# How is the Activation Energy of the Arrhenius equation related to the exponents in rate laws?

I know that activation energy for a particular reaction does not depend on the stoichiometry of the reaction.But how can that be justified from Arrhenius Equation ?Doesnt 'k' change with stoichiometry?Or am I going wrong?

Because we know that in all the kinetics equations, when the reaction stoichiometry changes the rate constant in the equations have to be altered accordingly.

Rate Law

For the reaction:

$\ce{aA + bB -> cP}$

The rate law is as follows:

$R = k[A]^x[B]^y$

where:

$a$,$b$,and $c$ are stoichiometric constants which have integer values
$[A]$ is the concentration of species A
$x$ is the order with respect to species A
$[B]$ is the concentration of species B
$y$ is the order with respect to species B
$k$ is the rate constant.

(1) Neither $x$ nor $y$ depend on $a$, $b$, or $c$.

(2) Neither $x$ nor $y$ is required to be an integer value.

(3) k is measured for a specific temperature. The Arrhenius Equation would predict how k would change with temperature. So k does not depend on any of $x$, $y$, $a$, $b$, or $c$.

Say the reaction was for some sort of gas reaction. I now add a catalyst. I get a new rate:

$R_2 = k_2[A]^{x_2}[B]^{y_2}$

There isn't any way to use the first rate equation to predict what the second rate equation will be. (You could of course try to analyze the function of the catalyst to make predictions on what the best catalyst would be...)

There is a way to think about the problem that will give you an intuition about the key relationships and why they are the way they are. Think about the individual molecular interactions that have to occur for a reaction to happen.

For example, consider some reaction where two molecules have to bang together to cause some chemical change. In many reactions they have to bang together with enough energy for the reaction to actually occur. This is the reason there is an activation energy. Too little energy in the collision and there isn't enough to initiate the chemical change (perhaps a bond needs to be stretched or a repulsive barrier has to be overcome). From this perspective it is easy to see that the activation energy doesn't depend on the concentration.

But the concentration of the molecules does influence the probability that a collision will occur. If there are a lot more of the relevant molecules around, collisions are far more likely and the rate of the reaction will be faster. But concentration doesn't influence the barrier to the reaction for individual collisions.

The high level equations for rates of reaction are, essentially, an embodiment of these ideas. But thinking one molecular collision at a time will give you the correct intuition about the factors that matter in the equation and how they interact.