The Maths of Recycled KitKats

Every so often I see a certain fun fact or news item about chocolates or soft drink do the rounds of the internet. The most recent example I saw said that when a KitKat fails quality control, it doesn’t get thrown out, instead it gets mashed up into the chocolate paste that fills the KitKat wafers, meaning KitKats are full of recycled KitKats.

It’s not only KitKats that do this. Most confectionaries, including other chocolates, caramel, cookies, soft drinks, you name it, are recycled in this way.

When these news items pop up on sites like Reddit, it’s usually not too long before someone asks the interesting question:

If this KitKat is made from recycled KitKats…
Which themselves were made from recycled KitKats…
Could the KitKat I’m eating contain some of the very first batch of KitKats?

Let’s find out.

We don’t know what fraction of the confectionary gets recycled and it’s going to vary from product to product, but as an example, let’s assume that 10% of KitKats fail quality control and are recycled. On day two, your KitKat batch contains 10% from the original batch and again, 10% of those KitKats fail and are recycled. So now you’ve got 10% of the original 10%. Each day, you only retain 10% of the previous batch, so each day you multiply your original amount remaining by 0.1.

If you start cooking on Monday, by the end of Friday you’ll have only:

0.1 × 0.1 × 0.1 × 0.1 × 0.1 = 0.1⁵ = 0.00001

Of your original batch left.

Because everything is made of atoms and molecules, nothing can be diluted indefinitely. Eventually, the last molecule remaining from your original batch will be in a KitKat that passes quality control and will leave the factory.

So, how long will it be before the last molecule from a whole vat of KitKat mixture disappears?

To answer that, we need to know how many molecules are in a KitKat. Chocolate is a complex mixture of a huge range of different chemicals that give it its texture and flavour, so it’s very difficult to work out how many individual molecules are in a certain amount of mixture. However, according to the nutrition information, a KitKat is 50% sugar, so I’ll use the molecular weight of pure sucrose to get an estimate for how many molecules are in a particular weight of KitKat.

A mol is the standard unit for measuring amounts of a chemical. One mol contains 6.022×10²³ molecules. A whole mol of sucrose weighs 342.3 grams.

An ordinary KitKat bar weighs 45 grams, which means it contains

(45/342.3) × 6.022×10²³ = 8 × 10²²

molecules of sugar.

That’s a huge number of molecules, but each day you multiply that number by 0.1, which means it starts to get a lot smaller very quickly.

After 23 days you would have only

0.1²³ × 8 × 10²² = 0.8

molecules left from a single KitKat. You can’t have a fraction of a molecule, so this means the average KitKat bar made 23 days later has a roughly 80% chance of having an original molecule in it.

After 24 days you would have

0.1²⁴ × 8 × 10²² = 0.08

molecules left, meaning you would need to eat 10 KitKat bars to have an 80% chance of eating just one molecule from the original batch.

After only 49 days, you would have

0.1⁴⁹ × 8 × 10²² = 8 × 10⁻²⁷

molecules left.

So, to have an 80% chance of having one original KitKat molecule, you will need 10²⁶ KitKats. Since each KitKat weighs 45 grams, that’s 4.5 × 10²⁷ grams of KitKats.

The entire Earth weighs about 5.9 × 10²⁷ grams. After only 49 days of recycling at a 10% KitKat failure rate, you would need to have a mass of KitKats nearly equal to the weight of the Earth to stand a good chance of having one molecule of the original batch.

So, the fact that everything, including KitKats, is made of molecules, and molecules are discreet units and cannot be infinitely diluted, means that the chances of any KitKat or other confectionery you choose containing any of the original batch are inconceivably tiny. But, depending on the fraction that fail and need recycling, it might contain some made two or three weeks ago.