ChE 551 Lecture 14
Potential Energy Surfaces
1
Last Time Collision Theory


Assumes reactions occur whenever
reactants collide
Key equations
k 0 = v A  BC  cA  BC
(7.26)
v ABC
Å  T 
 2.52  10


sec  300K 
13
(7.29)
1/2
 1AMU 


  ABC 
1/2
2
Preexponentials Really Used By The Same
Order As Collision Theory?
Table 7.2 a selection of the preexponentials reported by Wesley [1980]
Reaction
H+C2H6C2H5+H2
Preexponential
Å3/molecule Sec
1.6  1014
Reaction
Preexponential
Å3/molecule Sec
2.5  1013
O+C2H6OH+C2H5
H+CHH2+C
1.1  1012
O+C3H8(CH3)2CH+OH
H+CH4H2+CH3
1  1014
O2+HOH+O
1.4  1010
1.5  1014
O+H2OH+H
1.8  1013
OH+OHH2O+O
1  1013
O+OHO2+H
2.3  1013
OH+CH4H2O+CH3
5  1013
O+CH4CH3+OH
2.1  1013
OH+H2COH2O+HCO
5  1013
O+CH3H+CH3O
5  1013
OH+CH3H+CH3O
1  1013
O+HCOH+CO2
5  1012
OH+CH3H2O+CH2
1  1013
3
Comparisons Between Collision Theory And
Experiments
Table 7.3 Preexponentials calculated from equation (7.30) for a number of reactions
compared to experimental data.
Reaction
H  C2H6 C2H5  H2
H  CH  H 2  C
O  C 2 H 6  OH  C 2 H 5
OH  OH  H 2 O+ O
H  O 2  OH  O
Calculated
Calculated Preexponential
Preexponential
assuming bcoll=covalent radius
assuming bcoll=van
Der Waals radius
Experimental
Å3/molec sec
6.2 1014
Å3/molec sec
2.0 1014
Preexponential
4 1014
2.0 1014
1.1 1012
1.9 1014
7.6 1013
2.5 1013
1.25 1014
5.8 1013
1 1013
4.0 1014
2 1014
1.5 1014
1.6 1014
4
Why Does Collision Theory Fail
For Reaction 7.30?
Reaction 7.30 requires a special collision geometry:
•
CH3CH 2CH3 +O:  CH3 C HCH3 +•OH
(7.32a)
CH3CH 2CH3 +O:  CH 2CH 2CH3 +•OH (7.32b)
Configurations = e
S
kB
(7.33)
e
ΔS†
kB
configurations which lead to reactions
=
average number of configurations of the reactants
(7.34)
5
Energy
Next Few Lectures Will Cover Conventional
Transition State Theory
A
‡
Barrier
Reactants
Products
Reaction Cordinate
Figure 7.5 Polanyi’s picture of excited molecules.


Model reaction as motion over a
potential energy surface
Use stat mech to estimate key terms
6
Objective For Today

Overview of Potential Energy Surfaces



What do they look like
How to interpret the plots
How to interpret motion
7
Figure 7.6 PE Surface For
H + C2H6 →H2 + C2H5
transition
state
Energy
X
2.5
transition
state
2
*
H-H DISTANCE (ANGSTROMS)
3
C-H Dis
ta
nce
HH
X
Dis
tan
ce
Y
1.5
1
1
1.5
2
2.5
Y
3
C-H Distance (Angstroms)
8
Potential Energy Surfaces
For H+C2H6  H2 + C2H5, 27
degrees of freedom since 9 atoms
 3 translations 3 rotations, 21
others
Energy
ce
Y
X
Dis
tan

Saddle
point
C-H Dis
ta
H

Potential energy surface is
defined as the energy of the
system as a function of the
coordinates of all of the atoms in
a reaction
Many coordinates:
nce
H-

9
Simplified Potential Energy Surfaces

Textbook examples also usually
assume that bond angle
dependence is small
Y
ce
For A+BC  AB + C, 9 degrees of
freedom since 3 atoms
 3 translations 3 rotations, 3
others (AB distance, BC distance
and ABC bond angle).
Energy
X
Dis
tan

Saddle
point
C-H Dis
H

Only consider bonds that break
and form
Treat ligands as united atoms
tance
H-

10
Simplified Potential Energy Surfaces
Simplified example: analytical PE
surface
18.0
16.0
14.0
12.0
10.0
Energy
20.0
8.0
1
6.0
11
4.0
21
2.0
31
S31
S28
S25
S22
S19
S16
S13
S7
S10
S4
0.0
S1

