Concise Physical Chemistry

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158 CHEMICAL KINETICS

0.541

0

(.9⋅r)^2 ⋅e−.9⋅r

r^2 ⋅e−r

[1.1⋅(r)]^2 ⋅e−1.1⋅r

0 r 10

FIGURE 10.6 A Boltzmann distribution of molecular speeds. The area under the tail of the

curve beyondrincreases exponentially.

The quantitative relationship between the reaction rate constant and temperature is

k=ae−aH/RT

This can be appreciated by looking at a Boltzmann distribution of reactant molecules.

The area under the Boltzmann curve that surpasses the activation barrier atrincreases

sharply (exponentially) with temperature, leading to the observed exponential in-

crease ink(Fig. 10.6). The exponential relationship betweenkandTis called the

Arrhenius rate law.

10.7 COLLISION THEORY

One can calculate the number of molecules colliding with one another in a gaseous

sample at a specified temperature. SupposeZABis the collision frequency for a

collection of gas molecules capable of undergoing a bimolecular reaction. If all the

molecules were activated, the rate would be directly determined by the number of

collisions per unit time of molecules of kind A with those of kind B. Not all molecules

will be activated, however, and we must multiplyZABby the fraction that are. This

is the Boltzmann factore−aH/RT.

For a gas reaction taking place between molecules that are structurally compli-

cated, a probability factorPis introduced to account for collisions between activated

molecules that are not oriented toward each other in the proper way for reaction to

occur. Now

k=PZABe−aH/RT

where the integration factorais included in the empirical probability factorP.