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

1

• 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

2

• First order reactions
• Rate law is rate k A
• The integrate rate law can be expressed as
• A0 is A at t (time) 0
• At is A at t t
• e base of natural logarithms 2.71828

3

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

4

• A plot of lnAt 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.
5

• The simplest second-order rate law has the form
• The integrated form of this equation is

6

• Graphical methods can also be applied to
  second-order reactions
• A plot of 1/Bt 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).
7

• 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

8

First-order radioactive decay of iodine-131. The
initial concentration is represented by I0.
9

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

10

• 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

11

• 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

12

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.
13

• 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

14

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.
15

• 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

16

The potential-energy diagram for an exothermic
reaction. The extent of reaction is represented
as the reaction coordinate.
17

A successful (a) and unsuccessful (b) collision
for an exothermic reaction.
18

• 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.
19

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

Activation energy barrier for the forward and
reverse reactions.
20

• 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

21

Formation of the activated complex in the
reaction between NO2Cl and Cl.
NO2ClCl?NO2Cl2
22

• The activation energy is related to the rate
  constant by the Arrhenius equation
• k rate constant
• Ea activation energy
• e base of the natural logarithm
• R gas constant 8.314 J mol-1 K-1
• T Kelvin temperature
• A frequency factor or pre-exponential factor

23

• 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

24

• The activation energy can be related to the rate
  constant at two temperatures
• The reactions mechanism is the series of simple
  reactions called elementary processes
• The rate law of an elementary process can be
  written from its chemical equation

25

• 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

26

• 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

27

• 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

28

• The second step is the rate-limiting step, which
  gives
• N2O2 is a reactive intermediate, and can be
  eliminated from the expression

29

• The first step is a fast equilibrium
• At equilibrium, the rate of the forward and
  reverse reaction are equal

30

• Substituting, the rate law becomes
• Which is consistent with the experimental rate law

31

• 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

32

(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.
33

• 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

34

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

35

• 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

36

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