Chapter 14 Chemical Kinetics

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Chapter 14Chemical Kinetics
Chemistry, The Central Science, 10th
edition Theodore L. Brown H. Eugene LeMay, Jr.
and Bruce E. Bursten

• John D. Bookstaver
• St. Charles Community College
• St. Peters, MO
• ? 2006, Prentice Hall, Inc.

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Kinetics

• Studies the rate at which a chemical process
  occurs.
• Besides information about the speed at which
  reactions occur, kinetics also sheds light on the
  reaction mechanism (exactly how the reaction
  occurs).

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Factors That Affect Reaction Rates

• Physical State of the Reactants
• In order to react, molecules must come in contact
  with each other.
• The more homogeneous the mixture of reactants,
  the faster the molecules can react.

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Factors That Affect Reaction Rates

• Concentration of Reactants
• As the concentration of reactants increases, so
  does the likelihood that reactant molecules will
  collide.

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Factors That Affect Reaction Rates

• Temperature
• At higher temperatures, reactant molecules have
  more kinetic energy, move faster, and collide
  more often and with greater energy.

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Factors That Affect Reaction Rates

• Presence of a Catalyst SHOW MOVIE
• Catalysts speed up reactions by changing the
  mechanism of the reaction.
• Catalysts are not consumed during the course of
  the reaction.

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Reaction Rates

• Rates of reactions can be determined by
  monitoring the change in concentration of either
  reactants or products as a function of time.

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Reaction Rates
C4H9Cl(aq) H2O(l) ??? C4H9OH(aq) HCl(aq)

• In this reaction, the concentration of butyl
  chloride, C4H9Cl, was measured at various times.

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Reaction Rates
C4H9Cl(aq) H2O(l) ??? C4H9OH(aq) HCl(aq)

• The average rate of the reaction over each
  interval is the change in concentration divided
  by the change in time

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Reaction Rates
C4H9Cl(aq) H2O(l) ??? C4H9OH(aq) HCl(aq)

• Note that the average rate decreases as the
  reaction proceeds.
• This is because as the reaction goes forward,
  there are fewer collisions between reactant
  molecules.

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Reaction Rates
C4H9Cl(aq) H2O(l) ??? C4H9OH(aq) HCl(aq)

• A plot of concentration vs. time for this
  reaction yields a curve like this.
• The slope of a line tangent to the curve at any
  point is the instantaneous rate at that time.

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Reaction Rates
C4H9Cl(aq) H2O(l) ??? C4H9OH(aq) HCl(aq)

• All reactions slow down over time.
• Therefore, the best indicator of the rate of a
  reaction is the instantaneous rate near the
  beginning.

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Reaction Rates and Stoichiometry
C4H9Cl(aq) H2O(l) ??? C4H9OH(aq) HCl(aq)

• In this reaction, the ratio of C4H9Cl to C4H9OH
  is 11.
• Thus, the rate of disappearance of C4H9Cl is the
  same as the rate of appearance of C4H9OH.

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Reaction Rates and Stoichiometry

• What if the ratio is not 11?

2 HI(g) ??? H2(g) I2(g)

• Therefore,

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Reaction Rates and Stoichiometry

• To generalize, then, for the reaction

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Concentration and Rate

• One can gain information about the rate of a
  reaction by seeing how the rate changes with
  changes in concentration.

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Concentration and Rate

• Comparing Experiments 1 and 2, when NH4
  doubles, the initial rate doubles.

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Concentration and Rate

• Likewise, comparing Experiments 5 and 6, when
  NO2- doubles, the initial rate doubles.

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Concentration and Rate

• This means
• Rate ? NH4
• Rate ? NO2-
• Rate ? NH NO2-
• or
• Rate k NH4 NO2-
• This equation is called the rate law, and k is
  the rate constant.

