xα π α

∞ − =∫ 1

21

2t, 2

1 2( ) ( )

2 x

m(T ) a

h

π π

α

= = ⋅

32

t2

2( )

m(T ) a b c

h

π = ⋅ ⋅ ⋅

;n li=e manner,t, y) t, z )

32

2

2 ( )

m(T V

h

π = ⋅

%rom mathematic relations in A endi


24/72

" Example alculate the molecular artitionfunction ) for 0e in a cubical bo ith sides 14cmat 2F


25/72

$

# m =

"

N(T

"

n!T V ==

( )

⋅

×

=

−

>a

< mol=g1424 2. 2

23

1?

t "

T # N

)

%or ideal gas,


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8.#." Rotational partition function

The rotational energy of a linear molecule is given byr # % ( % G1)h2: + 2 & and each % level is 2 % G1 degenerate.

2

r 2( 1) 4 1 2h

% % % &

ε

π

= + = ⋅ ⋅ ⋅

, 2

, 24

(2 1) e ( 1)r i

(T r r i

i %

h) g e % % %

&(T

ε

π

∞−

=

= = + − + ∑ ∑define the c!aracteristic rotational temperature

2

2r

h &(

Θ π

=


27/72

r r

4

( 1)(2 1) e ( ) %

% % ) % T

Θ ∞=

+= + −∑ r T at ordinary tem erature, The summation can be

a ro imated by an integral

[ ]r 4 (2 1) e ( 1) : dr ) % % % T % Θ ∞

≈ + − +∫ Het % ( % G1)# x, hence % (2 % G1)d % #d x, then

2 2r r 4

e ( : )d r ) x T x T &(T hΘ Θ π ∞

≈ − = =∫


28/72

2

r 2r

T &(T )

h

π

Θ σ σ = =

%or a homonuclear diatomic molecule, such as - 2, it comes bac=to the same state after only 1 4 o rotation.

here - is called the symmetry number . - is the number ofindistinguishable orientations that a molecule can e hibit by

being rotated around symmetry a is. ;t is equal to unity forheteronuclear diatomic molecules and is equal to 2 formononuclear diatomic molecules.

%or 0 l, - # 1* and for l 2, - # 2.


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8.#.# ibrational partition function

v

1( ) 4,1, 2,

2v h vε ν = + = ⋅⋅⋅

'ibrational energies for one dimensional oscillator are

'ibration is non&degenerate, g #1. The artition function is

,

,4

1e :

2

i

(T i

i

) g e h (T ε

ν ∞−

=

= = − + ÷ ∑ ∑

v

v v

v

v


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31/72

" haracteristic vibrational tem eratures are usuallyseveral thousands of


32/72

At lo T

according to mathematics

v 1T

Θ >> ve ( ) 1T

Θ −


33/72

( )v,4 v4v :1e

1 e ( )(T ))

h

(T

ε ν

× ==− −

ta=e the ground energy level as 7ero,

%or J-, the characteristic vibrational tem erature is2+F4


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8.#.+ Electronic and nuclear partition function

e,4 e,1e e,4 e,1e ( ) e ( )) g g (T (T

ε ε = − + − + ⋅⋅⋅

e,4 e,1 e,1 e,4e,4

e,4

e ( ) 1 e ( ) B g

g (T g (T

ε ε ε −= − + − + ⋅⋅⋅

&1e,1 e,4( ) /44 =5 mol ,ε ε − = ⋅

e,4e e,4 e ( )) g (T

ε = −

Energy difference is large, so electrons are generally at groundstate, all terms e ce t first one in the summation e ression isnegligible.

,4 :4,4

e (T e e e) e ) g

ε = =


35/72

" ;f the quantum number of total angular momentumfor electronic motion is , the degeneracy is (2 G1).Then the electronic artition function can be

ritten as

" A rare e ce tion is halide atoms and J- molecule.

The difference bet een the ground state and thefirst e cited state of them are not so large, thesecond term in the summation has to beconsidered.