11
PE Surface
20.0
18.0
16.0
14.0
12.0
10.0
8.0
1
6.0
7
13
4.0
19
2.0
25
31
Spreadsheet
S31
S29
S27
S25
S23
S21
S19
S17
S15
S13
S11
S9
S5
S3
S1
37
S7
0.0
12
Numerical Values
r2\r1
0.5
0.7
0.9
1.1
1.3
1.5
1.7
1.9
2.1
2.3
2.5
2.7
2.9
3.1
3.3
3.5
3.7
3.9
4.1
0.5
20.0
20.0
20.0
20.0
20.0
20.0
20.0
19.1
15.5
12.8
10.7
9.1
7.9
7.0
6.3
5.7
5.3
4.9
4.6
0.70
20.0
20.0
20.0
20.0
20.0
16.8
12.4
9.2
6.9
5.2
4.0
3.1
2.4
1.9
1.5
1.2
0.9
0.7
0.6
0.90
20.0
20.0
20.0
20.0
13.8
9.7
6.9
5.1
3.9
3.1
2.5
2.1
1.8
1.6
1.5
1.4
1.3
1.2
1.2
1.10
20.0
20.0
20.0
13.1
8.8
6.3
5.0
4.2
3.8
3.7
3.6
3.6
3.7
3.7
3.8
3.8
3.9
3.9
3.9
1.30
20.0
20.0
13.8
8.8
6.3
5.2
4.9
5.0
5.2
5.6
6.0
6.3
6.6
6.8
7.0
7.2
7.4
7.5
7.6
1.50
20.0
16.8
9.7
6.3
5.2
5.1
5.7
6.4
7.2
8.0
8.7
9.3
9.8
10.2
10.5
10.8
11.0
11.2
11.3
1.70
20.0
12.4
6.9
5.0
4.9
5.7
6.9
8.1
9.4
10.4
11.4
12.2
12.8
13.3
13.8
14.1
14.4
14.6
14.8
1.90
19.1
9.2
5.1
4.2
5.0
6.4
8.1
9.8
11.4
12.7
13.8
14.8
15.6
16.2
16.7
17.1
17.4
17.7
17.9
2.10
15.5
6.9
3.9
3.8
5.2
7.2
9.4
11.4
13.2
14.7
16.0
17.0
17.9
18.6
19.2
19.6
20.0
20.0
20.0
2.30
12.8
5.2
3.1
3.7
5.6
8.0
10.4
12.7
14.7
16.4
17.8
19.0
19.9
20.0
20.0
20.0
20.0
20.0
20.0
2.50
10.7
4.0
2.5
3.6
6.0
8.7
11.4
13.8
16.0
17.8
19.3
20.0
20.0
20.0
20.0
20.0
20.0
20.0
20.0
Saddle Point
2.70
9.1
3.1
2.1
3.6
6.3
9.3
12.2
14.8
17.0
19.0
20.0
20.0
20.0
20.0
20.0
20.0
20.0
20.0
20.0
2.90
7.9
2.4
1.8
3.7
6.6
9.8
12.8
15.6
17.9
19.9
20.0
20.0
20.0
20.0
20.0
20.0
20.0
20.0
20.0
3.10
7.0
1.9
1.6
3.7
6.8
10.2
13.3
16.2
18.6
20.0
20.0
20.0
20.0
20.0
20.0
20.0
20.0
20.0
20.0
3.30
6.3
1.5
1.5
3.8
7.0
10.5
13.8
16.7
19.2
20.0
20.0
20.0
20.0
20.0
20.0
20.0
20.0
20.0
20.0
3.50
5.7
1.2
1.4
3.8
7.2
10.8
14.1
17.1
19.6
20.0
20.0
20.0
20.0
20.0
20.0
20.0
20.0
20.0
20.0
3.70
5.3
0.9
1.3
3.9
7.4
11.0
14.4
17.4
20.0
20.0
20.0
20.0
20.0
20.0
20.0
20.0
20.0
20.0
20.0
3.90
4.9
0.7
1.2
3.9
7.5
11.2
14.6
17.7
20.0
20.0
20.0
20.0
20.0
20.0
20.0
20.0
20.0
20.0
20.0
4.10
4.6
0.6
1.2
3.9
7.6
11.3
14.8
17.9
20.0
20.0
20.0
20.0
20.0
20.0
20.0
20.0
20.0
20.0
20.0
4.30
4.4
0.5
1.1
4.0
7.6
11.4
14.9
18.0
20.0
20.0
20.0
20.0
20.0
20.0
20.0
20.0
20.0
20.0
20.0
Spreadsheet
13
Top View
A+BC  AB + C
Reactants
S31
S28
S25
S22
BC Distance
Saddle Point
S19
S16
S10
S7
S1
37
34
31
28
25
22
19
16
13
10
7
4
1
20.0
18.0
16.0
14.0
12.0
10.0
8.0
6.0
4.0
2.0
0.0
S4
Products
S13
Spreadsheet
AB Distance
14
Barrierless Reaction
20.0
18.0
16.0
14.0
12.0
10.0
8.0
6.0
4.0
2.0
1
0.0
12
S22
S19
S16
S13
S7
S10
S4
S1
34
S25
S28
S31
-2.0
23
Spreadsheet
15
37
34
31
28
25
22
19
16
13
10
7
4
1
20.0
18.0
16.0
14.0
12.0
10.0
8.0
6.0
4.0
2.0
0.0
-2.0
Barrierless Reaction
S31
S28
S25
S22
S19
S16
S13
S10
S7
S4
Spreadsheet
S1
16
Well
37
34
31
28
25
22
19
16
13
10
7
4
1
20.0
18.0
16.0
14.0
12.0
10.0
8.0
6.0
4.0
2.0
0.0
-2.0
-4.0
-6.0
-8.0
-10.0
-12.0
-14.0
-16.0
-18.0
Attractive Interaction
S31
S28
S25
S22
S19
S16
S13
S10
S7
S4
Spreadsheet
S1
17
PE With Van der Waals Well
S31
Complex
S28
S25
S22
Saddle
Point
S19
S16
Complex
S13
S10
S7
Spreadsheet
S1
37
34
31
28
25
22
19
16
13
10
7
4
1
20.0
18.0
16.0
14.0
12.0
10.0
8.0
6.0
4.0
2.0
0.0
-2.0
-4.0
S4
18
PE For Series Reactions
S31
S28
Saddle
Point
S25
S22
S19
Intermediate
S16
Saddle
Point
S13
S10
S7
S1
37
34
31
28
25
22
19
16
13
10
7
4
1
20.0
18.0
16.0
14.0
12.0
10.0
8.0
6.0
4.0
2.0
0.0
-2.0
-4.0
-6.0
-8.0
-10.0
-12.0
S4
Spreadsheet
19
Why Do Plots Look The Way They Do?


