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Statistical Thermodynamics

Thermodynamics

(Bulk Properties)

Quantum Mechanics

Molecular structure,Spectroscopy

(individual properties)

StatisticalThermodynamics

Wave

function

Partition

function

Theories of Reaction Rates

bridge between microscopic properties and macroscopic reaction rate:result of many

microscopic collisionsCollision Theory based on kinetic theory fraction of collisions that are effective in

causing reaction

Transition-State Theory based on stat. mech. probability that a special state (transition

state) is occupied)

reaction dynamics, potential energy surfaces

Goal of Statistical Mechanics: describe macroscopic bulk Thermodynamic

properties in terms of microscopic atomic and molecular properties.


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Calculation ofProperties

Statistical Mechanics describes macroscopic bulk Thermodynamic properties in terms of microscopic atomic and

molecular properties. These microscopic properties are generally measured by spectroscopy.

Macroscopic: U, H, A, G, S, , p, V, T, CV, Cp

Microscopic: N particle monatomic gas


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Calculation of macroscopic properties

How d o w e calculate a macroscopic p roper ty, which is co nstant in t ime, from

a microscop ic prop erty that f luc tuates in t ime?Example: Pressure, which is a macroscopic property that arises from the microscopic

impulses of each molecule impacting the vessel's walls. The positions and velocities of

each molecule change on 10-1010-13s time scale (the duration of a collision)!

Alternative: Ensemble Average !

ENSEM LES An ensemble is a collection of all microstates of a system, consistent with the

constraints with which we characterize a system macroscopically.


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Distribution of molecular states


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ENSEM LES An ensemble is a collection of all

microstates of a system, consistent with theconstraints with which we characterize a system

macroscopically.

For example, a collection of all possible states of

the 1023molecules of gas in the container of volume

V temperature T is a statistical mechanicalensemble.

MICROCANONICAL ENSEMBLE: CONSTANT U,V,N

CANONICAL ENSEMBLE: CONSTANT N,V,T

Statistical Ensembles

GRAND CANONICAL ENSEMBLE: CONSTANT ,V,T


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Distribution of molecular states

Let, N no. of particles is distributed in various energy lavels.

No. of molecules: n0 n1 n2 n3 .... In Energy levels: E0 E1 E2 E3.Then, instantaneous configuration: {n0, n1, n2, n3, ...}

i in

N

nnn

NW

!

!

!!!

!

210

No. of ways it happens,

Weight of the

Configuration (W):

Examples:

For {5,0,0}: W = 1

For {3,2,0}: W = 10

(shown in fig.)

Most Probable configuration has the highest W.


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For system containing a large no. of units

(macroscopic system) , one configuration will

have vastly more associated permutations than

any other configuration. This configuration will

be the only one that is observed to an

appreciable extent.

j jWiW

iP

Specification of state of the system

tossing of coin

Describing the outcome of each experiment

10100


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The most probable / dominating distribution

i ii ii nNEnE ,

Most probable distribution

have largest W

For Constant E and N:

Boltzmann Distribution:

/kT

i

E

ePi

Pi: fraction of molecule in state i W = 181180 858 78 12870

= N! / (N1! N2! )


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Probability of microstates


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Probability of microstates

The probability of each microstate in the canonical ensemble (constant N,V, T )

is proportional to the exponential of the energy divided by the temperature.

In order to find the absolute probability of each microstate, we need to make

sure that the sum of all the probabilities is one. The normalization constant

for this is called the "canonical partition function," Q.

The summation over microstates is performed over all energies andparticle positions.


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Partition function

Once the partition function is defined, the probability of each microstate can

now be written explicitly:

Therefore, in the canonical ensemble, a general property F is given by


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The ensemble average

functionpartitioncanonicalthe

/;

:;/1 Z

kTi

EeZ

Z

iE

eP

kT

ii

i iAiPAA:ApropertyanyofAverage

At T = 0, Ei/kT = => Z= 1

At very large T, Ei/kT = 0 => Z 1+ 1+. = At intermediate T (let kT >> E1, E2) => Z~ 1+ 1+ 1 +0 +. = 3

1i iPClearly,

Z gives the number of thermally accessible states at the temperature of

interest.


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The Partition Function, examples

Vibrational:

Zv= 1 / [1exp(- h/kT)]

kTE

ekTE

ekTE

e

i

kTi

EeZ

/2

/1

/0

/

Z

iE

e

iP


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The Partition Function, examples

Vibrational:

Zv= 1 / [1exp(- h/kT)]

Translational:

ZT= (2mkT)3/2V / h3

Rotational:

ZR= kT / hB, : symmetry no.= 1 for unsymm. Lin. Rotor (HCl)

= 2 for symm. Lin. Rotor (H2, CO2)

Electronic:ZE is no t available in closed form

Molecular:

Z = ZTZRZVZEZS

kTE

ekTE

ekTE

e

i

kTi

EeZ

/2

/1

/0

/


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Thermodynamic properties

Z

eE

Epi

U i

iE

i

ii

Internal Energy:

BNVi

iE

BNVBNVi

iE

iE

ei

Ze

eE

i

Since

,,,

,

BNVBNVBNV T

ZkT

ZZ

ZU

,

2

,,

lnln1

Pressure: iiPpi

P

B

B

Nii

irev

Ni

VEP

dVPdw

dVVEdU

)/(

adiabatic)e(reversibl

)/(

BNTBNT V

Z

V

Z

ZP

,,

ln11


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Thermodynamic properties

BCNVTB

B

BNV

BNT

eZ

N

ZRT

ZkTA

ZkTUS

T

ZkTU

V

ZP

i

Ei

,,

,

2

,

ln

ln

ln/

ln

ln1

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