CHBE 452 Lecture 15
Theory Of Activation Barriers
1
Last Time Found Barriers To
Reaction Are Caused By





Uphill reactions
Bond stretching and distortion
Orbital distortion due to Pauli repulsions
Quantum effects
Special reactivity of excited states
2
Today Models

Polanyi


Marcus


Bond stretching + Linearization
Bond stretching + parabola
Blowers Masel

Bond streching + Pauli repulsions
3
E
Energy
Energy
Polayni's Model
a
Reactants
Products
B
rBH
H
rRH
R
rBH
Figure 10.3 A diagram illustrating how an upward
displacement of the B-H curve affects the
activation energy when the B-R distance is fixed.
B
rBH
H
rRH
R
rBH
Figure 10.4 A diagram illustrating a case where
the activation energy is zero.
4
Derivation Of Polayni Equation
Actual
Potential
BC
AB
Linear
Fit
E 1  E Reactant + Sl 1  rABC - r1 
(reactants)
Energy
(10.9)
Ea
E 2  E product + Sl 2  r2 - rABC 
Line 1
r1
(products)
(10.10)
r2
Line 2
rAB
Figure 10.6 A linear approximation to the Polanyi
diagram used to derive equation (10.11).
5
Case Where Polayni Works
(Over A Limited Range Of ΔH)
- 2 0 - 1 5 - 1 0
2 5
- 5
0
5
1 0
1 5
2 0
- 5
0
5
1 0
1 5
2 0
Ea , kcal/mole
2 0
1 5
1 0
5
0
- 2 0 - 1 5 - 1 0
 H r , kcal/mole
Figure 10.7 A plot of the activation barriers for the reaction R + H R  RH + R with R,
R = H, CH3, OH plotted as a function of the heat of reaction Hr.
6
Equation Does Not Work Over Wide
Range Of H
2
Log k ac
0
Marcus
Equation
-2
-4
-6
-2.4
-4.8
-7.2
 H r , kcal/mole
-9.6
Figure 10.11 A Polanyi plot for the enolization of
NO2(C6H4)O(CH2)2COCH3. Data of Hupke and Wu[1977]. Note Ln
(kac) is proportional to Ea.
7
Seminov Approximation: Use Multiple
Lines
100
Activation Barrier, kcal/mole
Activation Barrier, kcal/mole
40
Seminov
30
20
10
0
- 40
- 20
0
20
40
Heat Of Reaction, kcal/mole
Figure 11.11 A comparison of the
activation energies of a number of
hydrogen transfer reactions to those
predicted by the Seminov relationships,
equations (11.33) and (11.34)
Seminov
80
60
40
20
0
- 20
- 100
- 50
0
50
100
Heat Of Reaction, kcal/mole
Figure 11.12 A comparison of the
activation energies of 482 hydrogen
transfer reactions to those predicted by
the Seminov relationships, over a wider
range of energies.
8
Equation Does Not Work Over A Wide
Data Set
- 3 0
3 0
- 2 0
- 1 0
0
1 0
2 0
3 0
- 2 0
- 1 0
0
1 0
2 0
3 0
Ea , kcal/mole
2 5
2 0
1 5
1 0
5
0
- 3 0
 H r , kcal/mole
Figure 10.10 A Polanyi relationship for a series of reactions of the form RH + R
 R + HR. Data from Roberts and Steel[1994].
9
Key Prediction Stronger Bonds Are Harder
To Break
Strong Bond
Ea
rAB
Ea
Bond Energy
Energy
Weak Bond
rAB
Figure 10.8 A schematic of the curve crossing during the destruction of a weak bond
and a strong one for the reaction AB + C  A + BC.
10
Experiments Do Not Follow Predicted
Trend
7 5
7 0
Predicted Trend
Ea , kcal/mole
6 5
6 0
5 5
5 0
4 5
4 0
Experimental Trend
3 5
3 0
5 0
6 0
7 0
8 0
9 0
1 0 0
1 1 0
1 2 0
Figure 10.9 The activation barrier for the reaction X- +CH3X  XCH3 +
X-. The numbers are from the calculations of Glukhoustev, Pross and
Radam[1995].
11
Marcus: Why Curvature?
2
Log k ac
0
Marcus
Equation
-2
-4
-6
-2.4
-4.8
-7.2
 H r , kcal/mole
-9.6
Figure 10.11 A Polanyi plot for the enolization of
NO2(C6H4)O(CH2)2COCH3. Data of Hupke and Wu[1977]. Note Ln
(kac) is proportional to Ea.
12
Derivation: Fit Potentials To
Parabolas Not Lines
Actual
Potential
Approximation
E left
Energy
E right
1
Ea
E
1
E
r1
2
r2
rX
Figure 10.13 An approximation to the change in the potential
energy surface which occurs when Hr changes.
13
Derivation
Actual
Potential
E right (rX )  SS 2 (rX  r2 ) 2  H r  E 2
(10.22)
E left
E right
Energy
E left (rX )  SS1 (rX  r1 ) 2  E 1
(10.21)
Approximation
1
Ea
E
1
E
r1
2
r2
rX
Figure 10.13 An approximation to the
change in the potential energy surface
which occurs when Hr changes.
14
Solving
SS1 (r
‡
2
r1 ) + E1 = SS2 (r
‡
r2 ) 2 + E 2 + H r
(10.23)
SS1  SS2
(10.24)
Result
2
 H r  0
E A  1  0  E a
 4E a 
(10.33)
2
 H r  0
E A  1 
 Ea  w r
0

