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Chemistry 125: Lecture 37
December 8, 2010
Statistics / Equilibrium / Rate:
Boltzmann Factor, Entropy, Mass
Action, Transition State Theory
The values of bond dissociation energies and average bond energies, when corrected for certain
“effects” (i.e. predictable errors) can lead to understanding equilibrium and rate processes through
statistical mechanics. The Boltzmann factor favors minimal energy in order to provide the largest number
of different arrangements of “bits’ of energy. The slippery concept of disorder is illustrated using Couette
flow. Entropy favors “disordered arrangements” because there are more of them than there are of
recognizable ordered arrangements. The Law of Mass Action explains the exponentiation of
concentration in equilibrium equations. Equilibrium ideas allow understanding reaction rates by using
transition state theory instead of considering all trajectories on the potential energy surface. This
semester’s study of structure and energy leads to next semester’s study of reaction mechanisms
and synthesis.
For copyright
notice see final
page of this file
Exam Help Sessions
Friday, Dec. 10, 10:30-11:30 am (here)
Sunday, Dec. 12, 3-5 pm
Monday, Dec. 13, 7-9 pm
Wednesday, Dec. 15, 8-10pm
Thursday, Dec. 16, 7-9 pm
Bond Strengths in CH4
Heat of Atomization of CH4 = 397.5 kcal mol-1
(from heat of combustion, Chupka, etc.)
Average Bond Energy = 397.5 / 4 = 99.4 kcal mol-1
Individual Bond Dissociation
(from spectroscopy etc.)
Energies
CH3 -H
104.99 ± 0.03
CH2-H
110.4 ± 0.2
CH-H101.3 ± 0.3
C-H
Sum
80.9 ± 0.2
No individual bond
actually equals the
“average” C-H bond.
(because of changes in
hybridization, etc.)
from Barney Ellison
(good experiments!)
397.5 ± 0.6
& his friends
ve Bond Energies
“2nd C-O bond” 90-93 kcal/mole!
(Carbonyl group very “stable”)
“3rd bond” 54 kcal/mole
“2nd C-C bond” 63 kcal/mole
(p bond even weaker, since
sp2-sp2 s bond is stronger)
How good are these average bond energies
for calculating "Heats of Atomization”
for other molecules?
From Streitwieser, Heathcock, & Kosower
HAtomization by Additivity of
Average Bond Energies?
Seems Pretty Impressive!
Ave. Bond Energy (kcal/mole)
83 99 146 86 111
C-C C-H C=C C-O O-H
S Bond
Energies
H
Error
-4.3
-0.8
-5.9
-0.4
atomization kcal/mole
Error
%
Ethene
0
4
1
0
0
542
537.7
c-Hexane
6
12
0
0
0
1686
1680.1
c-Hexanol
6
11
0
1
1
1784
1778.6
-5.4
-0.3
a-Glucose
5
7
0
7
5
2265
2248.9 -16.1
-0.7
12.5
-29.5
-68.4
-240.7
How accurate must you be to be useful? ~10× better than
Kcalc = 10-(3/4)(Hcalc)
Kcalc = 10-(3/4)(Htrue + Herror)
Kcalc = Ktrue  10-(3/4)(Herror)
kcal error not % error
composition
determines K error factor
To keep error less than 10
need <1.3 kcal/mole error!
Ketone
"Enol"
O
O
ve Bond Energies
H
H
C
C
C
H
H
H H
H H
H
C
C
C
H
H
H
C=O 179
C-O
86
Bonds that change C-C 83
C=C 146
(the others should cancel
in taking the difference)
C-H
99
O-H 111
sum 361
sum 343
Can one sum bond
energies
to get
-(3/4)
18
-13.5
Kcalc = 10
= 10
accurate"Heats of Atomization"?
Kobs = 10-7 = 10-(3/4) 9.3
Ketone
Why is Enol
9 kcal/mole
"Too" Stable?
