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 à 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.

Experiment #

[A]

[B]

rate

1

1.00 M

0.50M

1.2 M/s

2

2.00 M

0.50M

4.8 M/s

3

2.00 M

1.00M

4.8 M/s

 

We know that the rate law has the form

 

So for experiment #1 the rate law takes the form

 

and for experiment #2 the rate law looks like

.

If we divide these two (with experiment #2 over experiment #1), we get something that looks like this:

 

This can be simplified in the following ways: we know that

 

and that = 1 (no matter what y is).

 

This gives us the equation

 

This can be further simplified by dividing on the left and simplifying the fraction on the right.

 

Which simplifies into

 

Therefore 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

 

(remember that we figured out x= 2 in the steps above).

Dividing the rate for reaction 3 by the rate for reaction 2 gives us the following equation:

 

As before, the k’s will cancel, as will the [A]'s, leaving the simplified equation:

 

 

which simplifies to

 

Given this, the only possible value of y is 0. Thus our rate law is now: .

 

This could, of course be written more simply as .

 

Knowing the values of the exponents, allows us to calculate the value of k

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