ChE 551 Lecture 11

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ChE 551 Lecture 11

• Review Of Statistical Mechanics

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Last Time We Started Stat Mech To Estimate
Thermodynamic Properties

• All thermodynamic properties are averages.
• There are alternative ways to compute the
  averages state averages, time averages, ensemble
  averages.
• Special state variables called partition
  functions.

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Properties Of Partition Functions

• The partition functions are like any other state
  variable.
• The partition functions are completely defined if
  you know the state of the system.
• You can also work backwards, so if you know the
  partition functions, you can calculate any other
  state variable of the system.

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Properties Of Partition Functions

• Assume m independent normal modes of a molecule
• qmolecular partition function
• qnpartition function for an individual mode
• gndegeneracy of the mode

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How Many Modes Does A Molecule Have?

• Consider molecules with N atoms
• Each atom can move in x, y, z direction
• ? 3N total modes
• The whole molecule can translate in x, y, z
• ? 3 Translational modes
• Non linear molecules can rotate in 3 directions
• ?3 rotational modes
• 3N-6 Vibrational modes
• Linear molecules only have 2 rotational modes
• 3N-5 vibrational modes

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Equations For Molecular Partition Function
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Equations For The Partition Function For
Translational, Rotational, Vibrational Modes And
Electronic Levels
qt?1-10?/ax
q
1
qt3?106-107
qr2?102-104
Where Sn is symmetry number
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Key Equations Continued



Rotation of a nonlinear
qr3?104-105

molecule with a
?
?
3
4
5
q
10
10

r
moment of inertia of I
a
,
I
, I
, about three
b
c
orthogona
l axes




qv?1-3

Vibration of a harmonic
?
?
q
1
3
v
oscillator when energy
levels are measured

where ?
is the
relative to the harmonic
vibrational frequency
oscillators zero point
energy



?
?
?
E
Electronic Level

?
?
?
?
q
exp

e
?
?
?
?
??
T
q
E)
exp(
?
(Assuming That the
B
e
Levels Are Widely
Spaced)

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Table 6.7 Simplified Expressions For Partition
Functions
Type of Mode Partition Function Partition Function after substituting values of kB and hp
Average velocity of a molecule
Translation of a molecule in thre dimensions (partition function per unit volue
Rotation of a linear molecule
Rotation of a nonlinear molecule
Vibration of a harmonic oscillator
6
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Example 6.C Calculate The Partition Function For
HBr At 300K
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How Many Modes In HBr

• Total Modes 6
• Translations 3
• Rotations 2
• Leaves 1 vibration

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The Translational Partition Function
From Pchem Where qt is the translational
partition function per unit volume, mg is the
mass of the gas atom in amu, kB is Boltzmanns
constant, T is temperature and hp is Planks
constant
6.C.1
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Simplification Of Equation 6.3.1
(6.C.2)
(6.C.3)
Combining 6.C.2 and 6.C.3
3
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Solution Continued

• Equation 6.C.4 gives qt recall mg81 AMU, T300K

3
3
(6.C.5)
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The Rotational Partition Function

• From P-chem for a linear molecule

kB
(6.C.6)
kB
Derivation
Algebra yields
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Derivation Of Simplified Function

(6.C.7)
(6.C.8)
(6.C.9)
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Calculation of Rotation Function Step Calculate
I

• From P-chem
• Where

(6.C.10)
?
(6.C.13)
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Step 2 Calculate qr2

• Substituting in I from equation (6.C.13) and Sn
  1 into equation 6.C.9 yields

(6.C.14)
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The Vibrational Partition Function

• From Table 6.6

(6.C.15) where qv is the vibrational partition
function, hp is Planks constant ? is the
vibrational frequency, kB is Boltzmanns constant
and T is temperature. Note
Derivation
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Simplified Expression For The Term In The Exponent
(6.C.16)
(6.C.17)
Therefore
(6.C.18)
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Evaluation Of h? For Our Case
(6.C.19)
Substituting
(6.C.19)
(6.C.15)
(6.C.20)
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Summary

• qT843/ , qr24.4 qv1
• Rotation and translation much bigger than
  vibration

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Example Calculate The Molecular Velocity Of HBr

• Solution

Derivation
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Derivation Of Expression For Molecular Velocities

• Use the classical partition function (replace
  sums by integrals). The expectation value of the
  molecular velocity, ?v?, is given by

(6.B.1)
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Solution Cont

• Consider a single molecule whose energy is
  independent of position. Substituting momentum p
  mass times the velocity, and canceling out all
  of the excess integrals yields

