Rates
Description of time dependence requires a
statement of the time derivatives of the
concentrations of reactants and products
Physical Chemistry
Lecture 5
Introduction to chemical kinetics
H2 ( gas) 
1
O ( gas) 
2 2
H2 O ( liquid )
Rates



Rate of change of [H2O]:
Rate of change of [H2]:
Rate of change of [O2]:
d[H2O]/dt
d[H2]/dt
d[O2]/dt
Rates are not independent because of mass
conservation

Thermodynamics and kinetics
Thermodynamics



Relative stability of states
Characterized by free-energy differences
Static comparisons of states
Kinetics


Changes of state over time
Several different topics
 Empirical description of the rates of reaction
 Determination of experimental parameters
 Microscopic theories of reaction
1 oxygen molecule disappears for every 2 hydrogen
molecules in forming 1 water molecule
Reaction velocity
The rates of appearance of
products and disappearance
of reactants are related by
stoichiometry of the reaction
Reaction velocity, v,
“normalizes” rates of
appearance of products and
disappearance of reactants
Stoichiometric coefficient, i



Found from balanced
equation
Positive for products
Negative for reactants
v 
1 d [i ]
 i dt
 H  1
 O  1/ 2
 H O  1
2
2
2
1
H 2 ( gas)  O2 ( gas )  H 2O(liq.)
2
Descriptions of reactions
Chemical reaction often given by an equation
of the type
H2 ( gas) 
1
O ( gas)  H2 O (liquid )
2 2
Rate law
Description of how the reaction velocity
depends on parameters such as
concentrations, temperature, pressure, etc.
v 
This description is incomplete

Only gives beginning and ending states
 Sometimes does not even describe the final state
correctly

Gives stoichiometry of change
Must describe time dependences of amounts
of materials to describe the reaction more
completely
f ([ Areact ], [ B prod ], T , P)
Function may be complex
Gives insight into the reaction’s progress


Reactions do not necessarily occur as implied by
the overall equation
Attempts to describe the features of the
underlying physical chemistry of reaction in detail
1
Determining reaction rate
Initial-order determination
Must determine how the reaction
velocity depends on concentration,
temperature, etc.
Major method for finding
reaction velocity under
limiting conditions
Follows the concentration
only at early times – initial
rate
May not completely
describe the reaction
Often done as a
convenience

Determine time dependence of
concentration
At any point, rate of loss of
reactant is slope of tangent to a
plot of reactant concentration
versus time


Changes from time to time
Depends on the instantaneous
concentration

Not always easy to monitor
concentration



Care must be exercised in
making general
conclusions based on
initial-rate studies
Must find measurable parameter
proportional to reactant
concentration
Sometimes monitor appearance of
product, rather than reactant
Order
Isolation method
In many situations, one may formally write the reaction
velocity approximately as a function of the form
v  k [ A]a [ B]b [ C ]c 
a, b, c are the orders of reaction under the conditions
 k is the rate constant
Many rate laws are more complex functions of concentration
than the simple product given above
 Example: Production of HBr from H2 and Br2

v HBr
 k
Orders are usually determined over a limited range of
conditions
Orders are not necessarily the same as the stoichiometric
coefficients for the reaction
Differential method of
determining order
ln(v)  k
 nln(C )
Example: Decomposition of
di-tert-butyl peroxide


Line slope = 1.04
Order with respect to DTBP is
close to 1 under these
conditions (and probably is 1)
When there are multiple
reactants, it helps to create
a situation in which one
reactant dominates the
measurement.


[ H2 ][ Br2 ]1/ 2
[ HBr ]
1 k'
[ Br2 ]
Calculate approximate
derivatives as ratios of
differences for specific
concentrations
Plot approximate derivatives
versus concentration
Follow reaction for a
limited time
Can optimize sensitivity
Ri

Ci 1  Ci
ti 1  ti
Ci

Ci 1  Ci
2
R1
R2
Create isolation by making
one material the limiting
reagent
Generally requires one to
have other reagents in
significant excess
 [ A] 
  1 
 [ A]2 
a
Can be used with an initialrate measurement

Ratios of reaction rates
allow one to determine
initial order
Determining initial order
Measure initial velocity as a function of the initial
amounts of reactants in the mixture
Example: OCl- + I-  OI- + Cl[OCl- ]
[I-]
[OH-]
0.0017
0.0017
1.00
0.0034
0.0017
1.00
0.0017
0.0034
1.00
0.0017
0.0017
0.50
Concentrations are in mol dm-3. Rate is mol dm-3 sec-1.
Initial rate
1.75  10-4
3.50  10-4
3.50  10-4
3.50  10-4
By comparison, one finds the initial rate equation
vinitial  k[OCl  ][ I  ][OH  ]1
k  60.55 s 1
2
Measurement methods
Concentration as a function of time

Example:
decomposition of ditert-butyl peroxide
slope  - k1
Rate constant for
this reaction under
these conditions is
k1 = 0.0193 min-1
from the slope of
the line
Chemical (wet) methods such as
titration



