MolecularThermodynamics_StatisticalThermodynamics

8/9/2019 MolecularThermodynamics_StatisticalThermodynamics

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15 Statistical thermodynamics 1:

the concepts

Statistical thermodynamics provides the link between the microscopic

properties of matter and its bulk properties.

Two key ideas:

First: Boltzmann distribution

Second: Ensemble


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Statistical thermodynamics:

The crucial step in going from the uantum mechanics of individual

molecules to the thermodynamics of bulk samples is to recognize that

the latter deals with the average behavior of large numbers of

molecules.


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The distribution of molecular states

!opulation

!rinciple of eual a priori probabilities

"ssumption: "ll possibilities for the distribution of energy are eually

probable.


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15.1 Configurations and weights

#a$ %nstantaneous configurations&onfiguration {3,2,0,…} in '( ways:


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&onfiguration { N 0 , N 1, …} in W different ways

W = N !

N 0 ! N 1! N 2 ! …

The weight of a configuration

ln W =ln N !−∑i

ln N i !

N i ln N i(¿− N i)

¿ ( N ln N − N )−∑i

¿

¿ N ln N −∑i

N i ln N i

)sing Stirling*s appro+imation:

ln x != x ln x− x


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#b$ The Boltzmann distribution

)nder the two restrictions:

&onstant total energy:

∑i

N i εi= E

&onstant total number of molecules:

∑i

N i= N


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The populations in the configuration of greatest weight depend on the

energy of the state according to the Boltzmann distribution:

N i N = e

− βε i

∑i

e− β εi

β= 1

k T

, The thermodynamic temperature is the uniue parameter that governs

the most probable populations of states at thermal euilibrium.


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15.2 The molecular partition function

!opulation of state:

pi=e− β εi

q

-efinition of molecular partition function:

q=∑i

e− β εi

q=∑level I

g I e− βε I


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#a$ "n interpretation of the partition function

limT →0 q=g0

That is at T =0 the partition function is eual to the degeneracy of the

ground state.

%n general.

limT → ∞

q=∞

/owever when the molecules have only a finite number of states the upper

limit of q is eual to the number of states.

"t T =0 only the ground state level is accessible and q=g0 .

"t very high temperature virtually all states are accessible and q is


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correspondingly large.


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!artition function of a uniform array of states:

q= 1

1−e− βε


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pi=e− β εi

q =(1−e− βε) e− βε i

!opulation of two0state system:


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p0=

1

(1+e− βε )

p1= e

− βε

(1+e− βε )


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#b$ "ppro+imations and factorizations

!opulation function for translation in one dimension:

q X =( 2πmh2 β )1/2

X =( 2πmkT h2 )1/2

X

εn1

, n2

,n3

=εn1

( X )+εn2

(Y )+εn3

(Z )

q=∑alln

e− β εn

1

( X )− βε n2

(Y )− βε n3

(Z )

=∑alln

e− βε n

1

( X )

e− βε n

2

( Y )

e− β εn

3

(Z )

¿(∑n1 e− βε n1

( X )

)(∑n2 e− β εn2

(Y )

)(∑n3 e− β εn3

(Z )

)=q X qY qZ

q=( 2πmh2 β )3 /2

XYZ =(2πmh2 β )3 /2

V

!artition function for translation in three dimension:

q= V

Λ3 Λ=h( β2πm )

1 /2

= h

(2πmkT )1 /2


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The nternal energy and the entropy

15.! The internal energy

The importance of the molecular partition function is that it contains all the

information needed to calculate the thermodynamic properties of a system

of independent molecules.

Statistical thermodynamics: q

1, 2uantum mechanics: wavefunction


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#a$ The relation between ) and

E (T )=∑i N i εi

E (T )= N

q∑

i

εi e− βε i

εi e− β εi=

− β

e− βε i

E (T )=− N q ∑

i

β e− βε i=

− N q

β∑

i

e− β εi=

− N q

q

β


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! (T )=! (0 )+ E (T )

%nternal energy in terms of the partition function:

! (T )=! (0 )− N q ( " q" β )V

%n an alternative version:

! (T )=! ( 0 )− N ( " lnq" β )V


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#b$ The value of β

Fundamentals F.3:

(T )= (0 )+ 32

n#T

4ustification '3.5:

(T )= (0 )+ 3 N 2 β

β= N

n#T =

n N $

n N $ kT =

1

kT


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15." The statistical entropy

Boltzmann formula of the entropy:%=k lnW

Entropy in terms of partition function:

% (T )= (T )− (0)

T + Nk ln q


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The temperature #ariation of the entropy of an e$ually spaced energy system


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The temperature #ariation of the entropy of a two%le#el system


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The canonical partition function

15.5 The canonical ensemble

The crucial new concept we need when treating systems of interacting

particles is the 6ensemble*.