C(sp2)-H
stronger than
C(sp3)-H
(they shouldn’t actually cancel)
"Enol"
••
O
O
H
H
C
C
C
H
H
H H
H H
H
C
C
C
H
H
H
C=O 179
C-O
86
C-C
83
C=C 146
+O H
"Resonance
C-H
99
O-H 111
Stabilization”
sum 361
sum C 343
from
H
H
Intramolecular
Kcalc = 10-(3/4) 18 =C10-13.5 C
HOMO-LUMO -7 H-(3/4)H9.3 H
Kobs = 10 = 10
Mixing
“Constitutional Energy” from
bond additivity needs correction
*
for effects such as:
• Resonance
••
HO C
H H
H
vs.
C
C
sp
sp
H
H
H
(HOMO/LUMO)
• Hybridization
• Strain
CH2
2
3
* Polite name
for error in
simplistic scheme
Energy determines what can
happen (equilibrium)
-E/kT
e
K=
-(3/4)E
= 10
kcal/mole
@ room Temp
and how fast (kinetics)
‡
13
-E
/kT
10 e
k (/sec) =
13-(3/4)E
= 10
‡
kcal/mole
@ room Temp
What's so great
about low energy?
Statistics
Gibbs 1902
1902
Exponents &
Three Flavors of Statistics
1) The Boltzmann Factor
2) The Entropy Factor
3) The Law of Mass Action
OnConsidered
the Relationship
the
between
the Second
Law
implications
of random
of energyand
ofdistribution
Thermodynamics
among realCalculation
atoms.
Probability
regarding
the oflaws
“I am conscious
beingof
Thermal
Equilibrium
only
an individual
struggling
Ludwig Boltzmann
1844 - 1906
weakly against the stream of
time. But (1877)
it still remains in
my power to contribute in
such a way that when the
theory of gases is again
revived, not too much will
have to be rediscovered.”
S = k ln W
Random Distribution of 3 “Bits” of Energy
among 4 “Containers”
How many
“complexions”
have N bits
in the first
container?
N
3
2
#
1
3
Random Distribution of 3 “Bits” of Energy
among 4 “Containers”
How many
“complexions”
have N bits
in the first
container?
N
3
2
1
#
1
3
6
Random Distribution of 3 “Bits” of Energy
among 4 “Containers”
How many
“complexions”
have N bits
in the first
container?
N
3
2
1
0
#
1
3
6 10
30 in 20
30 bits of energy
in 20 molecules
3 bits of energy
in 4 “molecules”
N
3
2
1
0
#
1
3
6 10
Eave = 1/2 kT
Boltzmann Constant
1.987 cal/moleK
(N)
(E)
(Note: temperature is average energy)
-E/kT
e
E
If all “complexions” for a given Etotal are equally likely,
Boltzmann
limit
shifting energy
to any oneshowed
degree ofExponential
freedom of any one
molecule isfor
disfavored.
reducing the energy
available
lots of By
infinitesimal
energy
bits
elsewhere, this reduces the number of relevant complexions.
Exponents &
Three Flavors of Statistics
1) The Boltzmann Factor
2) The Entropy Factor
3) The Law of Mass Action
Disorder and Entropy
"It is the change from an ordered
arrangement to a disordered arrangement
which is the source of the irreversibility.”
The Feynman Lectures on Physics, Vol. I, 46-7
Disorder and Entropy
Which is more ordered?
Disorder,
Reversibility,
& Couette Flow
Click for webpage and "Magic" movie
Couette
Flow
Syrup
If disorder
is in the eye of
the beholder, how
can it measure a
fundamental
property?
Top
View
Ink
line
The rotated state only seemed to be disordered.
“A disordered arrangement”
seems to be an oxymoron.
The situation favored at equilibrium has
particles that have diffused every whichaway.
“A disordered arrangement” is code for a
collection of random distributions whose
individual structures are not obvious.
It is favored at equilibrium, because it
includes so many individual distributions.