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Fig 14.3
Figure 14.3 Progress of a hypothetical reaction
A ??B. Each red sphere represents 0.01 mol A,
each blue sphere represents 0.01 mol B, and the
vessel has a volume of 1.00 L. (a) At time zero
the vessel contains 1.00 mol A (100 red spheres)
and 0 mol B (no blue spheres). (b) After 20 s the
vessel contains 0.54 mol A and 0.46 mol B. (c)
After 40 s the vessel contains 0.30 mol A and
0.70 mol B.
BACK
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PRACTICE EXERCISE For the reaction pictured in
Figure 14.3, calculate the average rate of
appearance of B over the time interval from 0 to
40 s. (The necessary data are given in the figure
caption.)
Answer 1.8 ? 10 2 M/s
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Fig 14.3
Figure 14.3 Progress of a hypothetical reaction
A ??B. Each red sphere represents 0.01 mol A,
each blue sphere represents 0.01 mol B, and the
vessel has a volume of 1.00 L. (a) At time zero
the vessel contains 1.00 mol A (100 red spheres)
and 0 mol B (no blue spheres). (b) After 20 s the
vessel contains 0.54 mol A and 0.46 mol B. (c)
After 40 s the vessel contains 0.30 mol A and
0.70 mol B.
BACK
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Fig 14.4
Figure 14.4 Concentration of butyl chloride
(C4H9Cl) as a function of time. The dots
represent the experimental data from the first
two columns of Table 14.1, and the red curve is
drawn to connect the data points smoothly. Lines
are drawn that are tangent to the curve at t 9
and t 600 s. The slope of each tangent is
defined as the vertical change divided by the
horizontal change ?C4H9Cl/?t. The reaction
rate at any time is related to the slope of the
tangent to the curve at that time. Because C4H9Cl
is disappearing, the rate is equal to the
negative of the slope.
BACK
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PRACTICE EXERCISE Using Figure 14.4, determine
the instantaneous rate of disappearance of C4H9Cl
at t 300 s.
Answer 1.1 ? 10 4 M/s
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(b) If the rate at which O2 appears, ?O2??t, is
6.0 ? 105 M/s at a particular instant, at what
rate is O3 disappearing at this same time,
?O3??t?
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Answers (a) 8.4 ? 10 7 M/s, (b) 2.1 ? 10 7 M/s
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Rate Laws

• A rate law shows the relationship between the
  reaction rate and the concentrations of
  reactants.
• The exponents tell the order of the reaction with
  respect to each reactant.
• This reaction is
• First-order in NH4
• First-order in NO2-

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Rate Laws

• The overall reaction order can be found by adding
  the exponents on the reactants in the rate law.
• This reaction is second-order overall.

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Integrated Rate Laws

• Using calculus to integrate the rate law for a
  first-order process gives us

Where
A0 is the initial concentration of A. At is
the concentration of A at some time, t, during
the course of the reaction.
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Integrated Rate Laws

• Manipulating this equation produces

ln At - ln A0 - kt
ln At - kt ln A0
which is in the form
y mx b
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First-Order Processes
ln At -kt ln A0

• Therefore, if a reaction is first-order, a plot
  of ln A vs. t will yield a straight line, and
  the slope of the line will be -k.

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First-Order Processes

• Consider the process in which methyl isonitrile
  is converted to acetonitrile.

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First-Order Processes

• This data was collected for this reaction at
  198.9C.

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First-Order Processes

• When ln P is plotted as a function of time, a
  straight line results.
• Therefore,
• The process is first-order.
• k is the negative slope 5.1 ? 10-5 s-1.

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Second-Order Processes

• Similarly, integrating the rate law for a
  process that is second-order in reactant A, we get

also in the form
y mx b
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Second-Order Processes

• So if a process is second-order in A, a plot of
  1/A vs. t will yield a straight line, and the
  slope of that line is k.

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Second-Order Processes
The decomposition of NO2 at 300C is described by
the equation
and yields data comparable to this
Time (s) NO2, M
0.0 0.01000
50.0 0.00787
100.0 0.00649
200.0 0.00481
300.0 0.00380
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Second-Order Processes

• Graphing ln NO2 vs. t yields

• The plot is not a straight line, so the process
  is not first-order in A.

Time (s) NO2, M ln NO2
0.0 0.01000 -4.610
50.0 0.00787 -4.845
100.0 0.00649 -5.038
200.0 0.00481 -5.337
300.0 0.00380 -5.573
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Second-Order Processes

• Graphing ln 1/NO2 vs. t, however, gives this
  plot.

• Because this is a straight line, the process is
  second-order in A.