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&a' The concept of ensemble


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The total energy of all the systems is E and because they are in thermal

euilibrium with one another they all have the same temperature T . This

imaginary collection of replications of the actual system with a common

temperature is called the canonical ensemble.

(efinitions of ensembles:

7icrocanonical ensemble: 8 9 E common

&anonical ensemble: 8 9 T common

rand canonical ensemble: µ 9 T common


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&b' (ominating configurations

~ E : the total energy of the ensemble~

N : the number of imaginary replications

Ei : the energy state of each member

~

N i : the number of members with energy Ei

N : the number of molecules in the actual system


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W : the total weigh of a configuration of the ensemble

~W =

N !~ N 0!~ N 1 !~ N 2! …

-efinition of canonical partition function:

N i~ N

=e− β Ei

& &=∑

i

e− β Ei


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#c$Fluctuations from the most probable distribution


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distinct from being in a specified state$ is given by N i~ N

=e− β Ei

& a sharply

decreasing function multiplied by a sharply increasing function.

Therefore the overall distribution is a sharply peaked function. ;e conclude

that most members of the ensemble have an energy very close to the mean

value.


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15.) The thermodynamic information in the partition function


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The internal energy

(T )= (0 )+ E (T )= (0 )+~ E(T )/

~

N as N → ∞

Because ~ pi=e− βE i

&

(T )= (0 )+∑i

~ pi E i= (0 )+ 1

&∑

i

Ei e− β Ei

%nternal energy in terms of the canonical ensemble:

(T )= (0 )− 1& ( " &" β )V = (0 )−(

" ln &

" β )V


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#a$ The entropy

The total weight of~

W of a configuration of the ensemble is theproduct of the average weight W of each member of the ensemble

~W =W N

/ence

%=k lnW =k ln~W 1/~ N = k ~

N ln~W

Entropy in terms of the canonical partition function:

% (T )= (T )− (0)

T +k ln &

ndependent molecules


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#a$ -istinguishable and indistinguishable molecules

=elation between & and q :For indistinguishable independent molecules>

&=q N / N !

For distinguishable independent molecules>

&=q N


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#b$ The entropy of a monatomic gas

Sackur0Tetrode euation for the entropy of a monatomic gas:

% (T )=n# ln( e5/2

V

n N $ Λ3 ) Λ=( h(2πmkT )1 /2 )

%n terms of pressure

% (T )=n#ln

(e

5/2kT

p Λ3

)


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The Sackur0Tetrode euation implies that when a monatomic perfect gas

e+pands isothermally from V i to V ' its entropy changes by

( %=n# ln (a V ' )−n# ln (a V i )=n# ln(V '

V i )


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1) Statistical thermodynamics 2: applications

"pply the concepts of statistical thermodynamics to the calculation of chemically significant uantities:

'. Establish the relation between thermodynamic functions and partition functions.

?. The molecular partition function can be factorized into contributions from each mode of motion and

establish the formulas for the partition functions for translational rotational vibrational modes of the

motion and the contribution of electronic e+citation.

5. Specific applications: the mean energies of modes of motion the heat capacities of substances and

residual entropies.

@. &alculate the euilibrium constant of a reaction and through that calculation understand some of the

molecular features that determine the magnitudes of euilibrium constants and their variation with

temperature.


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*undamental relations

1).1 The thermodynamic functions

− (0 )=−( " ln &" β )V %=

− (0 )T

+k ln&

#c$ /elmholtz energy

$= −T%

$− $ (0 )=−kT ln&

#d$ The pressure

$= −T%


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$= pV −%T

Because p=−( " $" V )T !ressure in terms of &

p=kT ( " ln &" V )T


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#e$ The enthalpy

) = + pV

Enthalpy in terms of &

) − ) (0 )=−( " ln&" β )V +kTV (" ln &

"V )T


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#f$The ibbs energy

*= ) −T%= $+ pV

ibbs energy in terms of &

*−* (0)=−kT ln &+kTV ( " ln&" V )T

*−* (0 )=−kT ln &+n#T


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Because &=q N / N !