4 2 1e) .= +


36/72

n,4n n,4 e ( )) g (T

ε = −

Juclear motion is al ays in the ground state at ordinarychemical and hysical rocess because of large energydifference bet een ground and first e cited state. ;ts

artition function has the form of

Juclear motion Juclear motion

4 ,4 2 1n n) g & = = +here & is a quantum number of nuclear s in.


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38/72

:

:

2

:2

:2

1

1

i

i

i

i

(T i

iV V

(T i

ii

(T i i

i

(T i i

iV

) g eT T

g e( T

g e(T

)(T g eT

ε

ε

ε

ε

ε

ε

ε

−

−

−

−

∂ ∂ = ÷ ÷∂ ∂ = − −

÷ ÷ =

∂ = ÷∂

∑

∑∑

∑Cubstitute this equation into equation ( ./ ), e have

8.+ !ermodynamic energy and partition function


39/72

2 2 lnV V

N ) )U (T N(T ) T T

∂ ∂ = = ÷ ÷∂ ∂

Cubstitute the factori7ation of artition function for )

2 ln t r v e nV

) ) ) ) )U N(T

T ∂ = ÷∂

-nly ) t is the function of volume, therefore


40/72

2 2 2

2 2

ln lnln

ln ln

t vr

V

e n

t r v e n

) d )d )U N(T N(T N(T T dT dT

d ) d ) N(T N(T

dT dT U U U U U

∂ = + + ÷∂

+ +

= + + + +;f the ground energy is s ecified to be 7ero, then

44 2 lnV

)U N(T T

∂= ÷∂ 4 :4Cubstitute # into this equation, it follo s that(T ) )e ε


41/72

4 4U U N ε = −;t tells us that the thermodynamic energy de ends

on the 7ero oint energy. N4

is the total energy of

system hen all articles are locali7ed in ground

state. ;t (denoted as U 4) can also be thought of as the

energy of system at 4


42/72

" U 4

can be e ressed as the sum of differentenergies

4 4 4 4 4 4

4 4 4

4 4

24 4

t r v e n

t t r r v v

e n

U U U U U U Nhv

U U U U U U

U U

= + + + +≈ = = −

= =


43/72


44/72

The calculation ofThe calculation of

4 2

2

ln

ln

r r r

V

r

)U U N(T T

T d

N(T N(T dT

σ

∂ = = ÷∂

Θ=

4r U

The degree of freedom of rotation for diatomic orlinear molecules is 2, the contribution to the energy ofevery degree is also K !T for a mole substance.

4


45/72

The calculation of The calculation of

4vU

4vU

4vU

4 :4 2 2

:

1lnln 1

11

v

v

T v

v

v T

d d ) eU N(T N(T dT dT

N( e

−Θ

Θ

−= =

= Θ −Lsually, v is far greater than T , the quantum effect

of vibration is very obvious. hen v:T MM1,Cho ing that the vibration does not have contributionto thermodynamic energy relative to ground state.

4

4vU ≈


46/72

" ;f the tem erature is very high or the v is verysmall, then v:T 1, the e onential function can be e ressed as

: 1T eT

Θ Θ ≈ +v v

4v :

1 11 1 1

T U N( N( N(T eT

Θ Θ Θ Θ = ≈ =− + −

vv v

v


47/72

" %or monatomic gaseous molecules e donot need to consider the rotation andvibration, and the electronic and nuclearmotions are su osed to be in their groundstates. The molar thermodynamic energy is

4,

32m mU !T U = +


48/72


49/72

f8 i d i i f i


50/72

8., -eat capacity and partition function8., -eat capacity and partition function

" The molar heat ca acity, / ',m , can be derivedfrom the artition function.

,U m/ V m T

V

∂=

∂

÷

ln2,

) !T

T V

V m T V

/ ∂ ÷∂

∂=∂ $e lace ) ith 4 :4 (T ) ) e ε −=

4ln2

,

) !T

T V V m T V

/ ∂ ÷ ÷∂

∂=∂

e can see from above equations that heat ca acity does notde ends on the selection of 7ero oint of energy.


