Chapter 12Chemical Kinetics

• The area of chemistry that concerns reaction rates.

• Goals: To understand the steps (reaction mechanism) by which a reaction takes place. Allows us to find ways to facilitate the reaction

Reaction Rate

Kinetics deals with the speed (rate) at which changes occur. The quantity that changes is the amount or concentration of a reactant or a product.

Change in concentration (conc.) of a reactant or product per unit time.

R ate = co n c o f A a t tim e co n c o f A a t tim e 2 1

2 1

t t

t t

A

t

Continued….

2NO2(g) 2NO(g) + O2(g)

The concentration of the reactant (NO2) decreases with time and the concentration of the products (NO and O2) increase with time.

A change can be positive (increase) or negative (decrease) leading to a positive or negative reaction rate. However, we will always define the rate as a positive quantity.

Figure 12.1 Definition of Rate (Conc. vs.Time)

Continued….

The concentration of NO2 decreases with time, [NO2] is a negative quantity. The reaction rate is a positive quantity

The concentration of reactants always decrease with time. The rate expression involving reactant will include a negative sign.

R ate =ch an g e in [N O ]

T im e e lap sed

N O ]

t

2 2

[

R ate = -[N O ]

t

2

Continued….

2NO2(g) 2NO(g) + O2(g)

The rate can also be defined in terms of the products. In doing so we must take into account the coefficients in the balance equation for the reaction. Stoichiometry determines the relative rates.

Rate of consumption = rate of production = 2(rate of production

of NO2 of NO of O2)

[N O ]

t

[N O ]

t

O ]

t

2 22

[

Rate Laws

2NO2(g) 2NO(g) + O2(g)

The reaction rate will depend on the concentrations of the reactants

Rate = k[NO2]n

The above expression shows how the rate depends on the concentration of reactants is called a rate law.

k = rate constant (proportionality constant)

n = rate order (integer or fraction including zero)

R ate = -[N O ]

tk [N O ]

22

n

Types of Rate Laws

Differential Rate Law: expresses how rate depends on concentration.

Integrated Rate Law: expresses how concentration depends on time.

Continued….

• We consider reactions where the reverse reaction is unimportant, rate laws involve only concentrations of reactants.

• Differential and integrated rate laws for a given reaction are related in a well-defined way, the experimental determination of either of the rate laws is sufficient.

• Experimental convenience dictates which types of rate law is determined experimentally.

Method of Initial Rates

Initial Rate: the “instantaneous rate” just after the reaction begins.

The initial rate is determined in several experiments using different initial concentrations.

Figure 12.3 A Plot of the Concentration of N2O5 as a Function of Time for the Reaction

Overall Reaction Order

Sum of the order of each component in the

rate law.

rate = k[H2SeO3][H+]2[I]3

The overall reaction order is 1 + 2 + 3 = 6.

First-Order Rate Law

For aA Products in a 1st-order reaction,

Integrated first-order rate law is

ln[A] = kt + ln[A]o

Rate = A

A

t

k

Continued….

ln[A] = -kt + ln[A]o

• The equation shows how the concentration of A depends on time. If the initial concentration of A and the rate constant k are known, the concentration of A at any time can be calculated.

• The above equation is the equation of straight line of the form y = mx + b, where a plot of y versus x is a straight line with slope m and intercept b.

Continued….

• The reaction is first order in A if a plot of ln[A] versus t is a straight line.

• The integrated rate law for a first order reaction can also be expressed in terms of a ratio of [A] and [A]o as follow:

ln[A ]

[A ]k t

o

Figure 12.4 A Plot of In(N2O5) Versus Time

Half-Life of a First-Order Reaction

The time required for a reactant to reach half its original concentration is called the half-life of a reactant and is designated by the symbol t1/2.

t1/2 = half-life of the reaction, k = rate constant

For a first-order reaction, the half-life does not depend on concentration.

ln[A ]

[A ]k t w h en t = t th en [A ] =

[A ]

2

ln[A ]

[A ] / 2k t ln (2 ) = k t

t =ln (2 )

kt =

0 .6 9 3

k

o1 / 2

o

o

o1 / 2 1 / 2

1 / 2 1 / 2

Figure 12.5 A Plot of (N2O5) Versus Time for the Decomposition Reaction of N2O5

Second-Order Rate LawFor aA products in a second-order reaction,

Integrated rate law is

A plot of 1/[A] versus t will produce a straight line with a slope equal to k

The above equation shows how [A] depends on time and can be used to calculate [A] at any time t, provided k and [A]o are known

R ate = A

A

t

k 2

1

A +

1

A o

kt

Half-Life of a Second-Order ReactionWhen one half-life of the second order reaction has elapsed (t = t1/2), by definition, [A] = [A]o/2 then the integrated rate law becomes

t1/2 = half-life of the reaction, k = rate constant,

Ao = initial concentration of A

The half-life is dependent upon the initial concentration.

1

[A ] / 2k t +

1

[A ]

2

[A ] A ]k t

1

[A ]k t

so lv in g fo r t g iv es th e ex p ress io n

t =1

k [A ]

o1 / 2

o o o1 / 2

o1 / 2

1 / 2

1 / 2o

1

[

Figure 12.6 (a) A Plot of In(C4H6) Versus t (b) A Plot of 1/(C4H6) Versus t

Zero-Order Rate Laws

The rate law for a zero-order reaction isRate = k[A]o = k(1) = k

For a zero-order reaction, the rate is constant. It does not change with concentration as it does for first-order or second-order reactions.

