MF111 Study Guide 4
Kinetics
Learning outcomes

Understand the factors that affect the rate of reaction

Calculate the average and instantaneous rates

Distinguish between zero, first and second order reactions

Apply differentiated and integrated rate equations

Apply the concept of pseudo-first and pseudo-second order kinetics

Predict the reaction order based on the rate-limiting step

Relate the concepts of activation energy, temperature and catalysis
Overview
Chemical kinetics, also known as reaction kinetics, is the study of rates of chemical processes.
Chemical kinetics include investigations of how different experimental conditions can influence
the speed of a chemical reaction and yield information about the reaction’s mechanism and
transition states, as well as the construction of mathematical models that can describe the
characteristics of a chemical reaction. In 1864, Peter Waage and Cato Guldberg pioneered the
development of chemical kinetics by formulating the law of mass action, which states that the
speed of a chemical reaction is proportional to the quantity of the reacting substances.
Factors that affect the rate of a reaction
The nature of the reaction
Some reactions are naturally faster than others. For instance, sodium tarnishes almost instantly
when exposed to moisture and air. Iron forms rust when exposed to moisture and air but the
reaction takes a much longer period (i.e. days) as iron does not lose electrons as easily as sodium.
Physical state of reactants
A reaction cannot occur if the reactants cannot come into contact (i.e. no contact, no reaction). A
reaction between reactants in the same phase (i.e. gas/gas or liquid/liquid) is called a
homogeneous reaction. In such cases, gases react more readily than liquid followed by solids
because it is much easier for reactants to react in the gaseous state. For a heterogeneous reaction
(reactants in two different phases), the collisions between the reactants can occur only at
interfaces between phases (i.e. the surface area of the more condensed phase). Figure 4.1 shows
how smaller particle size increases total surface area and its implications.
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Figure 4.1 An explosion in a wheat grain elevator in New Orleans, 1977 (right). A mixture of fine grain
particles and air was ignited by a spark. Diagram on the left shows how total surface area increases with
reduced size.
Concentration
Reaction rate increases with concentration, as described by the rate law and explained by the
collision theory. The more reactant molecules that collide per unit time, the more often a reaction
between them can occur. As reactant concentration increases, the frequency of collision increases.
Figure 4.2 illustrates the different rate of reactions of concentrated and dilute solutions.
Figure 4.2 Concentrated sulphuric acid is able to dehydrate sucrose into elemental carbon in a very rapid
exothermic reaction. In a dilute solution, the reaction proceeds so slowly that almost no changes are
observed.
Temperature
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Conducting a reaction at a higher temperature delivers more energy into the system and this
often increases the reaction rate by causing more collisions between particles, as explained by
the collision theory. Temperature increases also result in a greater number of particles achieving
the necessary activation energy for a success full reaction. The influence of temperature is
described by the Arrhenius equation. Rule of thumb: rates for many reactions double with every
10 degrees Celsius increase in temperature.
Cold
Hot
Figure 4.3 Elemental magnesium reacts with hot water to form magnesium hydroxide (purple). In cold
water, no changes are observable.
Rate and rate laws
The rate of reaction and rate law are the most important concepts in this topic. It is important to
note that the two terms are very similar but not the same:
Rate of reaction: The speed of a reaction measured in amount of reactant consumed per unit
time or in amount of product produced per unit time e.g. mol/min or mmol/s.
Rate law: An expression of the dependence of the rate of reaction on the concentrations of
reactants. There are two forms of rate law i.e. differential rate law and integrated rate law.
Expressing the rate of reaction
Figure 4.4 shows a simple reaction in which reactant A is converted into product B. The rate of
reaction can be expressed as the amount A consumed or B produced per unit time:
Rate  
[ A] [ B]

t
t
Note:  [A] ([A2] − [A1]) is a negative value, but the rate
whether expressed in terms of [A] or [B] is a positive value.
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For an equation where reactants are not converted into products in a 1 to 1 ratio (e.g.
A → 2B), the rate would be expressed as:
Rate  
[ A] 1 [ B]

t
2 t
Therefore, in a more complicated reaction such as aA + bB → cC + dD, the rate would be
expressed as:
Rate  
[ A]
[ B] [C ] [ D]



a.t
b.t
c.t
d .t
Rate 
[B ]
b.t
Rate  
[ A]
a.t
Figure 4.4 The rate of reaction of a hypothetical reaction where A is converted into B.
Measuring the rate of reaction
Another key feature of Figure 4.4 is that the lines representing [A] and [B] are not linear and this
leads to the following methods of expressing rate of reaction:
Average rate: The change in molar concentration of either reactants or products in unit over a
period of time.
Instantaneous rate: The change in molar concentration of either reactants or products at a given
point in time
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The average rate is simply the slope of a line such as the one representing [B] in Figure 4.4. It
can be calculated using this simple formula:
Rate 
[ B] [ B 2 ]  [ B1 ]

t
t 2  t1
However, because the line is not linear i.e. rate of reaction changes over time, the average rate
may not accurately represent the observed rate of reaction. Recall: concentration of reactants
influences the rate. Therefore, the instantaneous rate is often a better representation of rate
especially over a large time frame i.e. large ∆t. The instantaneous rate is often determined by
calculating the slope of a tangent at a given point of time as shown in Figure 4.5.
Concentration [C] (M)
6
A+B→C
5
4
3
2
Rate 
1
[C ] [C 2 ]  [C1 ] 5.15  3.24


