893 KWH = 3.6⁶ joules, one sievert is equal to 1 joule of energy, therefore we require 3.6⁶ sieverts, which is 3.6¹² microsieverts, multiply by ten is 3.6¹³ microsieverts, which means that it takes 3.6¹³ bananas to power the average American household for a month. (That's a lot of bananas)Thank you for reading this calculation. It's probably wrong (the joule to sievert conversion is really iffy). Please correct me on any errors. Thank you :)

First off, I love this question! Your approach is quite logical, although there are a few notes that may improve the accuracy. As you point out, the conversion from sievert to joules is pretty iffy. Sieverts measure dosage of radiation, or how much energy your body absorbs from the banana’s radioactive decay while you eat it. The unit is normalized by the mass of the person eating the banana and isn’t quite related to the true amount of absorbed radiation, which makes it difficult to convert to something like power output. For the 0.1 microsieverts you referenced, this is also the total radiation dose for an adult human over a span of 50 years (assuming the potassium stays in their system that long), which further complicates things and isn’t necessarily accurate. Also, while there’s nothing wrong with using kilowatt-hours per month, it makes the math easier if you just convert this to watts or kilowatts (893kwh/month = 1.22kilowatts).

Let’s try a slightly different approach and see how it compares to your answer. As you note, the reason why bananas are radioactive is because they contain potassium. And more specifically, that potassium naturally contains a few different isotopes. Most of it is potassium-39 and potassium-41, which are stable. But a tiny fraction (0.012%) is potassium-40, which is radioactive and decays into other more stable elements over time. When the potassium-40 atoms undergo radioactive decay, they release a ton of energy! There are a couple different types of radioactive decay that can happen, but on average, each potassium-40 atom releases about 1.326 MeV (or 2.125^-13 joules) when it decays. Of course, the potassium-40 in a banana does not all decay at once. If it did, all bananas would explode with the energy of about 40g of TNT! Rather, the atoms fall apart randomly and at a pretty slow rate, as described by potassium-40’s half-life (the time it takes for half of the atoms in a sample to decay). For potassium-40, this is about 1.25 billion years (3.95¹⁶ seconds). Based on this rate of decay, we can calculate an average energy output of potassium-40:

Power = (2.125^-13 J/atom) * (6.022²³ atoms/mole) * ln(2) / (3.95¹⁶ seconds) = 2.25^-6 watts per mole potassium-40

Or converting to power per gram of potassium-40,
power = 5.63^-8 watts per gram potassium-40

So to power a house, we need 2.17¹⁰ grams of potassium-40. And if we know that the natural abundance of potassium-40 is about 0.012%, we can calculate that we need roughly 1.81¹⁴ grams of potassium. Each banana only contains about 0.422 g potassium, so this would mean we need about 429 trillion bananas (4.29¹⁴) to power a house! Which is actually not too far off from your value given the accuracy of this sort of estimation!

Of course the idea of collecting all that radioactivity with perfect efficiency from an absolutely enormous pile of bananas is quite unlikely, so you would probably need even more bananas to account for inefficiencies. Also, one interesting outcome of this problem is that because the half-life of potassium-40 is so long, this pile of bananas would power your house well beyond your lifespan! So if you can manage to find 429 trillion bananas, you won’t need to keep restocking every month (as long as you can deal with the smell of a few hundred trillion rotting bananas).

Hi Kang, excellent question. You put great effort in the calculation of banana radioactivity. Well done using conversions to calculate the joules. Your calculation was well thought out, especially taking into account .1 microsieverts. However, you're right about the iffy joule sievert conversion. Its the fault of the unit here. It's important to understand the sievert unit represents the biological affects. Sievert means the health effect of 1 joule of radiation energy on 1kg of human tissue. That's because we care about people eating enough potassium rich bananas but not too many bananas. So, let's use a slightly different calculation.

Thinking about energy, a bananas chemical energy is more useful in calculating the bananas ability to power a household. How do we power a house without energy conversion? Well, remember, practically a banana must generate electricity. We must convert the radiation energy or chemical energy into electrical energy!

There's a very cool application I suggest you read about. Quite a mouthful, a radioisotope thermoelectric generator! Radioactive material makes heat, the heat converts to electricity! Incredible work by NASA to power Mars rovers! It even powers satellites. Click this link to learn more!

My brother Jim sees one problem:
1 sievert = 1 joule/kilogram
So it is not a direct conversion.

The sievert represents the equivalent biological effect of the deposit of a joule of radiation energy in a kilogram of human tissue. Measuring radiation and its effects can be awfully messy.

I'll add that you get a lot more radiation by going from sea level to someplace higher up, like Denver, than you do when eating a banana.