Entropy is Counting in Disguise.
Free Energy & 1.377 entropy units
K = e-G/RT
e-(H - TS)/RT
e-H /RT e TS/RT
-H
/RT
S/R
e
e
e-H /RT e R ln 2/R
e-H /RT e ln 2 Y
e-H /RT x 2
1.377 e.u. (R ln 2)
is a common S.
e.g.
Conclusions:
difference in
1.377 e.u. just means
entropy
a factor between
of two.
gauche
and
K depends
on anti
T
butane
because
of
X
X S.
H, not
Y
X
G (and S) sometimes
obscure what is
fundamentally simple.Y
Anti
Gauche Gauche
Exponents &
Three Flavors of Statistics
“there’s a divinity
that shapes our ends”
-H/RT
R
1)Same
The
Boltzmann
Factor
e
thing:
Hamlet V:2
k is per individual molecule
R is per mole (= k  NA)
from counting random arrangements
of a fixed number of energy bits
k =W
2) The Entropy Factor eTS/kT
from counting W, the number of
molecular structures being grouped
3) The Law of Mass Action
Cyclohexane Conformers
10.8
7.0
kcal/mole
5.5
few
quantum
states
0
few
"structures"
Chair
(stiff)
many "structures"
many quantum states
Twist-Boat
(flexible)
few
quantum
states
Both classical and quantum views suggest a statistical
"entropy" factor (of ~7) favoring twist-boat.
This reduces the room-temperature Boltzmann "enthalpy" bias
of 10-(3/4) 5.5 (= 14,000) in favor of chair to about 2,000.
few
"structures"
Chair
(stiff)
Experimental Entropy
Although we discuss entropy theoretically
(in statistical terms), physical chemists
can measure it experimentally.
The entropy of a perfectly ordered crystalline
material at zero Kelvin is zero ( ln 1 ).
As the material is warmed it gains entropy in
increments of (Heat Absorbed)/Temperature.
S = H/T
“Floppy” molecules with closely spaced energy levels
absorb more energy, and at lower temperatures, and thus
gain more S on warming. Cf. Ethane rotation - Lecture 33
K = e-G/RT
Exponents &
Three Flavors of Statistics
1) The Boltzmann Factor e -H/RT
from counting random arrangements
of a fixed number of energy bits
2) The Entropy Factor eTS/kT = W
weighted
from counting W, the number of
quantum states being grouped
3) The Law of Mass Action
from counting molecules per volume
Law of Mass Action
Late 1700s : Attempts to assemble.
hierarchy of “Affinities”
Early 1800s : Amounts [concentration] can
shift reaction direction away.
from “affinity” prediction. …
Mid 1800s : Equilibrium “K” as balance of
forward and reverse rates...
Law of Mass Action
[concentration]
2A
[A2]
2
[A]
A2
= K
[A2] = K [A] 2
Where does
the exponent
come from?
Randomly
Distributed
“Particles”
# Particles # Dimers
50
1
100
9
150
19
200
35
250
59
# of Dimers
Randomly
Distributed
“Particles”
Increasing concentration
increases both the number
number
of units and the fraction
fraction
of units that have near
neighbors.
