statmech-1
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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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