Second-order reactions
A second-order reaction is one whose rate depends on the
square of the concentration of a single reactant , or on the concentrations of two reactants, each raised to the first power
For example, for the second order reaction 2A  B,
Rate
 k
 
2
Or, for A + B  C,
Rate
 k
  
We will only concern ourselves with second-order rate laws of the first form (i.e., Rate = k[A] 2 )
Just as we did for the first-order case, we can write an integrated second-order rate law :
1
 
 kt

1
 
0
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Again, this equation has the general form of a line: a plot of 1/[A] vs. time will be linear with a slope = k and intercept = 1/[A]
0
if a reaction is second-order
This provides us with a method to determine reaction orders and rate constants experimentally
Collect concentration vs time data; if the reaction is firstorder, a plot of ln [ ] vs time will be linear; for a secondorder reaction, a plot of 1/[ ] vs time will be linear
What if neither plot is linear?
E.g., the following data were obtained at 300 o C for the reaction
2NO
2
(g)

2NO(g) + O
2
(g) time (s) [NO
2
]
0.0
5.0
10.0
15.0
20.0
0.100
0.017
0.0090
0.0062
0.0047
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Is the reaction first or second-order in NO
2
?
What is the value of the rate constant?
In this case, we must plot the data in the first and secondorder forms.....
0
-1
-2
-3
-4
-5
-6
0 5 10
Time (s)
15 20 25
Does the first-order treatment appear to linearize the data?
60

250
200
150
100
50 y = 10.16x + 9.1984
0
0 5 10
Time (s)
15
Find the rate constant from the plot.......
61
20 25

Temperature dependence of reaction rates
The kinetic rate constant k for all reactions is a constant at
a given temperature
However, k is temperature-dependent: consider the second-order gas-phase reaction
CO(g) + NO
2
(g)  CO
2
(g) + NO(g)
What does a plot of k vs. temperature look like for this reaction?
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35
30
25
20
15
10
5
0
600 650 700 750
Temperature (K)
800 850
As k increases with T, the reaction rate increases with T
Why? How to explain the dependence of k on T?
We use the collision model to account for the dependence of reaction rate on temperature
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Collision Model
Based on kinetic-molecular theory
Molecules must collide to react
Model helps explain dependence of rate on concentration and temperature
As reactant concentration increases, the frequency of collisions increases, and hence the reaction rate increases
As temperature increases, average molecular speeds increase
As molecules move faster, collisions become more energetic, and reaction rate increases
So... the collision model pretty much explains it all, right?
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Consider the reaction between hydrogen and iodine to form HI:
H
2
(g) + I
2
(g)

2HI(g)
At STP, each molecule undergoes ~ 10 10 collisions/sec -- yet only ~ 1 in 10 13 collisions forms products !
So, only a very small fraction of collisions actually lead to reaction
Why?
Simple collisions between reactant molecules aren’t enough to produce a reaction
Molecules must collide in the proper orientation for reaction to occur
E.g. with H
2
and I
2
collisions, what possible orientations are possible? Which would be most efficient at producing product?
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In addition to orientation, we also use the concept of activation energy to explain the temperature dependence of reaction rates
Arrhenius: Molecules must have a certain minimum amount of energy in order to react
Collision model: this energy comes from the kinetic energies of the colliding molecules
Upon collision, KE can be converted into other types of energy, i.e., into vibrational energy which can stretch and break a bond
If molecules are moving too slowly (low KE), they bounce off each other when colliding
Kinetic energy not converted and molecules bounce off each other without reacting
Conclude that colliding molecules must have a total KE
 some minimum value in order to react
This minimum energy required in order to initiate a reaction is known as the activation energy E a
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We represent this graphically: energy profile for an exothermic reaction
E.g., For the decomposition of H
2
O
2
(aq) to H
E a
= 75.3 kJ/mol and overall
2
O(l) and O
2
(g),

E = -98.1 kJ/mol.
Sketch the energy profile for this reaction
Note that the reactants pass through an activated complex at the top of the activation energy barrier
Activated complex (transition state): high-energy, transient arrangement of reactants
What is E a
for the reverse reaction?
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How to calculate E a
? How to relate k to E a
and temperature?
From above: k varies nonlinearly with temperature
Arrhenius equation: gives the dependence of k on:
Activation energy Ea
Temperature T k

Ae

E a
RT
Here, A= frequency factor – A is essentially constant
R = gas constant (8.314 J/mol K)
Note the exponential behavior:
As E a
increases, k decreases: rate decreases
As T increases, k increases: rate increases
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Arrhenius equation: k

Ae

E a
RT
Put in linear form: take ln of both sides..... ln k
 
E a
RT
 ln A
In this form: equation of a line........
Plot lnk vs. 1/T: slope = -E a
/R.........
Now: suppose we know E a
and k at some temperature T
1
, and we wish to find k at some other temperature T
2
.....
We have ln k
1
 
E a
RT
1
 ln A
And ln k
2
 
E a
RT
2
 ln A
Subtract (and assume that E a
and A do n’t depend on T)..... ln k
1
 ln k
2
 
E a
RT
1




E a
RT
2


Clean this up a little... k ln k
2
1

E a
R


1
T
2

1
T
1


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Based on their activation energies and

E values, and assuming that all frequency factors (A) are the same, which of the following reactions would be fastest and which be slowest?
(a) E a
= 45 kJ/mol,

E = -25 kJ/mol
(b) E a
= 35 kJ/mol,

E = -10 kJ/mol
(c) E a
= 55 kJ/mol,

E = 10 kJ/mol
The rate of the reaction
CH
3
COOC
2
H
5
(aq) + OH (aq)

CH
3
COO (aq) +C
2
H
5
OH(aq)
Was measured at several temperatures, and the following data collected:
T ( o C) k (M -1 s -1 )
15
25
0.0521
0.101
35
45
0.184
0.332
Find the value of E a
for the reaction.
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