Time (s) NO2, M 1/NO2
0.0 0.01000 100
50.0 0.00787 127
100.0 0.00649 154
200.0 0.00481 208
300.0 0.00380 263
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Check Each box contains 10 spheres. The rate law
indicates that in this case B has a greater
influence on rate than A because B has a higher
reaction order. Hence, the mixture with the
highest concentration of B (most purple spheres)
should react fastest. This analysis confirms the
order 2 lt 1 lt 3.
PRACTICE EXERCISE Assuming that rate kAB,
rank the mixtures represented in this Sample
Exercise in order of increasing rate.
Answer 2 3 lt 1
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• PRACTICE EXERCISE 14.11H2 I2 ? 2 HI Rate k
  H2 I2
• (a) What is the reaction order of the reactant H2
  in Equation 14.11? (b) What are the units of the
  rate constant
• for Equation 14.11?

Answers (a) 1, (b) M1 s1
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SAMPLE EXERCISE 14.6 Determining a Rate Law from
Initial Rate Data
Using these data, determine (a) the rate law for
the reaction, (b) the magnitude of the rate
constant, (c) the rate of the reaction when A
0.050 M and B 0.100 M.
Solution Analyze We are given a table of data
that relates concentrations of reactants with
initial rates of reaction and asked to determine
(a) the rate law, (b) the rate constant, and (c)
the rate of reaction for a set of concentrations
not listed in the table. Plan (a) We assume that
the rate law has the following form Rate
kAmBn, so we must use the given data to
deduce the reaction orders m and n. We do so by
determining how changes in the concentration
change the rate. (b) Once we know m and n, we can
use the rate law and one of the sets of data to
determine the rate constant k. (c) Now that we
know both the rate constant and the reaction
orders, we can use the rate law with the given
concentrations to calculate rate.
Solve (a) As we move from experiment 1 to
experiment 2, A is held constant and B is
doubled. Thus, this pair of experiments shows how
B affects the rate, allowing us to deduce the
order of the rate law with respect to B. Because
the rate remains the same when B is doubled,
the concentration of B has no effect on the
reaction rate. The rate law is therefore zero
order in B (that is, n 0).
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(a) Determine the rate law for this reaction. (b)
Calculate the rate constant. (c) Calculate the
rate when NO 0.050 M and H2
0.150 M.
Answers (a) rate kNO2H2 (b) k 1.2
M2s1 (c) rate 4.5 ? 104 M/s
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Solution Analyze We are given the rate constant
for a reaction that obeys first-order kinetics,
as well as information about concentrations and
times, and asked to calculate how much reactant
(insecticide) remains after one year. We must
also determine the time interval needed to reach
a particular insecticide concentration. Because
the exercise gives time in (a) and asks for time
in (b), we know that the integrated rate law,
Equation 14.13, is required. Plan (a) We are
given k 1.45 yr1, t 1.00 yr, and
insecticide0 5.0 ? 107 g/cm3, and so
Equation 14.13 can be solved for
1ninsecticidet. (b) We have k 1.45yr1,
insecticide0 5.0 ? 107 g/cm3, and
insecticidet 3.0 ? 107 g/cm3, and so we can
solve Equation 14.13 for t.
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Check In part (a) the concentration remaining
after 1.00 yr (that is,1.2 ? 107 g/cm3) is less
than the original concentration (5.0 ? 107
g/cm3), as it should be. In (b) the given
concentration (3.0 ? 107 g/cm3) is greater than
that remaining after 1.00 yr, indicating that the
time must be less than a year. Thus, t 0.35 yr
is a reasonable answer.
Answer 51 torr
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Is the reaction first or second order in NO2?
Solution Analyze We are given the concentrations
of a reactant at various times during a reaction
and asked to determine whether the reaction is
first or second order. Plan We can plot lnNO2
and 1/NO2 against time. One or the other will
be linear, indicating whether the reaction is
first or second order.
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SAMPLE EXERCISE 14.8 continued
Solve In order to graph lnNO2 and 1/NO2
against time, we will first prepare the following
table from the data given
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SAMPLE EXERCISE 14.8 continued
As Figure 14.8 shows, only the plot of 1/NO2
versus time is linear. Thus, the reaction obeys a
second-order rate law Rate kNO22. From the
slope of this straight-line graph, we determine
that k 0.543 M1 s1 for the disappearance of
NO2.
PRACTICE EXERCISE Consider again the
decomposition of NO2 discussed in the Sample
Exercise. The reaction is second order in NO2
with k 0.543 M1s1. If the initial
concentration of NO2 in a closed vessel is 0.0500
M, what is the remaining concentration after
0.500 h?
Answer Using Equation 14.14, we find NO2
1.00 ? 103 M
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Half-Life

• Half-life is defined as the time required for
  one-half of a reactant to react.
• Because A at t1/2 is one-half of the original
  A,
• At 0.5 A0.