[D] = K
# of Particles
# Particles # Dimers
[P] 2
50
1
100
9
150
19
200
35
250
59
Equilibrium, Statistics & Exponents
Particle Distribution : Law of Mass Action
[A2]
= K
2
[A]
Energy Distribution : H , Boltzmann Factor
-H/RT
Ke
Counting Quantum States :
S
S/R
Ke
Free energy determines what
can happen (equilibrium)
-G/RT
e
K=
-(3/4)G
= 10
Energy &
Entropy
kcal/mole
@ room Temp
But how quickly
will it happen? (kinetics)
Visualizing Reaction
Classical Trajectories
&
The Potential Energy Surface
Time-Lapse
“Classical”
(Molecular Mechanics)
faster
rotating Too
rapidly
rotating
&
Complicated
slowly
vibrating
(for our purposes)
Trajectory for
non-reactive
collision
of 13 atoms
6 molecules
40 Dimensions
slower
(3n + time)
by E. Heller
Potential
Energy
Rolling Ball
Maps A-B
Vibration
A-B Distance
Potential Energy
“Surface” for Stretching
Diatomic Molecule A-B
Plateau
Pass
+
(Transition State
or Transition Structure)
Potential Energy
Surface
for Linear *
Triatomic A-B-C
* So 2-D specifies structure
Valley
Cliff
Vibration of A-B
with distant
C spectator
Potential Energy
Surface
for Linear
Triatomic A-B-C
Vibration of B-C
with distant
A spectator
Slice and
fold back
Unreactive
Trajectory:
(A bounces off
vibrating B-C)
Potential Energy
Surface
for Linear
Triatomic A-B-C
C flies away
from
vibrating A-B
Reactive
Trajectory
Potential Energy
Surface
for Linear
Triatomic A-B-C
“classical” trajectory
(not quantum)
A approaches
non-vibrating B-C
H3 Surface
Henry Eyring
(1935)
Transition State
(“Lake Eyring”)
Crazy angle of axes means that classical trajectories can be modeled by rolling marble.
H + H-Br
“I wanted
to catch a
little one”
John McBride (1973)
Studying Lots of
Random Trajectories
Provides Too Much Detail
Summarize Statistically
with Collective
Enthalpy (H) & Entropy (S)
“steepest descent”
path
(not a trajectory)
Slice along
this path,
then flatten
and tip up
to create…
Transition “State”
G
Starting
Materials
Products
“Reaction Coordinate” Diagram
(for one-step transfer of atom B)
Not a realistic trajectory, but rather
a sequence of three species
each with H and S, i.e. Free Energy (G)
Free Energy determines
what can happen (equilibrium)
-G/RT
Since the transition state
e
(universal)
K=
-(3/4)G
= 10 the velocity is not universal,
Velocity and
of ts theory
is not truly in equilibrium
kcal/mole
with starting materials,
and
@ room Temp
the theory is approximate.
how rapidly (kinetics)
‡
13
-G
/RT
10 e
k (/sec) =
13-(3/4)G
= 10
‡
Amount
of ts
kcal/mole
@ room Temp
(1953)
(2007)
(1959)
Physical
Organic
Chemistry
Paul D. Bartlett
1907-1997
http://osulibrary.oregonstate.edu/specialcollections/coll/pauling/bond/audio/1997v.1-bookdunitz.html
1939
Jack Dunitz: At the time when I was reading that book I was
wondering whether chemistry was really as interesting as I had
hoped it was going to be. And I think I was almost ready to give
it up and do something else. I didn't care very much for this
chemistry which was full of facts and recipes and very little
thought, very little intellectual structure. And Pauling's book gave
me a glimpse of what the future of chemistry was going
to be and particularly, perhaps, my future.
The Chemical Bond
Is there an Atomic Force Law?
Feeling & Seeing Molecules and Bonds
Understanding Bonding & Reactivity
through H = E
How chemists learned to treasure
Composition, Constitution,
Configuration, Conformation
and Energy
Some
Big Questions:
The Chemical
Bond
Is there an Atomic Force Law?
How does science know?
Feeling & Seeing Molecules and Bonds
Compared to what?
Understanding Bonding & Reactivity
Were
chemical
through
H =bonds
E
discovered
or invented?
How
chemists learned
to treasure
Composition, Constitution,
WouldConfiguration,
we even have
chemical bonds
Conformation
without our particular
chemical history?
and Energy
End of Lecture 37
Dec. 8, 2010
Copyright © J. M. McBride 2009, 2010. Some rights reserved. Except for cited third-party materials, and those used by
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J. M. McBride, Chem 125. License: Creative Commons BY-NC-SA 3.0