 you can't pull it out of thin air. Before we look at some rate data, I want to show you the maths that we used to derive the rate law from experiments. The logic is simple and it's well illustrated by some data on a geometrical shape. So for the purposes of this analogy you need to imagine that you don't know the equation that describes the relationship between the volume of a rectangular prism and its length, width and height. Imagine instead that you've been given this 3D object and that you can modify its length and width only. The height is fixed and that you have to find an equation for its volume. You do some experiments. You vary the length and the width and you measure the volume and you come up with this data in the table here. You look through the data and you can see some relationships. If you look at experiments one and two you'll see that the width was held constant but the length was doubled. When you do that you can see that the volume doubles also. So we can say that the volume must be proportional to the length. You double the length. You double the volume. Note this is not the same as saying it's equal to the length. It's clearly not. But rather that when we multiply the length by a factor the volume gets multiplied by the same factor. Equally you can see that the volume must be proportional to the width. Look at experiments two and three. When the length is held constant but the width is halved the volume halves. You can see a similar relationship between experiments three and four. The width is quadrupled and the volume is also quadrupled. So far so good but we can go further. If the volume is proportional to both the length and the width then we can say that it must be proportional to the product of the length and the width. Well this makes sense. For instance we know that the volume doubles when the length doubles and it doubles when the width doubles. So if we double both the length and the width the volume will double and double again. That is it'll be multiplied by four. If you compare experiments one and four you can see this happening. Alternatively look at experiments one and three. You can see that the length doubles so that will double the volume. But you can also see that the width halves so that will halve the volume. So the combined effect is a doubling and a halving of the volume so it stays the same. Okay but we've got one last problem. Although we know that the volume is proportional to the product of the length and the width it's not equal to the product of the length and the width. Looking at experiment one clearly two times five is not 30 so there's something missing. What's missing is a proportionality constant an unchanging value that's put into the expression to turn it into a true equation with both sides having the same numerical value. At the moment we don't know the value of this constant so we'll just call it k. c already means the speed of light so k is often used to indicate constant. And we write our equation like this volume equals k times length times width. Now since we have some experimental data we can find out what k is because all we have to do is substitute some values into our new equation and divide the volume value by length times width and that will give us k. And if you look at all the data you can see that in every case volume divided by length times width equals three. So for this system the value of k is three. So we now have an equation that expresses precisely how the length and the width of this object are related to its volume. Obviously in this particular case the constant is the height of the shape and for the purposes of this analogy I've fixed the height at three. It could really have any value. However the point is the process of determining the equation.