4E a 
(10.31)
15
Qualitative Features Of The Marcus
Equation
2
Log k ac
0
Marcus
Equation
-2
-4
-6
-2.4
-4.8
-7.2
-9.6
 H r , kcal/mole
Figure 10.11 A Polanyi plot for the enolization of
NO2(C6H4)O(CH2)2COCH3. Data of Hupke and Wu[1977].
Note Ln (kac) is proportional to Ea.
16
Marcus Is Great For Unimolecular
Reactions
11
10
Marcus
Equation
10
10
Data
9
rate, sec-1
10
8
10
7
10
6
10
5
10
0
5 0
1 0 0
1 5 0
2 0 0
2 5 0
 Hr , kcal/mole
Figure 10.17 The activation energy for intramolecular electron transfer
across a spacer molecule plotted as a function of the heat of reaction.
Data of Miller et. al., J. Am. Chem. Soc., 106 (1984) 3047.
17
Marcus Not As Good For Bimolecular
Reactions
11
10
-1
rate, lit mol sec
-1
10
10
Data
9
10
8
Marcus
Equation
10
7
10
6
10
5
10
- 2 5
0
2 5
5 0
7 5
1 0 0 1 2 5
 Hr , kcal/mole
Figure 10.18 The activation energy for florescence quenching of a
series of molecules in acetonitrite plotted as a function of the heat of
reaction. Data of Rehm et. al., Israel J. Chem. 8 (1970) 259.
18
Comparison To Wider Bimolecular
Data Set
E OA
Activation Barrier, kcal/mole
1 0 0
8 0
6 0
4 0
Marcus
Blowers
Masel
2 0
0
- 2 0
- 1 0 0
- 5 0
0
5 0
1 0 0
Heat Of Reaction, kcal/mole
Figure 10.29 A comparison of the barriers computed from Blowers and Masel's model
to barriers computed from the Marcus equation and to data for a series of reactions of
the form R + HR1  RH + R1 with wO = 100 kcal/mole and = 10 kcal/mole.
19
Why Inverted Behavior?
Ea
Medium distance
rAB
Ea =0
Smaller Distance
rAB
Inverted
Region
Even Smaller Distance
rAB
At small distances bonds need to stretch to to TS.
20
Limitations Of The Marcus Equation

Assumes that the bond
distances in the molecule
at the transition state does
Ea heat of Ea =0
not change as the
reaction changes


Good assumption for
unimolecular
reactionsSmaller Distance
Medium distance
In bimolecular
reactions
rAB
rAB
bonds can stretch to lower
barriers
Inverted
Region
Even Smaller Distance
rAB
21
Explains Why Marcus Better For
Unimolecular Than Bimolecular
11
11
10
10
Marcus
Equation
10
8
10
7
10
-1
rate, lit mol sec
Data
9
10
rate, sec-1
10
-1
10
10
Data
9
10
8
7
10
6
6
10
5
10
10
10
Marcus
Equation
10
5
0
5 0
1 0 0
1 5 0
 Hr , kcal/mole
2 0 0
2 5 0
- 2 5
0
2 5
5 0
7 5
1 0 0 1 2 5
 Hr , kcal/mole
22
Blowers-Masel Derivation