*−* (0)=− NkT ln q+kT ln N !+n#T

¿−n#T ln q+kT ( N ln N − N )+n#T

¿−n#T ln q N

ibbs energy of independent molecules:

*−* (0)=−n#T lnqm

N $


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1).2 The molecular partition function

The energy of a molecule is the sum of contributions from its different modes of motion:

ε i=εiT +εi

#+εiV +ε i

E

The electronic contribution is not actually a 6mode of motion* but it is convenient to

include it here.

The separation of terms in the euation above is only appro+imate #e+cept for translation$

because all modes are not completely independent but in most cases it is satisfactory.


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The separation of the electronic and vibrational motions is Austified provided only the ground

electronic state is occupied #for otherwise the vibrational characteristics depend on the

electronic state$ and for the electronic ground state and the Born0ppenheimer

appro+imation is valid.

The separation of the vibrational and rotational modes is Austified to the e+tent that the

rotational constant is independent of the vibrational state.


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iven that the energy is a sum of independent contributions the partition function factorizes

into a product of contributions:

q=∑i

e− β εi= ∑

i(all +ae+

)

e− βε i

T − β ε i #− β εi

V − βε i E

¿qT

q #

qV

q E


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#c$ The translational contribution

Translational contribution toq

:q

T = V

Λ3 Λ=h( β2 πm)

1 /2

= h

(2 πmkT )1/2

qT

→ ∞ a + T → ∞

"t room temperature qT -2.1028 for an ? molecule in a vessel of volume

'(( cm5.

The thermal wavelength Λ should be much less than the average

separation of the molecules in the sample in order for the appro+imations

that led to the e+pression of qT are valid.

#d$ The rotational contribution


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The partition function of a nonsymmetrical #"B$ linear rotor:

q #=∑

/

(2/ +1)e− βh0 1/ (/ +1)


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=otational contribution to q in the high temperature limit #linear rotors$:

q #=

T

2 3 #

2 :+4mme5i0 n6m7e5

8ha5a0e5i+i059ai9nal empe5a65e 3 # :

3 #=h0

~1

k


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=otational contribution to q in the high temperature limit #nonlinear

molecules$:

q #=

1

2 ( kT h0 )3 /2

( π ~ $~1~8 )1 /2


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#e$ The vibrational contribution

The vibrational energy levels:

E:=(:+12 )h0~; :=0,1,2,…

9ibrational contribution to q :

The partition function:

qV =∑

:

e− β:h0~;=∑

:

(e− βh0~; ):

qV =

1

1−e− βh0~;


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9ibrational contribution to q in the high temperature limit:

qV =

1

βh0~;=

kT

h0~;

q

V

=

T

3V

8ha5a0e5i+i0 vi75ai9nal empe5a65e 3V :

3V =h0~;

k


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#f$The electronic contribution

%f we denote the energies of two levels as E1 /2=0 and E3 /2=ε the partition


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function is:

q E= ∑

ene5g4level+

g > e− βε >=2+2e

− βε


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#g$ The overall partition function

The overall partition function for a diatomic molecule with no low0lying

electronically e+cited states and T ≫3 #

q=g E( V Λ3 )( T

2 3 # )( 1

1−e−3V /T )ne5he a++6mpi9n+ :

he 59ai9nallevel+ a5e ve5 4 0l9+e 9gehe5

he vi75ai9nallevel+a5eha5m9ni0


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+sing statistical thermodynamics

&alculate partition functions

, "ny thermodynamic uantities

, aining insight into a variety of physical chemical biological processes

Four important properties:

'. 7ean energies

?. /eat capacities

5. Euations of state

@. 7olecular interactions in liuids


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1).! ,ean energies

The mean energy is the sum of contributions from:

i. Translation

ii. =otation

iii. 9ibration

7ean energy of a mode of motion:

⟨ ε ? ⟩=−1q

? ( " q ?

