# 0.15 Reaction rates  (Page 8/10)

 Page 8 / 10

Although we will not show it here, kinetic molecular theory shows that the fraction of molecules with energygreater than ${E}_{a}$ at temperature $T$ is proportional to $e^{-\left(\frac{{E}_{a}}{RT}\right)}$ . This means that the reaction rate and therefore also the rateconstant must be proportional to $e^{-\left(\frac{{E}_{a}}{RT}\right)}$ . Therefore we can write

$k(T)=Ae^{-\left(\frac{{E}_{a}}{RT}\right)}$
where $A$ is a proportionality constant. If we take the logarithm of both sides of , we find that
$\ln k(T)=-\left(\frac{{E}_{a}}{RT}\right)+\ln A$
This equation matches the experimentallyobserved . We recall that a graph of $\ln k$ versus $\frac{1}{T}$ is observed to be linear. Now we can see that the slope of that graph is equal to $-\left(\frac{{E}_{a}}{R}\right)$ .

As a final note on , the constant $A$ must have some physical significant. We have accounted for the probability ofcollision between two molecules and we have accounted for the energetic requirement for a successful reactive collision. We havenot accounted for the probability that a collision will have the appropriate orientation of reactant molecules during the collision.Moreover, not every collision which occurs with proper orientation and sufficient energy will actually result in a reaction. There areother random factors relating to the internal structure of each molecule at the instant of collision. The factor $A$ takes account for all of these factors, and is essentially the probability that a collision with sufficient energy for reactionwill indeed lead to reaction. $A$ is commonly called the frequency factor .

## Observation 4: rate laws for more complicated reaction processes

Our collision model in the previous section accounts for the concentration and temperature dependence of thereaction rate, as expressed by the rate law. The concentration dependence arises from calculating the probability of the reactantmolecules being in the same vicinity at the same instant. Therefore, we should be able to predict the rate law for anyreaction by simply multiplying together the concentrations of all reactant molecules in the balanced stoichiometric equation. Theorder of the reaction should therefore be simply related to the stoichiometric coefficients in the reaction. However, shows that this is incorrect for many reactions.

Consider for example the apparently simple reaction

$2I\mathrm{Cl}\left(g\right)+{H}_{2}\left(g\right)\to 2H\mathrm{Cl}\left(g\right)+{I}_{2}\left(g\right)$
Based on the collision model, we would assume that the reaction occurs by $2I\mathrm{Cl}$ molecules colliding with a single ${H}_{2}$ molecule. The probability for such a collision should be proportional to $\left[I\mathrm{Cl}\right]^{2}\left[{H}_{2}\right]$ . However, experimentally we observe (see ) that the rate law for this reaction is
$\mathrm{Rate}=k\left[I\mathrm{Cl}\right]\left[{H}_{2}\right]$
As a second example, consider the reaction
$N{O}_{2}\left(g\right)+CO\left(g\right)\to NO\left(g\right)+C{O}_{2}\left(g\right)$
It would seem reasonable to assume that this reaction occurs as a single collision in which an oxygen atom isexchanged between the two molecules. However, the experimentally observed rate law for this reaction is
$\mathrm{Rate}=k\left[N{O}_{2}\right]^{2}$
In this case, the $\left[CO\right]$ concentration does not affect the rate of the reaction at all, and the $\left[N{O}_{2}\right]$ concentration is squared. These examples demonstrate that the rate law for a reaction cannot be predicted from the stoichiometriccoefficients and therefore that the collision model does not account for the rate of the reaction. There must be somethingseriously incomplete with the collision model.

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