## The special case of 1 st order reactions

First order reactions have an interesting aspect to them. We know that for a first order reaction, if the concentration is doubled, the rate will double. It stands to reason then, that if the concentration is cut in half, the rate will be cut in half.

The result of that simple logic is very interesting, however. As any reaction occurs the reactant is used up and the reaction slows. For a first order reaction the rate of slowing is very predictable.

Imagine a first order reaction in a container that holds 40 molecules of the reactant. Now imagine that the reaction starts fast enough to use 20 of those molecules in the first minute. After that minute the concentration will be half of what it was (20 molecules) and, as a result, will be going half as fast. That means in the second minute it will only use up 10 molecules (leaving 10 behind). Now of course the concentration has again been cut in half, so the rate will also be cut in half. In the third minute, the reaction will use 5 of the remaining 10 molecules.

In other words, in each minute, exactly half of what remained was used up. Another way to say that is that it took a certain amount of time (in this case) a minute to use up half of the reactant, no matter how much was present at the beginning of that minute. That’s pretty weird! The rate is changing constantly, but the amount of time to use up half is always the same. This is called the half-life. For those of you who know some advanced mathematics, you may recognize that the constantly changing rate would require some calculus to fully analyze, but the end result is the same. it always takes the same amount of time to use half of whatever you have.

Let’s generalize a little. All first order reactions have a half-life that can be calculated. The half-life differs from reaction to reaction, but in every case the half life is the time it takes for half of what is present to react.

If it helps, we can understand this with an analogy.

Understanding this aspect of first order reactions is important because radioactive decay occurs in first order reactions. As a result, these reactions can be used to determine the age of artifacts and other things.