(6.B.2)
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Next Substitute For U

(6.B.3)
(6. B.5)
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Performing The Algebra Noting ? 1/kBT Yields
P-Chem expression for molecular velocity
(6. B.8)
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Simplified Expression

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Next Derive Adsorption Isotherm

• Consider adsorption on a surface with a number of
  sites
• Ignore interactions
• Calculate adsorption concentration as a function
  of gas partial pressure

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Solution Method

• Derive an expression for the chemical potential
  of the adsorbed gas as a function of the gas
  concentration
• Calculate canonical partition function
• Use AkBT ln(Qcanon) to estimate chemical
  potential
• Derive an expression for the chemical potential
  of a gas
• Equate the two terms to derive adsorption
  isotherm

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Solution Step 1 Calculate The Canonical
Partition Function

• According to equation (6.72),
• qPartition for a single adsorbed molecule on a
  given site
• gathe number of equivalent surface
  arrangements.

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Step 1A Calculate ga

• Consider Na different (e.g., distinguishable)
  molecules adsorbing on So sites. The first
  molecule can adsorb on So sites, the second
  molecule can adsorb on (So-1) sites, etc.
  Therefore, the total number of arrangements is
  given by

(6.83)
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Next Now Account For Equivalent

• If the Na molecules are indistinguishable,
  several of these arrangements are equivalent.
• Considering the Na sites which hold molecules.
  If the first molecule is on any Na of these
  sites, and the second molecule is on any Na-1 of
  those sites, etc., the arrangement will be
  equivalent. The number of equivalent
  arrangements is giving by Na(Na-1)(Na-2)1Na!
• (6.84)
• Therefore, the total number of inequivalent
  arrangements will be given by
• (6.85)

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Step 1b Combine To Calculate

• Combining equations (6.72) and (6.85)
• (6.86)
• where qa is the molecular partition function for
  an adsorbed molecule.

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Step 2 Calculate The Helmholtz Free Energy

• The Helmholtz free energy at the layer, As is
  given by
• (6.87)
• Combining equations (6.86) and (6.87) yields
• (6.88)

kB
kB
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Use Stirlings Approximation To Simplify Equation
(6.88).
For any X. If one uses equation (6.89) to
evaluate the log terms in equation (6.88), one
obtains
kB
(6.90)
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Step 3 Calculate The Chemical Potential Of The
Adsorbed Layer

• The chemical potential of the layer, µs is
  defined by
• (6.91)
• substituting equation (6.90) into equation (6.91)
  yields
• (6.92)

kB
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Step 4 Calculate The Chemical Potential For The
Gas

• Next, lets calculate µs, the chemical potential
  for an ideal gas at some pressure, P. Lets
  consider putting Ng molecules of A in a cubic box
  that has longer L on a side. If the molecules
  are indistinguishable, we freeze all of the
  molecules in space. Then we can switch any two
  molecules, and nothing changes.

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Step 4 Continued

• There are Ng! ways of arranging the Ng molecules.
  Therefore, 
• (6.93)
•  substituting equation (6.93) into equation
  (6.91) yields
•  
• (6.94)
•  where Ag is the Helmholtz free energy in the gas
  phase, and qg is the partition function for the
  gas phase molecules.

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Lots Of Algebra Yields
kB
(6.95)
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Step 5 Set ?g ?a To Calculate How Much Adsorbs

• Now consider an equilibrium between the gas phase
  and the adsorbed phase. At equilibrium
• (6.96)
• substituting equation (6.92) and (6.95) into
  equation (6.96) and rearranging yields
• Taking the exponential of both sides of Equation
  (6.97)

(6.97)
(6.98)
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Note That Na Is The Number Of Molecules In The
Gas Phase

• Na is the number of adsorbed molecules and
  (So-Na) is the number of bare sites.
  Consequently, the left hand side of equation
  (6.98) is equal to KA, the equilibrium constant
  for the reaction
• Consequently

(6.99)
(6.100)
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If we want concentrations, we have to divide all
of the terms by volume
Partition function per unit volume
Memorize this equation
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Table 6.7 Simplified Expressions For Partition
Functions
Type of Mode Partition Function Partition Function after substituting values of kB and hp
Average velocity of a molecule
Translation of a molecule in thre dimensions (partition function per unit volue
Rotation of a linear molecule
Rotation of a nonlinear molecule
Vibration of a harmonic oscillator
6
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Summary

• Can use partition functions to calculate
  molecular properties
• Be prepared to solve an example on the exam

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Question

• What did you learn new today