# Beer's Law: A Relationship between gunk and light

In order to design a machine that calculates the amount of gunk by measuring how much light gets through, we need to know what the relationship between gunk and light is. Luckily, the relationship is actually DIRECTLY PROPORTIONAL, and all we need to figure out is the slope. So let's write the equation, keeping in mind that ...

Instead of x, we say c (for concentration).

Instead of y, we say OD (for optical density, which is the technical way of saying “how much light the gunk absorbed”).

Instead of m (which is the slope, which is a rate), we say e (which stands for “extinction coefficient”, i.e., the amount of light that is extinguished by each extra bit of gunk). So we get:

Just to make it a little trickier, the amount of light absorbed also depends on how big your fish tank is. A 2 foot tank will absorb twice as much light as a 1 foot tank. Another DIRECTLY PROPORTIONAL relationship. So, to be really correct, we have to write the equation like this:

where "l" stands for the “length of the path” that light follows.

Luckily, those people that designed the spec were pretty smart: they keep the distance that light has to travel constant, and they make it equal to 1 cm, which means in practice we can just use the easier equation, OD = ec.

By the way, this particular equation has a name that's pretty easy to remember: Beer's Law. Yes, there was a person named Beer (Herr Professor Beer, actually).