• The relationship between concentration and time can be derived from the rate law and calculus

• Integration of the rate laws gives the integrated rate laws

• Integrate laws give concentration as a function of time

• Integrated laws can get very complicated, so only a few simple forms will be considered

• First order reactions– Rate law is: rate = k [A]– The integrate rate law can be expressed as:

• [A]0 is [A] at t (time) = 0

• [A]t is [A] at t = t

• e = base of natural logarithms = 2.71828…

kt

t

eAAktA

A 0t0 ][][or

][

][ln

• Graphical methods can be used to determine the first-order rate constant, note

bmxy

AktA

ktAA

ktAA

ktA

A

t

t

t

t

]ln[]ln[

]ln[]ln[

]ln[]ln[

][

][ln

0

0

0

0

• A plot of ln[A]t versus t gives a straight line with a slope of -k

The decomposition of N2O5. (a) A graph of concentration versus time for the decomposition at 45oC. (b) A straight line is obtained from a logarithm versus time plot. The slope is negative the rate constant.

• The simplest second-order rate law has the form

• The integrated form of this equation is

2][ rate Bk

tBB

BB

ktBB

t

t

at time ofion concentrat the][

ofion concentrat initial the][

][

1

][

1

0

0

• Graphical methods can also be applied to second-order reactions

• A plot of 1/[B]t versus t gives a straight line with a slope of k

Second-order kinetics. A plot of 1/[HI] versus time (using the data in Table 15.1).

• The amount of time required for half of a reactant to disappear is called the half-life, t1/2

– The half-life of a first-order reaction is not affected by the initial concentration

ktkt

A

A

AAtt

ktA

A

t

t

2lnor

][

][ln

ngsubstituti ,][2

1][ ,at

][

][ln :law rateorder -First

2/12/102

10

02/1

0

First-order radioactive decay of iodine-131. The initial concentration is represented by [I]0.

– The half-life of a second-order reactions does depend on the initial concentration

02/12/1

0

2/1002

1

02/1

0t

][

2lnor

][

1

][

1

][

1

ngsubstituti ,][2

1][ ,at

][

1

][

1 :law rateorder -Second

Bktkt

B

ktBB

BBtt

ktBB

t

• One of the simplest models to explain reactions rates is collision theory

• According to collision theory, the rate of reaction is proportional to the effective number of collisions per second among the reacting molecules

• An effective collision is one that actually gives product molecules

• The number of all types of collisions increase with concentration, including effective collisions

• There are a number of reasons why only a small fraction of all the collisions leads to the formation of product:– Only a small fraction of the collisions are

energetic enough to lead to products– Molecular orientation is important because a

collision on the “wrong side” of a reacting species cannot produce any product

• This becomes more important as the complexity of the reactants increases

The key step in the decomposition of NO2Cl to NO2 and Cl2 is the collision of a Cl atom with a NO2Cl molecules. (a) A poorly orientated collision. (b) An effectively orientated collision.

– The minimum energy kinetic energy the colliding particles must have is called the activation energy, Ea

– In a successful collision, the activation energy changes to potential energy as the bonds rearrange to for products

– Activation energies can be large, so only a small fraction of the well-orientated, colliding molecules have it

– Temperature increases increase the average kinetic energy of the reacting particles

Kinetic energy distribution for a reaction at two different temperatures. At the higher temperature, a larger fraction of the collisions have sufficient energy for reaction to occur. The shaded area under the curves represent the reacting fraction of the collisions.

• Transition state theory explains what happens when reactant particles come together

• Potential-energy diagrams are used to help visualize the relationship between the activation energy and the development of total potential energy

• The potential energy is plotted against reaction coordinate or reaction progress

The potential-energy diagram for an exothermic reaction. The extent of reaction is represented as the reaction coordinate.

A successful (a) and unsuccessful (b) collision for an exothermic reaction.

• Activation energies and heats of reactions can be determined from potential-energy diagrams

Potential-energy diagram for an endothermic reaction. The heat of reaction and activation energy are labeled.