Take aliquots as a function of time
Analyze separately
Limited time scales (> 1-10 s)
Physical methods




Spectrometry, electrical
conductivity, pressure, etc.
Usually measured in situ
Requires one to find a physical
property proportional to the
concentration of a reactant (or
perhaps a product)
Can often measure faster than
chemical methods




Faster reactions are observed, down
to 10-15 s
Stopped-flow reactions
Flash photolysis
Relaxation methods
Integrated rate law – second
order in one reactant
Integrated rate laws
Gives an expression for the time
dependence of the concentration,
rather than derivative of the
concentration with time
Can be used under a variety of
conditions



First-order rate law
Second-order rate law can
also be integrated

Linear plot of 1/[A(t)] versus
t
Often see reported rate
constant for disappearance
of A


Limiting reagent
Single molecule
Fast reactions


Concentration is an
exponential function
of time
Logarithm of the
concentration is
linear in time
 Usually plot linear
form
 Slope gives –k1
keff = 2 k2
Exercise caution in
assessing reported rate
constants
 Is it the inherent rate
constant?
 Is it the rate constant for
the disappearance of the
reactant?
Integrated first-order rate law
Rate linear in
reatant
concentration
Integration to find
Reactant concentration as a
function of time
v 
d [ A]
dt
[ A]

v  
1 d [ A]
2 dt
 k 2[ A]2
1
d [ A]   2k 2 dt
[ A]2
[ A]
t
1
d [ A]   2k 2  dt
[ A]2
[ A ( 0 )]
0

1
[ A(t )]

1
[ A(0)]
 2k 2 t
Second-order rate law
Example:
 k1[ A]
1
d [ A]   k1dt
[ A]
[ A( 0 )]
2 A  Products


t
1
d [ A]    k1dt
[ A]
0
[ A(t )]  [ A(0)]exp(k1t )
ln[ A(t )]  ln [ A(0)]  k1t
Collision-induced
decomposition of
diacetylene, DA
Hou and Palmer, 1965
Linear plot of [DA]-1
versus t
keff = 6.79 x107
cm3/mol-sec

Note units of the rate
constant
3
Second-order rate law, first order
in two different reactants
A 
Integration relies on the
stoichiometry of the
reaction

Every time one molecule v
of A reacts, one molecule
of B reacts
Complex function


 
B  Products
d [ A]
 k 2 [ A][ B]
dt
Half life
Can describe time
dependence in several
different ways


Integration gives a complex result
Must measure both
 [ B] [ A(0)] 
ln
concentrations as

 [ A] [ B(0)] 
functions of time
Is not appropriate to the
situation in which the
initial concentrations of
the two reacts are the
same

[ B(0)]  [ A(0)]k2t
Rate constant, k
Half-life, t1/2, time for one
half of reactant to
disappear
 Half-life depends on the
initial concentration for
all orders except first

Other times that describe
the amount left




[B] = [B(0)] >> [A]
Equation simplifies
Reverts to a quasi-first
order equation for [A]
“Rate constant” is
[B(0)]k2
v  
reactant
 Time to use up 90% of
reactant
d [ A]
 k 2 [ A][ B]
dt
 [ B] [ A(0)] 
 
ln
 [ A] [ B(0)] 
Second order in one reactant
1
t1/2 
keff [ A(0)]
 Time to use up 10% of
Determining kinetic
parameters
Simplification of second-order
rate law
Use of isolation
Have one material in
great excess
First order
ln 2

k
t1/ 2
Two conceptual steps

[ B(0)]  [ A(0)]k 2t
 [ B(0)] [ A(0)] 
  [ B(0)]k 2t
ln
 [ A] [ B (0)] 

Find parameter proportional to concentration
Find appropriate function of time to allow
evaluation of time course
Phase
Parameter often measured
Parameter needed
Gas
P, total pressure
Pi, partial pressure of
reactant
Total optical absorption
Absorption of a single
component
Total conductance
Conductance of a single
component
Total volume
Volume change of a
single component
Titration
Concentration of a
single component
 [ A(0)] 
  [ B (0)]k 2t
ln
 [ A] 
 [ A] 
   [ B(0)]k 2t
ln
 [ A(0)] 
Solution
Integrated rate laws for other
reactant orders
Integration gives a
d [ A]
  k [ A]n
general form for all
dt
orders (except 1)
The power of the
1
1

 kt
function of
[ A(t )]n 1
[ A(0)]n 1
concentration linear
in time is related to
order of reaction for
the conditions under
which the system is
observed
Summary
Chemical change quantified by the mathematical
results of chemical kinetic calculations
Rate constant and order characterize a reaction

Usually seek to determine these quantities
Determining rates and velocities



Differential method
Integrated-rate-law method
Half-life method
Results often limited to a particular time scale or
situation


Initial reaction
Isolation of one reactant
4