 0.191
t
t 2  t1
15  5
0
0
5
10
Time (s)
15
20
Figure 4.5 The instantaneous rate at 10 s of a hypothetical reaction where A reacts with B to produce C
Differential rate law
The equation below shows the differential rate law of a reaction with only one reactant (A):
Reaction: aA → products
Rate  
 [ A]
 k[ A]n
t
The differential rate law shows how the rate of reaction changes according to the concentration.
The rate constant (k) depends on the intrinsic reactivity of the reaction but it can be influenced
by temperature i.e. it is only constant if temperature is constant. The reaction order (n) is the
degree to which the rate of the reaction depends on the concentration of each reactant. To a
certain extent n can be predicted based on the coefficient, a (i.e. if a is 2, n would also be 2) but
this must be verified by experimental data (cases where n ≠ a will be described later)
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Zero-order reactions
A zero-order reaction is independent of concentration i.e. the rate of reaction remains constant
over time (Figure 4.6). Figure 4.7 shows the reactant and product concentration changing linearly
over time.
Rate  
 [ A]
 k[ A]0  k
t
Remember: Any number to the power of zero = 1
2.5
Rate (M/s)
2
1.5
y =2
1
0.5
0
0
5
10
15
[Reactant]
12
14
12
10
8
6
4
2
0
10
y = -2x + 12
[Product]
[Reactant]
Figure 4.6 The rate of reaction of zero order reactions are independent of reactant concentration
y = 2x
8
6
4
2
0
0
2
4
6
0
2
Time (s)
4
6
Time (s)
Figure 4.7 Graph of [reactant] as a function of time (left) is a straight line; the slope is –k while the yintercept is the initial concentration of reactant. Graph of [product] as a function of time (right) is a
straight line with a slope of +k.
First-order reactions
The reaction rate of a first-order reaction is directly proportional to the concentration of one of
the reactants i.e. the rate of reaction is halved when the concentration of the reactant is halved.
Figure 4.9 shows the plot of reactant and product concentration against time for first-order
reactions, both of which are not linear.
Rate  
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 [ A]
 k[ A]1  k[ A]
t
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Rate (M/s)
8
y = 0.6931x
6
4
2
0
0
5
10
15
[Reactant]
14
12
10
8
6
4
2
0
[Product]
[Reactant]
Figure 4.8 The rate of reaction of first-order reactions
is linearly proportional to reactant concentration
0
2
4
14
12
10
8
6
4
2
0
6
0
2
Time (s)
4
6
Time (s)
Figure 4.9 Plot of reactant (left) and product (right)
concentration as a function of time for first-order reactions
Second-order reactions
The reaction rate of a second-order reaction is directly proportional to the square of the
concentration of the reactant (Figure 4.10). If the concentration of reactant doubles, the rate of
the reaction increases by a factor of four. Like first-order reactions, plot of reactant and product
concentration against time for second order reactions are not linear (Figure 4.11).
Rate  
 [ A]
 k[ A]2
t
16
Rate (M/s)
14
12
10
y = 0.5167x 2
8
6
4
2
0
0
2
4
6
[Reactant]
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14
12
10
8
6
4
2
0
[Product]
[Reactant]
Figure 4.10 The rate of reaction of first-order reactions
is proportional to the square of reactant concentration
First order
Second order
0
2
4
Time (s)
6
8
14
12
10
8
6
4
2
0
First order
Second order
0
2
4
6
8
Time (s)
Figure 4.11 Plot of reactant (left) and product (right) concentration as a function of time for first- and
second-order reactions. Note that the initial rates of reaction for second-order reactions are a lot faster
than first-order reactions.
Initial rate
To observe how concentration affects the rate, the rate of reaction is often determined
experimentally by measuring the initial rate of reaction using different concentration of reactants.
It can be observed from Figure 4.11 that the slopes of the line for first- and second-order
reactions are approximately linear when time is close to zero.
Integrated rate law
Unlike the differential rate law that shows the relationship between rate and concentration, the
integrated rate law shows the relationship between concentration and time. Integrated rate laws
are used to find the time needed to reach a certain concentration or the concentration present
after a given time. Integrated rate laws also allow us to determine the reaction order and rate
constants graphically. Point to ponder: Tangents can be inaccurate. How can the integrated rate
law be used with the differential rate law to calculate the instantaneous rate?
The equations below show how a first-order rate law is integrated. The reactant concentration is
represented by [A]. You do not need to know how to do the integration (this is not calculus) but
you would need to know how to apply the integrated rate law equations. These equations are
listed in Table 4.1.
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

