# Problem du Jour

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

58

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

59

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?

62

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

63

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?

64

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?

65

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

66

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?

67

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

68

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



69

# Problems du Jour

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.

70

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