51/72

" Electrons and nucleus are in ground state

442 2

, , ,

4ln2,

lnln vr

V V

V t V r V v

) t !T T

V V m T

V

))/ !T !T

T T T T / / /

∂ ÷ + + ÷∂

∂=

∂

∂∂∂ ∂ ÷ ÷∂ ∂ ∂ ∂

= + +

h l l fTh l l i f dd


52/72

The calculation of The calculation of ',t',t , , ',r ',r and and ',v',v

" (1) The calculation of / ',t

32

2

2( )t

m(T ) V

h

π =

4 3ln22,

) t !T !T

V V m T

V

/ ∂ ÷ = ÷∂

∂=

∂


53/72

" (2) The calculation of / ',r

2

2 (linear molecules)r r

&(T T )

hπ σ σ

= = Θ4

2,

ln r V r

V

)/ !T !

T T

∂∂= = ÷∂ ∂

;f the tem erature is very lo , only the lo est rotation state isoccu ied and then rotation does not contribute to the heatca acity.

h l l fTh l l i f /


54/72

The calculation ofThe calculation of / / ',v',v

T

(T

e)e)

v

4,v

11v4v Θ−

−==

ε

22

vv, 1vv

−ΘΘ

− Θ

= T T V eeT !/

4

2 v,v lnV

V V

)d / !T dT T

∂= ÷∂


55/72

v v v v

v

2 22 2v v

,v

2v

1

4

T T T T V

T

/ ! e e ! e eT T

! eT

− −Θ Θ Θ Θ

Θ−

Θ Θ = − ≈ ÷ ÷ ÷ ÷ Θ = ≈ ÷

;t sho s that under general conditions, the contribution to heatca acity of vibration is a ro imately 7ero.

Nenerally, v:T MM1, equation becomes


56/72

v

1 vT e

T

Θ Θ ≈ + ÷

hen tem erature is high enough,

v v2 2

v v,v

T T V / ! e !e !

T T

−Θ ΘΘ Θ ≈ = ≈ ÷ ÷


57/72

32V

/ !=

, , 4 2V V t V v/ / / != + + =

;n gases, all three translational modes are active and theircontribution to molar heat ca acity is

The number of active rotational modes for most linearmolecules at normal tem erature is 2

122V / ! != × =

;n most cases, vibration has no contribution to the heat ca acity,

8 8 E d i i f i


58/72

8.8 Entropy and partition function

8.8.% Entropy and microstateolt&mann formula

( # 1.3 4+2 14 &23 5 < &1

As the tem erature is lo ered, the 0 , and hence the 1 of thesystem decreases. ;n the limit T O4, 0 #1, so ln 0 #4, because

only one configuration is com atible ith E #4. ;t follo sthat 1 O4 as T O4, hich is com atible ith the third la ofthermodynamics.

ln1 ( Ω =


59/72

" %or e am le

maln ln ln 2 2

3 3 Ω = ≈∑

maln1 ( 3 =

/

4

144.41

14=

/

4

ln144.F+ 1

ln14= ≈

hen N a roaches infinity, maln

1

ln

3

Ω

=

8 8 2 Entrop and partition f nction8 8 2 Entropy and partition function


60/72

8.8.2 Entropy and partition function8.8.2 Entropy and partition function

" %or a non&locali7ed system, the most robabledistribution number is

" Lsing Ctirling equation ln N P# N ln N 4 N and @olt7manndistribution e ression

"

P

ini

2i i

g 3

n=∏

ln ( ln ln P) 2 i i ii

3 n g n= −∑

:i (T i i

N n g e

)ε −=


61/72

" e have,

4 4

ln ( ln ln )

( ln ln ln )

ln

ln ln (non&localised system)or

ln (non&localised system)

5 i i i i ii

i ii i i i i i

i

5

3 n g n n n

n N n g n n g n

) (T ) U

N N N (T

) U 1 ( 3 N( N( N T

) U 1 N( N(

N T

ε

= − +

= − − + +

= + +

= = + +

= + +

∑∑


62/72

" %or locali7ed system

" Entro y does not de end on the selection of 7ero ointenergy .