The integrated rate law for a zero-order reaction is

[A] = -kt + [A]o

continued…

[A] = -kt + [A]o

In this case a plot of [A] versus t gives a straight line of slope –k.

[A] = [A]o/2, when t = t1/2

[A]o/2 =-kt1/2 + [A]o

Solving for t1/2 gives,

t1/2 = [A]o/2k

Figure 12.7 A Plot of (A) Versus t for a Zero-Order Reaction

Rate Laws for Reactions with More Than One Reactant

A + B + C Product

Rate = k[A]n[B]m[C]p

For such reaction, concentration of one reactant remain small compared with the concentrations of the others. So the rate law reduce to

Rate = k`[A]n

Where, k` = k[B]m[C]op and [B]o>>[A]o and [C]o>>[A]o

The value of n can be obtained by determining whether a plot of [A] versus t is linear (n = 0), a plot of ln[A] versus t is linear (n = 1), or a plot of 1/[A] versus t is linear (n = 2). The value of k` is determined from the slope.

A Summary

1. Simplification: Conditions are set such that only forward reaction is important.

2. Two types of rate law: differential rate law and integrated rate law

3. Which type? Depends on the type of data collected - differential and

integrated forms can be interconverted.

A Summary (continued)

4. Most common: method of initial rates.

5. Concentration v. time: used to determine integrated rate law, often graphically.

6. For several reactants: choose conditions under which only one reactant varies significantly (pseudo first-order conditions).

Reaction Mechanism

The series of steps by which a chemical reaction occurs.

A chemical equation does not tell us how reactants become products - it is a summary of the overall process.

The purpose for studying kinetics is to learn as much as possible about the steps involved in a reaction.

Reaction Mechanism (continued)

The reaction

has many steps in the reaction mechanism.The rate law for this reaction is known from experiment to be

Rate = k[NO2]2

The balanced equation tells us the reactants, the products, the stoichiometry.

NO2(g) + NO2(g) NO3(g) + NO(g)

NO3(g) + CO(g) NO2(g) + CO2(g)

N O + C O N O + C O2 (g ) (g ) (g ) 2 (g )

k 1 k 2

Often Used Terms• Intermediate: formed in one step and used up in a

subsequent step and so is never seen as a product. (neither a reactant nor a product)

• Molecularity: the number of species that must collide to produce the reaction indicated by that step.

• Elementary Step: A reaction whose rate law can be written from its molecularity.

uni, bi and termolecular

• The sum of the elementary steps must give the overall balanced equation

• The mechanism must agree with the experimentally determined rate law.

Rate-Determining Step

Multistep reaction often have one step that is much slower than all the others. Reactants can become products only as fast as they can get through this slowest step. The overall reaction can be no faster than the slowest or rate determining step.In a multistep reaction, it is the slowest step. It therefore determines the rate of reaction.

Overall rate = k1[NO2]2

N O + N O N O + N O slo w (ra te d e te rm in in g )

N O + C O N O + C O fast

2 (g ) 2 (g ) 3 (g ) (g )

3 (g ) (g ) 2 (g ) 2 (g )

Collision Model

Key Idea: Molecules must collide to react.

However, only a small fraction of collisions produces a reaction. Why?

Arrhenius: An activation energy (threshold energy) must be overcome to produce a chemical reaction.

2BrNO(g) 2NO(g) + Br2(g)

Figure 12.11 Change in Potential Energy

Arrhenius Equation

Collisions must have enough energy to produce the reaction (must equal or exceed the activation energy).

Orientation of reactants must allow formation of new bonds necessary to produce products.

Figure 12.13 Several Possible Orientations for a Collision Between Two BrNO Molecules

Arrhenius Equation (continued)

k = rate constant, A = frequency factor

Ea = activation energy, T = temperatureR = gas constant

ln(k) = -Ea/R(1/T) + ln(A)A plot of ln(k) versus 1/T gives a straight line, slope = -Ea/R and intercept = ln(A)

k Ae E RT a /

Figure 12.14 Plot of In(k) Versus 1/T for the Reaction 2N2O5(g) g) + O2(g)

Arrhenius Equation (continued)

Activation energy (Ea) can also be calculated from the values of k at only two temperatures

At temperature T1, where the rate constant is k1,

At temperature T2, where the rate constant is k2,

Ea can be calculated from k1 and k2 at temperature T1 and T2

ln (k ) = -E a

R TA )1

1 ln (

ln ln

ln ln ln ln

ln

(k )E a

R T(A )

(k )- (k )= -E a

R T(A )

E a

R T(A )

E a

R T

R a

R T

k

k

E a

R T T

22

2 12 1 2 1

2

1 21

1 1

Catalysis

• Catalyst: A substance that speeds up a reaction without being consumed it self.

• Enzyme: A large molecule (usually a protein) that catalyzes biological reactions.

• Homogeneous catalyst: Present in the same phase as the reacting molecules.

• Heterogeneous catalyst: Present in a different phase than the reacting molecules.

Figure 12.15 Energy Plots for a Catalyzed and an Uncatalyzed Pathway for a Given Reaction