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Half-Life

• For a first-order process, this becomes

ln 0.5 -kt1/2
-0.693 -kt1/2
NOTE For a first-order process, the half-life
does not depend on A0.
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Half-Life

• For a second-order process,

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Check At the end of the second half-life, which
should occur at 680 s, the concentration should
have decreased by yet another factor of 2, to
0.025 M. Inspection of the graph shows that this
is indeed the case.

• PRACTICE EXERCISE
• (a) Using Equation 14.15, calculate t1/2 for the
  decomposition of the insecticide described in
  Sample Exercise 14.7. (b) How long does it take
  for the concentration of the insecticide to reach
  one-quarter of the initial value?

Answers (a) 0.478 yr 1.51 ? 107 s (b) it
takes two half-lives, 2(0.478 yr) 0.956 yr
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Temperature and Rate

• Generally, as temperature increases, so does the
  reaction rate.
• This is because k is temperature dependent.

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The Collision Model

• In a chemical reaction, bonds are broken and new
  bonds are formed.
• Molecules can only react if they collide with
  each other.

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The Collision Model

• Furthermore, molecules must collide with the
  correct orientation and with enough energy to
  cause bond breakage and formation.

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Activation Energy

• In other words, there is a minimum amount of
  energy required for reaction the activation
  energy, Ea.
• Just as a ball cannot get over a hill if it does
  not roll up the hill with enough energy, a
  reaction cannot occur unless the molecules
  possess sufficient energy to get over the
  activation energy barrier.

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Reaction Coordinate Diagrams

• It is helpful to visualize energy changes
  throughout a process on a reaction coordinate
  diagram like this one for the rearrangement of
  methyl isonitrile.

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Reaction Coordinate Diagrams

• It shows the energy of the reactants and products
  (and, therefore, ?E).
• The high point on the diagram is the transition
  state.

• The species present at the transition state is
  called the activated complex.
• The energy gap between the reactants and the
  activated complex is the activation energy
  barrier.

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MaxwellBoltzmann Distributions

• Temperature is defined as a measure of the
  average kinetic energy of the molecules in a
  sample.

• At any temperature there is a wide distribution
  of kinetic energies.

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MaxwellBoltzmann Distributions

• As the temperature increases, the curve flattens
  and broadens.
• Thus at higher temperatures, a larger population
  of molecules has higher energy.

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MaxwellBoltzmann Distributions

• If the dotted line represents the activation
  energy, as the temperature increases, so does the
  fraction of molecules that can overcome the
  activation energy barrier.

• As a result, the reaction rate increases.

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MaxwellBoltzmann Distributions

• This fraction of molecules can be found through
  the expression
• where R is the gas constant and T is the Kelvin
  temperature.

f e-Ea/RT
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SAMPLE EXERCISE 14.10 Relating Energy Profiles to
Activation Energies and Speeds of Reaction
Consider a series of reactions having the
following energy profiles
Assuming that all three reactions have nearly the
same frequency factors, rank the reactions from
slowest to fastest.
Solution The lower the activation energy, the
faster the reaction. The value of ?? does not
affect the rate. Hence the order is (2) lt (3) lt
(1).
PRACTICE EXERCISE Imagine that these reactions
are reversed. Rank these reverse reactions from
slowest to fastest.
Answer  (2) lt (1) lt (3) because Ea values are
40, 25, and 15 kJ/mol, respectively
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Arrhenius Equation

• Svante Arrhenius developed a mathematical
  relationship between k and Ea
• k A e-Ea/RT
• where A is the frequency factor, a number that
  represents the likelihood that collisions would
  occur with the proper orientation for reaction.