Fit potential energy surface to empirical
equation.
Solve for saddle point energy.
Only valid for bimolecular reaction.
23
Derive Analytical Expression

0

2
 w O + 0.5H r  VP - 2w O  H r 
Ea  
 VP 2  4 w O 2 +  H r 2


H r

When H r / 4 E oA  1
When - 1  H r / 4 E oA  1
When H r / 4 E oA  1
(10.63)
Bond energies
wO  wCC  wCH  / 2
(10.64)
Only parameter
 wO  EO

A
VP  2 w O 

O
 wO  EA 
(10.65)
24
Comparison To Experiments
E OA
Activation Barrier, kcal/mole
1 0 0
8 0
6 0
4 0
Marcus
Blowers
Masel
2 0
0
- 2 0
- 1 0 0
- 5 0
0
5 0
1 0 0
Heat Of Reaction, kcal/mole
Figure 10.29 A comparison of the barriers computed from Blowers and Masel's model to
barriers computed from the Marcus equation and to data for a series of reactions of the
form R + HR1  RH + R1 with wO = 100 kcal/mole and Eao = 10 kcal/mole.
25
Activation Energy , kcal/mole
Comparison Of Methods
1 0 0
Marcus
Blowers
Masel
5 0
Marcus
Polanyi
Blowers
Masel
0
Polanyi
- 1 0 0
- 5 0
0
5 0
1 0 0
Heat of Reaction, Kcal/mole
Figure 10.32 A comparison of the Marcus Equation, The Polanyi
o
relationship and the Blowers Masel approximation for E a =9 kcal/mole
and wO = 120 kcal/mole.
26
Example: Activation Barriers Via The
Blowers-Masel Approximation
Use a) the Blowers-Masel approximation b)
the Marcus equation to estimate the
activation barrier for the reaction
H + CH3CH3  H2 + CH2CH3
27
Solution: Step 1 Calculate VP
VP is given by:
 w o  E 0A 

VP  2 w o 
0
 wo  EA 
(11.F.2)
Where E 0A is the intrinsic barrier given in
Table 11.3. According to table 11.3E 0A =10
kcal/mole. One could calculate a value of
wo from data in the CRC. However, I
decided to assume a typical value for a C-H
bond, i.e. 100 kcal/mole. I chooseH r =-2
kcal/mole from example 11.C
28
Step 1 Continued
Substituting into equation (11.F.2) shows
 100kcal / mole  10kcal / mole
VP  2100kcal / mole
  244.4kcal / mole
 100kcal / mole  10kcal / mole
(11.F.3)
29
Step 2: Plug Into Equation 11.F.1


2


V

2
w


H
p
o
r

E a   w o  0.5 * H r 
 ( V ) 2  4 w 2  H 2 
 o  r 
 p
(11.F.1)
 244.4kcal/ mole2100kcal/ mole  2kcal/ mole2 
 9.0Kcal/ mole
Ea 100kcal/ mole0.5* 2kcal/ mole
2
2


(Vp)2 4wo  Hr 


By comparison the experimental value from Westley is
9.6 kcal/mole.
30
Solve Via The Marcus Equation:
Ea 
0 
E A 1 

H 

0
4E A 
2
(11.F.4)
From the data above E 0A =10 kcal/mole,
H r =-2 kcal/mole. Plugging into equation
(11.F.4)
2

 2 kcal / mole 
E a  10kcal / mole1 
  9.0 kcal / mole
 410kcal / mole
which is the same as the Blowers-Masel approximation.
31
Summary




Polayni: fair approximation – good over a
limited range of H
Marcus: Great for unimolecular
Blowers-Masel Great for bimolecular
Only small differences between methods
unless magnitude of heat of reaction is
larger than 3-4 times Ea0
32
Query

What did you learn new in this lecture?
33