How many bananas would you need to eat to die of radiation poisoning?

We've all heard someone tell us that bananas are "radioactive." I always thought it was odd that a popular fruit and one of the fruits that I eat most was in fact radioactive. So I decided that it would make perfect sense to answer a question thats been nagging me for a while:

How many bananas would you need to eat to die of radiation poisoning?

Most people know that bananas are full of potassium. In fact a single banana contains approximately 450 mg of potassium. Of this potassium about 0.01% of it contains potassium-40 a highly radioactive isotope.

Essentially from the information above we can say that bananas have around 45 micrograms of radioactive material.

Before we can do any equations or draw any conclusions we must first learn more about potassium-40, our radioactive isotope. Since it is radioactive it has a half-life of 10*(9) years or  3 X 10^16 seconds.

Beta Decay of a potassium-40: Beta Decay essentially means radiatiove decay when an electron is released. Beta decays make up 89% of all decays for potassium-40. There are other types of decay (alpha, beta, gamma).

Basically all of these very confusing values contribute to the idea that potassium-40 has an extremely long half life and will take a long time to break down.

The q-value of the Beta Decays is 1311 KeV or 1.18*10^-13 joules. Essentially this value represents the energy released from the parent isotope (potassium-40) to the daughter isotope. When the parent isotope begins to break down, it emits energy. This energy represents the q-value.

We know that our half life is 1.28 X 10(9), which is a very large half-life. Basically this means that the isotope will take a much longer time to break down. 

To calculate the amount of atoms of potassium-40 we do: 

45 micrograms (amount of radioactive material per banana) of potassium 40 = 1.125 micromoles = 6.77*10^17 atoms of potassium 40. We get this by multiplying the number of micromoles by avogadro's constant (the amount of atoms in one mole).

Take the energy per decay: 1.18 X 10(-13) and divide it by the half life decay rate: 3 X10^16. Then multiply by the number of atoms of potassium-40 in a banana: 6.77 X10^17. This will give us the energy released per second per banana which is 2.66 X 10^-12 joules per second per banana.

Grays is a type of unit that measures absorbed dose. To begin feeling sick you would need approx 2 grays, and to die you would need closer to 3-5 grays of radiation. Grays are equivalent to joules per kilo. Therefore 5 grays would require 5 joules per kilo. Lets say we are dealing with an 80 kilo man (176 lb). You would need 400 joules per second.

400 Joules per second divided by 2.66 X 10^-12 (energy released per second per banana) = 1.503 X 10^14. You would need to eat 150 trillion bananas in a second!

Obviously this is not a logical possibility. Firstly it is unlikely that there are even 154 trillion bananas on the planet. Another reason this is not feasible is the simple fact that bananas still have 450 mg of potassium. Potassium is a metal, therefore if you were to eat anywhere near to this amount of bananas you would die because of huge electrolyte imbalance. Lastly potassium in high doses is extremely dangerous (KCl, Potassium Chloride is used for lethal injections).

Logan Konarek