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Arrhenius Equation

• Taking the natural logarithm of both sides, the
  equation becomes
• ln k -Ea ( ) ln A

y mx b
Therefore, if k is determined experimentally at
several temperatures, Ea can be calculated from
the slope of a plot of ln k vs. 1/T.
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SAMPLE EXERCISE 14.11 Determining the Energy of
Activation
The following table shows the rate constants for
the rearrangement of methyl isonitrile at various
temperatures (these are the data in Figure 14.12)
(a) From these data, calculate the activation
energy for the reaction. (b) What is the value of
the rate constant at 430.0 K?
Solution Analyze We are given rate constants, k,
measured at several temperatures and asked to
determine the activation energy, Ea, and the
rate constant, k, at a particular
temperature. Plan We can obtain Ea from the
slope of a graph of ln k versus 1/T. Once we know
Ea, we can use Equation 4.21 together with the
given rate data to calculate the rate constant at
430.0 K.
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SAMPLE EXERCISE 14.11 continued
Solve (a) We must first convert the temperatures
from degrees Celsius to kelvins. We then take the
inverse of each temperature, 1/T, and the natural
log of each rate constant, ln k. This gives us
the table shown at the right
A graph of ln k versus 1/T results in a straight
line, as shown in Figure 14.17.
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We report the activation energy to only two
significant figures because we are limited by the
precision with which we can read the graph in
Figure 14.17.
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SAMPLE EXERCISE 14.11 continued
Thus,
Note that the units of k1 are the same as those
of k2.
PRACTICE EXERCISE Using the data in Sample
Exercise 14.11, calculate the rate constant for
the rearrangement of methyl isonitrile at 280C.
Answer 2.2 ? 102s1
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Reaction Mechanisms

• The sequence of events that describes the actual
  process by which reactants become products is
  called the reaction mechanism.

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Reaction Mechanisms

• Reactions may occur all at once or through
  several discrete steps.
• Each of these processes is known as an elementary
  reaction or elementary process.

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Reaction Mechanisms

• The molecularity of a process tells how many
  molecules are involved in the process.

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Multistep Mechanisms

• In a multistep process, one of the steps will be
  slower than all others.
• The overall reaction cannot occur faster than
  this slowest, rate-determining step.

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Slow Initial Step
NO2 (g) CO (g) ??? NO (g) CO2 (g)

• The rate law for this reaction is found
  experimentally to be
• Rate k NO22
• CO is necessary for this reaction to occur, but
  the rate of the reaction does not depend on its
  concentration.
• This suggests the reaction occurs in two steps.

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Slow Initial Step

• A proposed mechanism for this reaction is
• Step 1 NO2 NO2 ??? NO3 NO (slow)
• Step 2 NO3 CO ??? NO2 CO2 (fast)
• The NO3 intermediate is consumed in the second
  step.
• As CO is not involved in the slow,
  rate-determining step, it does not appear in the
  rate law.

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Fast Initial Step
2 NO (g) Br2 (g) ??? 2 NOBr (g)

• The rate law for this reaction is found to be
• Rate k NO2 Br2
• Because termolecular processes are rare, this
  rate law suggests a two-step mechanism.

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Fast Initial Step

• A proposed mechanism is

Step 2 NOBr2 NO ??? 2 NOBr (slow)
Step 1 includes the forward and reverse reactions.
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Fast Initial Step

• The rate of the overall reaction depends upon the
  rate of the slow step.
• The rate law for that step would be
• Rate k2 NOBr2 NO
• But how can we find NOBr2?

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Fast Initial Step

• NOBr2 can react two ways
• With NO to form NOBr
• By decomposition to reform NO and Br2
• The reactants and products of the first step are
  in equilibrium with each other.
• Therefore,
• Ratef Rater