 [ A]
 k[ A]
t
 [ A]
[ A]
[ A ]t


[ A ]0
 kt
 [ A]
[ A]
t
 k  t
t0
 (ln[ A]t  ln[ A]0 )  kt  kt0
ln[ A]0  ln[ A]t  kt
Table 4.1 Summary of differential and integrated rate laws for zero-, first- and second-order reactions.
Reaction
order
Differential
rate law
Zero

Integrated
rate law
 [ A]
[ A]t  [ A]0  kt
k
t
 [ A]

 k[ A] ln[ A]  ln[ A]  kt
t
0
t
First

Second
 [ A]
 k[ A]2 1  1  kt
t
[ A]t [ A]0
Linear
kinetic plot
Slope of
kinetic plot
y-intercept
[A]t vs t
−k
[A]0
ln[A]t vs t
−k
ln[A]0
1
vs t
[ A]t
+k
1
[ A] 0
The linear kinetic plot for each type of reaction is very useful for determining the rate constant k
and for verifying the reaction order e.g. if ln[A]t vs t does not give a linear line, then the reaction
is not first order (Figure 4.12).
Time (s)
[A]
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6
3
1.5
0.75
0.375
−k = −0.6931
k = 0.6931
4
y = -0.6931x + 2.4849
2
Ln[A ]
0
1
2
3
4
5
ln[A]0 = 2.489
[A]0 = 12
0
0
2
4
6
-2
Time (s)
Figure 4.12 Linear kinetic plot of ln[A]t vs t of a first-order reaction.
A linear line is proof that the reaction is indeed first order.
Multiple reactants
Most reactions often involve more than one reactant:
Reaction: A + B → C
Rate  
 [C ]
 k[ A]n [ B]m
t
The exponents m and n are independent of one another. Therefore, the reaction can be first order
with respect to one reactant and second order to another i.e. k[A][B]2. To simplify interpretation
of experimental data, the equation can be simplified if the concentration of one of the reactants is
kept constant e.g. if [B] is in large excess to [A].
Rate  
 [C ]
 k[ A]n [ B]m  k '[ A]n
t
When [B] is in excess, [B] would be constant and k’ = k[B]m. The resulting equation would be a
pseudo-first or pseudo-second order reaction depending on whether n is 1 or 2. This concept
would be used many times to simplify interpretation of data.
Reaction order predictions
As mention before, the order of a rate law can be predicted based on the coefficient of the
reactants. Below is an example involving the decomposition of NO2Cl:
Reaction: 2NO2Cl → 2 NO2 + Cl2
Predicted rate law: k[NO2Cl]2 (wrong)
However, in this case, the prediction is wrong because the decomposition of NO2Cl is a multistep reaction involving the following equations:
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NO2Cl → NO2 + Cl (slow)
NO2Cl + Cl → NO2 + Cl2 (fast)
Predicted rate law: k[NO2Cl] (correct)
Most reactions comprise of multiple elementary reactions i.e. steps. In multi-step equations, the
rate law has to be predicted based on the slowest reaction, which is also known as the ratedetermining step or the rate-limiting step. It is impossible to guess the number of steps involved
in a reaction and thus all predictions must be verified using experimental data.
The Arrhenius equation
A minimum energy (activation energy, Ea) is required for a collision between molecules to result
in a chemical reaction (Figure 4.13). Reacting molecules must have enough energy to overcome
electrostatic repulsion and a minimum amount of energy to break chemical bonds so that new
ones may be formed.
Figure 4.13 Energy diagram showing the change in potential energy as reactants are converted into
products.
Activation energy is not influenced by temperature. However, at higher temperatures a greater
proportion of molecules achieve the minimum kinetic energy required for a reaction (Figure
4.14). The Arrhenius equation shows the relationship between the rate constant (k) with
activation energy (Ea) and temperature (T in kelvin):
k  Ae  ( Ea / RT )
ln k  ln A 
Ea 1

R T
R is the molar gas constant (8.314 J K-1mol-1)
A is the collision frequency factor
(constant, determined experimentally)
Although the activation energy cannot be influenced by temperature, it can be influenced by the
presence of a catalyst. A catalyst reduces the activation energy and increases the rate of reaction
by allowing a larger proportion of molecules to achieve the necessary kinetic energy (Figure
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4.15). The opposite of a catalyst is an inhibitor which raises activation energy. Both catalyst and
inhibitors are used to promote desired reactions and inhibit unwanted side-reactions. Enzymes
are an example of biological catalyst. They are specific to molecules of a particular shape
analogous to a lock and key (Figure 4.16).
Figure 4.14 Effects on temperature on the kinetic energy of molecules in relation to the activation energy
of a reaction.
Figure 4.15 Effects of a catalyst on a reaction.
Figure 4.16 Enzymes are biological catalysts.
Reading material

Chapters 16.1 to 16.8
Silberberg, M.S. (2006). Chemistry: The Molecular Nature of Matter and Change. 4th Ed.
McGraw Hill.
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