How fast will my coffee get cold?

With a not so difficult mathematical formula you can calculate exactly at what time will your cup of coffee (or tee) get cold (or to a certain temperature). It all depends on the room temperature, if it's colder the coffee will get colder faster and if it's a hot summer's day you coffee will stay warm for longer.

Let's take a look at the formula:







This is called a first-order differential equation. The T refers to the temperature at any given time t and A is the room temperature. The constant k indicates the rate at which the temperature will change.

Let's solve this equation:


Now we need to A (room temperature), B (which we can find by knowing the initial temperature of the coffee) and k (which is a bit more difficult to find).

Let's say A = 22˚C and that since water boils at 100˚C and we immediately make our coffee, let's assume that T(0) = 90˚C (Taking into account that it will cool off a bit when pouring and stirring).

Now we have the following:

To find the value of k we need to do some experimentation, where we determine the temperature of the coffee after some specific time. So let's say that after t=2 minutes we find that the temperature of the coffee is now 85˚C (I'm just guessing a value, for exact results one would have to take the actual temperature of the coffee after two minutes).

So now we find k:

Hence the equation becomes:

If I now want to know the temperature of my coffee after 20 minutes, let t = 20, then

and you will see that as time goes by the temperature will drop at a slower rate until it reaches the room temperature.

Happy coffee-drinking!