• Reactions generally have different activation energies in the forward and reverse direction

Activation energy barrier for the forward and reverse reactions.

• The brief moment during a successful collision that the reactant bonds are partially broken and the product bonds are partially formed is called the transition state

• The potential energy of the transition state is a maximum of the potential-energy diagram

• The unstable chemical species that “exists” momentarily is called the activated complex

Formation of the activated complex in the reaction between NO2Cl and Cl. NO2Cl+ClNO2+Cl2

• The activation energy is related to the rate constant by the Arrhenius equation

k = rate constant

Ea = activation energye = base of the natural logarithmR = gas constant = 8.314 J mol-1 K-1

T = Kelvin temperatureA = frequency factor or pre-exponential factor

RTEaAek /

• The Arrhenius equation can be put in standard slope-intercept form by taking the natural logarithm

• A plot of ln k versus (1/T) gives a straight line with slope = -Ea/RT

xmby

TREAk

RTEAk

a

a

)/1()/(lnln

or /lnln

• The activation energy can be related to the rate constant at two temperatures

• The reaction’s mechanism is the series of simple reactions called elementary processes

• The rate law of an elementary process can be written from its chemical equation

121

2 11ln

TTR

E

k

k a

• The overall rate law determined for the mechanism must agree with the observed rate law

• The exponents in the rate law for an elementary process are equal to the coefficients of the reactants in chemical equation

22

32

]k[NO rate

NONO2NO

:process Elementary

• Multistep reactions are common

• The sum of the element processes must give the overall reaction

• The slow set in a multistep reaction limits how fast the final products can form and is called the rate-determining or rate-limiting step

• Simultaneous collisions between three or more particles is extremely rate

• A reaction that depended a three-body collision would be extremely slow

• Thus, reaction mechanism seldom include elementary process that involve more than two-body or bimolecular collisions

• Consider the reaction

• The mechanism is thought to be

tal)(experimen ][Hk[NO]rate

O2HN2H2NO

22

222

• The second step is the rate-limiting step, which gives

• N2O2 is a reactive intermediate, and can be eliminated from the expression

(fast) OH N H ON

(slow) OH ONHON

(fast) ON 2NO

2222

22222

22

]][HON[ rate 222k

• The first step is a fast equilibrium

• At equilibrium, the rate of the forward and reverse reaction are equal

222

222

22

2

NO][]ON[

or ]ON[NO][

thus]ON[se)rate(rever

NO][rd)rate(forwa

r

f

rf

r

f

k

k

kk

k

k

• Substituting, the rate law becomes

• Which is consistent with the experimental rate law

]H[NO]['rate

or ]H[NO][rate

]H[]ON[rate

22

22

222

k

k

kk

k

r

f

• A catalyst is a substance that changes the rate of a chemical reaction without itself being used up– Positive catalysts speed up reactions– Negative catalysts or inhibitors slow reactions

• (Positive) catalysts speed reactions by allowing the rate-limiting step to proceed with a lower activation energy

• Thus a larger fraction of the collisions are effective

(a) The catalyst provides an alternate, low-energy path from the reactants to the products. (b) A larger fraction of molecules have sufficient energy to react when the catalyzed path is available.

• Catalysts can be divided into two groups– Homogeneous catalysts exist in the same phase

as the reactants– Heterogeneous catalysts exist in a separate

phase

• NO2 is a homogeneous catalyst for the production of sulfuric acid in the lead chamber process

• The mechanism is:

• The second step is slow, but is catalyzed by NO2:

4223

322

12

22

SOHOHSO

SOOSO

SOOS

222

1

322

NOONO

SONOSONO

• Heterogeneous catalysts are typically solids

• Consider the synthesis of ammonia from hydrogen and nitrogen by the Haber process

• The reaction takes place on the surface of an iron catalyst that contains traces of aluminum and potassium oxides

• The hydrogen and nitrogen binds to the catalyst lowering the activation energy

322 2NHN3H

The Haber process. Catalytic formation of ammonia molecules from hydrogen and nitrogen on the surface of a catalyst.