44

ln ln (localised system)

or

ln (localised system)

5

U 1 ( 3 N( )

T

U 1 N( )

T

= = +

= +


63/72

"%actori7ing the artition function into differentmodes of motions and using

" e can give

4 4 4 4 4 4

t r v e nU U U U U U = + + + +

t r v e n1 1 1 1 1 1 = + + + +


64/72

4 4

ln t t t

) U 1 N( N(

N T = + +

4 4

ln v vv) U

1 N( N T

= +

4 4

ln e ee) U

1 N( N T

= +4 4

ln e et ) U

1 N( N T

= +

4 4

ln r r r ) U

1 N( N T

= +

%or identical article system, entro ies for every mode ofmotion can be e ressed as

8 8 3 $alculation of statistical entropy8 8 3 $alculation of statistical entropy


65/72

8.8.3 $alculation of statistical entropy8.8.3 $alculation of statistical entropy

"At normal condition electronic and nuclearmotions are in ground state, and in general

hysical and chemical rocess the contribution tothe entro y by t o modes of motion =ee s

constant. Therefore only translational, rotationaland vibrational entro ies are involved incom utation of statistical entro y.

vt r 1 1 1 1 = + +

8 8 3 $alculation of statistical entropy8 8 3 $alculation of statistical entropy


66/72

8.8.3 $alculation of statistical entropy8.8.3 $alculation of statistical entropy

34 22

2( )t t

m(T ) ) V

h

π = = ×

4 4

ln t t t ) U

1 N( N( N T

= + +

( )3:2

32ln

2t m(T V 1 N( N( Nh

π = +

(%) $alculation of S t

4 32t

U N(T =


67/72

( )1, 3 ln : =g mol ln( : a) 24.?232 2

m t 1 ! # T "− = × + − +

"%or ideal gases, the Sac/ur0 etrode equation isused to calculate the molar translational entro y.

(2) $alculation of(2) $alculation of SS


68/72

(2) $alculation of(2) $alculation of S S rr

" %or linear molecules

" hen all rotational energy levels are accessible

" e obtain

4 4

ln r r r ) U

1 N( N T

= +

4 4:r r r r ) ) T U N(T σ = = Θ =

ln( : )r r 1 N( T N( σ = Θ +, lnm r

r

T 1 ! !

Θ σ

= +

(3) $alculation of(3) $alculation of SS


69/72

(3) $alculation of(3) $alculation of S S **

"Cubstitute

" ;nto the follo ing equation

( ) ( )1 1: :4 41 and 1v vT T v v r ) e U N( e

− −−Θ Θ= − = Θ −

( ) ( )

4 4v v

1 1: :1

v

ln :

ln 1 1v vv

T T

1 N( ) U T

N( e N( T e− −−Θ Θ−

= += − + Θ −

( ) ( )v v1 1: :1

m,v vln 1 1T T 1 ! e ! T eΘ Θ Θ

− −− −= − + −


70/72

8.1 Ot!er t!ermodynamic functions and partition8.1 Ot!er t!ermodynamic functions and partition


71/72

functionsfunctions

"1 6, 7, and )

[ ]

( ) ( )

ln

ln ln ln

ln ln ln ln P

ln (for identical articles)Pln (locali7ed system)

N

N

) U 6 U T1 U T N( N(

N T

) N(T N(T (T N ) N N N N

(T N ) N N N (T N ) N

) 6 (T N 6 (T )

= − = − + + ÷

= − − = − − +

= − − − = − −

= −= −


72/72

( )

2

ln

lnln : P (non&locali7ed system)

lnln (locali7ed system)

ln ln

T T

N

T

N

T

V T

6 )7 6 "V " N(T V V

)7 (T ) N N(TV

V )

7 (T ) N(TV V

U "V

) ) N(T N(TV

T V

∂ ∂ = + = − = ÷ ÷∂ ∂ ∂ = − + ÷∂ ∂ = − + ÷∂

= +∂ ∂ = + ÷ ÷∂ ∂

7. polyatomic ideal gas 3

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