" β )V ? =T , # , V ,∨ E


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#c$ The mean translational energy

For a one0dimensional system of length X

qT =

X

Λ


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"t high temperature (T ≫3 # )

q #=

T

2 3 #


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qV =

1

1−e− βh0~;

qV

β =

β ( 11−e− βh0~; )=−h0~; e− βh0

~;

(1−e− βh0~; )2


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7ean vibrational energy:

⟨ εV ⟩=−1q

V

qV

β =−(1−e− βh0

~; ){−h0~; e− βh0~;

(1−e− βh0~; )2 }=h0

~; e− βh0~;

1−e− βh0~;

⟨ εV ⟩= h0~

;e

βh0~;−1

The zero0point energy1

2 h0~; can be added to the right side if the mean energy

is to be measured from ( rather than the lowest attainable level #the zero0point

level$.


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"t high temperature when T ≫3V or βh0~;≪1 the e+ponential functions can be

e+panded (e x -1+ x+…) and all but the leading terms discarded.

7ean vibrational energy #high temperature limit$

⟨ εV ⟩= h0~;

e βh0~;−1

= h0~;

(1+ βh0~;+…)−1- 1

β=kT

, "grees to the classical euipartition theorem too.


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1)." -eat capacities

#a$ The individual contributions

The constant0volume heat capacity:

8 V =( " "T )V

T =

β

T

β=−1

kT 2

β=−kβ2

β


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Translational contribution to 8 V

8 V ,mT = N $( ⟨ ε

? ⟩ T )V = N $

( 32 kT ) T

=3

2 #

%n the same way when T ≫3 #

rotational contribution to 8 V

8 V ,m # = # '95 a linea5 59a5

8 V ,m # =

3

2 # '95 a n9nlinea5 m9le06le+


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9ibrational contribution to 8 V

8 V ,mV = #' ( T ) ' (T )=(3V T )

2

( e−3V /2T

1−e−3

V /2T )

2

;here 3V =h0~

; /k


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#b$ The overall heat capacity

;hen euipartition is valid #when the temperature is well above the

characteristic temperature of the mode T ≫3 ? $ we can estimate the heat

capacity by counting the numbers of modes that are active.

Total heat capacity #at high temperatures$

8 V ,m❑ =12 (3+; #

⋆

+2;V ⋆

) #

where ; ? ⋆ is the number of active modes for each 7 mode

%n most case ;V ⋆ -0 .


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1).5 $uations of state

1).) ,olecular interpretations in li$uids


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"n A8lB3 molecule can adopt four orientation with the same energy #with the F

atom at any of the four corners of a tetrahedron$:%m (0 )= # ln 4=11.5/ C

−1m9l

−1 is in good agreement with the e+periment value #

10.1/ C −1

m9l−1 $

For 8B the measured residual entropy is 5/ C −1 m9l−1 which is very close to # ln 2

the value e+pected for a random structure of the form = = =8B8BB88BB8B8===


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1). $uilibrium constants

ibbs energy of independent molecules:

*−* (0)=−n#T lnqm

N $

#a$ The relation between C and the partition function

*mϑ (/ )−*m

ϑ (/ , 0 )=− #T ln qmϑ

N $


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The euilibrium constant for the reaction a $+7 1 → 0 8 + D in terms of partition

function:

C =(q8 ,m

ϑ

N $

)

0

(q D ,m

ϑ

N $

)

( q $ ,mϑ

N $ )a

( q1 ,mϑ

N $ )7

e− (5 E 0/ #T

C ={∏/ ( q/ ,m

ϑ

N $ ); /

}e− (5 E 0/ #T

(5 E0=0 mϑ

(8 ,0)+ mϑ

( D ,0 )−a mϑ

( $ ,0 )−7 mϑ

( 1 ,0 ),


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#b$ " dissociation euilibrium

"n euilibrium in which a diatomic molecule:

X 2 (g )⇌2 X (g ) C =

p X 2

p X 2

pϑ

C =( q X, m

ϑ

N $ )2

q X

2, m

ϑ

N $

e

− (5 E 0 #T

= ( q X ,m

ϑ )2

q X

2,m

ϑ N $

e

− (5 E0 #T

¿ C =( q8 , m

ϑ

N $ )0

( q D, mϑ

N $ )

( q $, mϑ

N $ )a

( q1, mϑ

N $ )7

e

− (5 E0 #T


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#c$ &ontributions to the euilibrium constant

&onsider a simple #⇌ gas0phase euilibrium:

N p

N #=

q

q #e− (5 E0 / #T

C =q

q #e− (5 E0 / #T


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MolecularThermodynamics_StatisticalThermodynamics

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