Physical Chemistry , 1st ed.







(

(

l

1

n

/T

k

)

)



E

R

A

—that is, that this derivative has some constant value given by the expression
EA/R—it is better to assume that the slope of the plot of ln kversus 1/Tis
dependent on temperature, also. That is, we also include a temperature term,
so the equation becomes







(

(

l

1

n

/T

k

)

)



E

R

AmT (20.53)

It is the convention to use a negative sign on this additional term. When we go
through and do the rearrangements and integrations like we did for equation
20.50, we ultimately get

kATmeEA/RT (20.54)

Defined like this,mis usually some negative number. For m0, equation
20.54 reduces into the Arrhenius equation.
How can we justify the Arrhenius equation beyond some energy difference
between the reactants and a transition state? The pre-exponential factor, which
is a constant for a given reaction (that is, it does not depend on temperature)
must have a value that is dictated by the specifics of the reaction itself, like the
nature of the reactants and how they interact on a molecular level.
What are the specific characteristics of a gas-phase molecular interaction
that determine if the molecules react? One of the most obvious is the number
of molecules that are colliding. The number of collisions is calculable from ki-
netic theory; we covered this topic starting in section 19.4. For example, at the
end of that section we showed that the total number of collisions per second
per unit volume, represented by Z, is given by

Z (20.55)

where d 1 and d 2 are the diameters of gas particles in species 1 and 2, and is
the reduced mass of two particles of those species. If we suggest that the effect
of temperature changes is minor compared to the exponential term in the
Arrhenius equation (that is, the eEA/RTterm), and if the densities of the two
species  1 and  2 are converted to concentrations and separated from the rest
of the expression, then we can argue that the rest of the expression is approx-
imately constant:

Z 1  2 (20.56)

constant

This one factor, a collision frequency factor,is one major contribution to the
pre-exponential constant A.
A second contribution to the value ofAis the orientation of the two reac-
tant species with respect to each other, and what fraction of collisions are ori-
ented properly so that bond rearrangement might occur (if the molecules have
enough energy—but that’s the consideration of the exponential term in the
Arrhenius equation). Figure 20.17 shows an example of how we can argue for
a steric factoras a contribution to the pre-exponential factor A. In one case, the



d 1 

2

d 2



2



8 kT



 (^12)
 1  (^2) 
d 1 
2
d 2

2

8 kT

 (^12)
20.6 Temperature Dependence 705
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