How do we calculate x, y, and k?

In order to determine these values, we need to start with some data.

Let’s assume that a chemist runs the reaction 2 A + 3 B → C + 3 D three times.

Each time, she measures the starting concentrations of A and B and the rate of the reaction. The data she obtained is in the table below.

We know that the rate law has the form:

Now, to solve for x, we need to pick a pair of reactions in which the concentration of one reactant changes and the other does not. We'll use reactions 1 and 2. 

So, for experiment #1, our equation looks like:

and for experiment #2, it looks like this:

If we divide these 2 (putting #2 over #1)

This can be simplified in two ways. we know both that

So, our equation simplifies down to:

This simplifies to

and then to             Thus x = 2.

Calculating y is done the same way, but in this case we want to pick a pair of reactions where the concentration of B changes but the concentration of A does not.

In this example, that occurs between reactions 2 and 3. We know that the rate law takes the form

Dividing the equation for reaction 3 by reaction 2 (and remembering that we already know that x=2) leaves us this:

As before, the k's will cancel as will the [A]'s, leaving: 1=[2]^y.The only possible value of y is 0. That means that our rate law is:

Now that we know the values of the exponents (x and y), we can solve for k.