Balance between attractive forces and
Pauli repulsions
Attractive forces



Van der Waals Interactions (Correlation)
Bond formation
Repulsive forces

Pauli repulsions (quantized electron-electron
repulsions)
20
Ne-Ne Interaction
Separated
Neons
Ne
Ne
Ne
Ne
Ne-Ne
Collision
Ne
AntiBonding
Ne
Bonding
21
100
10
50
5
Ne 2
0
0
-50
-5
F2
-100
Energy, Kcal/mole
Energy. Kcal/mol
Ne-Ne Potential
-10
-150
-15
0
1
2
3
4
5
Distance Angstroms
22
F-F interaction
Separated
Fluorines
F
F
F2
F
F
Pure Quantum Effect
23
100
10
50
5
Ne 2
0
0
-50
-5
F2
-100
Energy, Kcal/mole
Energy. Kcal/mol
F-F Potential
-10
-150
-15
0
1
2
3
4
5
Distance Angstroms
24
Morse Potential
100
10
50
5
Ne 2
0
0
-50
-5
F2
-100
Energy, Kcal/mole
Where w=bond energy
r=distance between atoms
ro=Equilibrium distance
X=range parameter
Energy. Kcal/mol
V(r)=W(exp(-2x(r-ro)-2exp(-x(r-ro)))
-10
-150
-15
0
1
2
3
4
5
Distance Angstroms
25
Cl + F2 Interaction
Separated
Reactants
During
Reaction
Cl
F
F
Non-bonding
Lobe
Fluorine-Fluorine
Bond
Cl
F
Non-bonding
Lobe
F
Fluorine-Fluorine
Bond
26
Cl + F2 Potential
Energy
RClF
RFF
27
Interaction During
H + C2H6 →CH4 + CH3
Bonds Break:
Transition
State
Nonbonding
Lobe Pushs
Into C-C Bond
Reactants Come
Together,
Nonbonding
Lobe Distorts
H
CH3 CH 3
H
C C
Separated Reactants
Reactio
n Prog
ress
New Bonds
Form
C-C bond
Non-bonding Lobes
Reactants
Begin To
Separate
CH4
CH3
Products
28
Analytical PE Surface
Table 7.G.1 The module used to calculate the function in
equation 7.G.1
Public Function v(r1, r2, r0, a, w, vp, wa, hr) As Variant
v = w * (Exp(-2 * a * (r1 - r0)) - 2 * Exp(-a * (r1 - r0)))
v = v + (w + hr) * (Exp(-2 * a * (r2 - r0)) - 2 * Exp(-a * (r2 - r0)))
v = v + vp * Exp(-a * (r1 + r2 - 2 * r0))
v=v+w
v = v + wa * Exp(-4 * a * a * ((r1 - r0) ^ 2 + (r2 - 3 * r0) ^ 2))
v = v + wa * Exp(-4 * a * a * (((r1 - 3 * r0) ^ 2) + ((r2 - r0) ^ 2)))
If (v > 20 + Abs(hr)) Then
v = 20 + Abs(hr)
End If
End Function
29
Summary




PE surface plot of energy vs internal
coordinates of reactive complex.
Attractive interaction due to bonding and
Van der Waals.
Repulsions due to Pauli repulsions
(quantized electron-electron repulsions).
Net yields saddle point if reaction not too
exothermic.
30
Question

What did you learn new in this lecture?
31