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Fast Initial Step

• Because Ratef Rater ,
• k1 NO Br2 k-1 NOBr2
• Solving for NOBr2 gives us

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Fast Initial Step

• Substituting this expression for NOBr2 in the
  rate law for the rate-determining step gives

k NO2 Br2
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(c) The intermediate is O(g). It is neither an
original reactant nor a final product, but is
formed in the first step of the mechanism and
consumed in the second.
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(a) Is the proposed mechanism consistent with the
equation for the overall reaction? (b) What is
the molecularity of each step of the mechanism?
(c) Identify the intermediate(s).
Answers (a) Yes, the two equations add to yield
the equation for the reaction. (b) The first
elementary reaction is unimolecular, and the
second one is bimolecular. (c) Mo(CO)5
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Solution Analyze We are given the equation and
asked for its rate law, assuming that it is an
elementary process. Plan Because we are assuming
that the reaction occurs as a single elementary
reaction, we are able to write the rate law using
the coefficients for the reactants in the
equation as the reaction orders. Solve The
reaction is bimolecular, involving one molecule
of H2 with one molecule of Br2. Thus, the rate
law is first order in each reactant and second
order overall Rate kH2Br2
Comment Experimental studies of this reaction
show that the reaction actually has a very
different rate law Rate kH2Br21/2 Because
the experimental rate law differs from the one
obtained by assuming a single elementary
reaction, we can conclude that the mechanism must
involve two or more elementary steps.
Answers (a) Rate kNO2Br2 (b) No, because
termolecular reactions are very rare
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(a) Write the equation for the overall reaction.
(b) Write the rate law for the overall reaction.
(b) The rate law for the overall reaction is just
the rate law for the slow, rate-determining
elementary reaction. Because that slow step is a
unimolecular elementary reaction, the rate law is
first order Rate kN2O
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The experimental rate law is rate kO3NO2.
What can you say about the relative rates of the
two steps of the mechanism?
Answer Because the rate law conforms to the
molecularity of the first step, that must be the
rate-determining step. The second step must be
much faster than the first one.
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Solution Analyze We are given a mechanism with a
fast initial step and asked to write the rate law
for the overall reaction. Plan  The rate law of
the slow elementary step in a mechanism
determines the rate law for the overall reaction.
Thus, we first write the rate law based on the
molecularity of the slow step. In this case the
slow step involves the intermediate N2O2 as a
reactant. Experimental rate laws, however, do not
contain the concentrations of intermediates, but
are expressed in terms of the concentrations of
starting substances. Thus, we must relate the
concentration of N2O2 to the concentration of NO
by assuming that an equilibrium is established in
the first step. Solve The second step is rate
determining, so the overall rate is Rate
k2N2O2Br2
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What is the expression relating the concentration
of Br(g) to that of Br2(g)?
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Catalysts

• Catalysts increase the rate of a reaction by
  decreasing the activation energy of the reaction.
• Catalysts change the mechanism by which the
  process occurs.

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Catalysts

• One way a catalyst can speed up a reaction is by
  holding the reactants together and helping bonds
  to break.

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Enzymes

• Enzymes are catalysts in biological systems.
• The substrate fits into the active site of the
  enzyme much like a key fits into a lock.

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The decomposition reaction is determined to be
first order. A graph of the partial pressure of
HCOOH versus time for decomposition at 838 K is
shown as the red curve in Figure 14.28. When a
small amount of solid ZnO is added to the
reaction chamber, the partial pressure of acid
versus time varies as shown by the blue curve in
Figure 14.28.

• (a) Estimate the half-life and first-order rate
  constant for formic acid decomposition.
• (b) What can you conclude from the effect of
  added ZnO on the decomposition of formic acid?

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SAMPLE INTEGRATIVE EXERCISE continued
Solution (a) The initial pressure of HCOOH is
3.00 ? 102 torr. On the graph we move to the
level at which the partial pressure of HCOOH is
150 torr, half the initial value. This
corresponds to a time of about 6.60 x 102s, which
is therefore the half-life. The first-order rate
constant is given by Equation 14.15 k
0.693/t1/2 0.693/660 s 1.05 ? 103 s1.
(b) The reaction proceeds much more rapidly in
the presence of solid ZnO, so the surface of the
oxide must be acting as a catalyst for the
decomposition of the acid. This is an example of
heterogeneous catalysis.
(c) If we had graphed the concentration of
formic acid in units of moles per liter, we would
still have determined that the half-life for
decomposition is 660 seconds, and we would have
computed the same value for k. Because the units
for k are s1, the value for k is independent of
the units used for concentration.
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SAMPLE INTEGRATIVE EXERCISE continued