>> Key Terms (indicated in blue within the text below):

Kinetics is a term that relates to how fast a reaction occurs. The rate of reaction is measured as the change in concentration of a product or reactant ([X]) over a given time (t). The rate of reaction for reactants is negative, since reactants are disappearing, and positive for products, which are appearing. Rate can be measured as average rate using the equation

rate of X

=

[X] t

=

([X]f – [X]i) (tf – ti)

(Equation 14.6)

Rate decreases over time. Therefore instantaneous rate, the rate at any given time, is sometimes used. The instantaneous rate can be determined from a tangent line at the relevant instant of time on a graph of concentration versus time. The instantaneous rate at the start of the reaction (t = 0) is called the initial rate.

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The rates for each product and reactant are related by the stoichiometry of the reaction. For example, a product with a stoichiometric coefficient of 2 appears twice as fast as a product with a stoichiometric coefficient of 1. The relationship between rates of each reactant (A and B) and products (C and D) depends on the stoichiometric coefficient (lowercase of letter representing products or reactants) according to

–1 a

[A] t

=

–1 b

[B] t

=

+1 c

[C] t

=

+1 d

[D] t

The relationship between concentration and rate is called the rate law. The rate is proportional to the product of the concentration of reactants raised to some exponent. It has the form

rate = k[A]^m[B]ⁿ         (Equation 14.10)

The proportionality constant (k) of this equation is called the rate constant. The exponents on the reactant concentration are called the order. With the form given, m is the order in A and n is the order in B. The sum of the exponents (m + n) is called the overall order. The order of the reaction is normally an integer or simple fraction.

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The rate law can be integrated to get a relationship between time (t) and concentration. For a first–order reaction with a single reactant (rate = k[X]), the integrated rate law is

ln[X] = –kt + ln[X]₀         (Equation 14.11)

where [X]₀ is the initial concentration of X. The integrated rate law for a second–order reaction with a single reactant (rate = k[X]²) is

1 [X]

= kt +

1 [X]0

(Equation 14.15)

A reaction that is first order in two reactants (rate = k[X][Y]) can be expressed as a pseudo–first–order reaction if the concentration of one reactant is significantly greater than that of the other. For example, if [Y] is much greater than [X], the rate law can be expressed as

rate = k'[X], where k' = k[Y]         (Equation 14.18)

Thus first–order relationships can be used. It is also possible to have a zero–order reaction (rate = k). For zero–order reactions, the integrated rate law is

[X] = –kt + [X]₀

Although there are many other forms of the rate law, the integrated rate laws are too complex to consider at this point.

Another way to express the rate of reaction is with the half–life. Half–life is the time required for the reactant concentration to decrease to half its initial value ([X] = 1/2[X]₀. The integrated rate laws can be used to relate the half–life (t_1/2) to rate constant (k) and initial concentration ([X]₀). For a first–order reaction,

t1/2

=

ln 2 k

=

0.693 k

(Equation 14.12)

A first–order reaction is not dependent on concentration of reactant. All nuclear reactions are first order reactions and the rates of nuclear reactions are commonly designated by the half–life.

The half–life of a second–order reaction is

t1/2

=

1 k[X]0

(Equation 14.16)

and that for a zero–order reaction is

t1/2

=

[X]0 k

The rate law is determined experimentally, rather than from the chemical reaction. This is because the overall chemical reaction does not necessarily reflect the way in which the reaction occurs. A mechanism is the step–by–step sequence by which a chemical reaction occurs. Each of these elementary steps goes at a specific rate. The rate depends on the slowest elementary step of the mechanism. Thus the rate law is determined by the slowest, rate–determining, elementary step rather than by the overall reaction.

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Reactions occur when bonds are broken and formed. The substance formed during this process, as bonds are breaking and forming, is called an activated complex. In some steps, an unstable substance that later undergoes further reaction is formed. This product of one elementary step that is used up in a subsequent step is called an intermediate.

Bonds breaking and forming usually occur as a result of a collision. The number of molecules participating in the collision is called the molecularity of the step. If only one molecule is involved, the step is unimolecular. If two molecules collide, the step is bimolecular. In the unlikely event that three molecules collide simultaneously, the step is called termolecular.

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For bond breakage to occur in the collision, the molecules must have sufficient kinetic energy. The energy required to get a reaction going is called the activation energy (E_a). At higher temperatures, more molecules will have sufficient energy to overcome the activation energy; therefore the reaction will occur at a faster rate. An energy profile diagrams the changes in energy (as H or G) versus the progress of the reaction (reactants to products). The activation energy appears as a hill between the reactants and the products. The substance at the top of the hill is called the transition state. The energy difference between the reactants and the transition state is the activation energy. The energy difference between products and reactants is the H (or G) for the reaction.

The relationship between temperature (T) and rate constant (k) is described by the Arrhenius equation

k = Ae^–Ea/RT          (Equation 14. 21)

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where E_a is the activation energy, R is the gas constant (8.314 J/mol•K) and A is the frequency factor. The frequency factor is related to how successful the collisions between molecules are. For a collision to result in a reactions, not only must there be sufficient kinetic energy for the bonds to break, but the molecules must collide in the proper orientation. The frequency factor takes orientation into account.

One way to increase the rate of a reaction is to add a catalyst. A catalyst increases the rate of reaction without itself being consumed. It does this by lowering the activation energy, often by directing the orientation of the colliding molecules. Catalysts are categorized as homogeneous, being in the same phase as the reactants and products, or heterogeneous, in a different phase from the reactants and products.