Due in part to the recent events at Fukushima Daiichi, I have become more interested in understanding the physics of radioactive decay, both as it occurs in nature and as it has been harnessed in technology. One of the problems I ran into is that nuclear physics often uses its own terminology, and it is not always easy to compare those terms with more commonly used terms.
For example, radioactive decay often has its energy measured in electronvolts, defined as the amount of energy needed to drive an electron against a current of one volt.1 If we imagine an electron as a ball in a river, one electronvolt is the amount of energy needed to keep that ball still against a current of one volt.2 Most of the energies produced in nuclear decays are on the order of a Megaelectronvolt, or a million electron volts3, which is a great amount of energy. However, how do we compare that amount of energy to more commonly used units of energy? Luckily, physics has many different ways to convert energy from unit to unit.
And in this case, the formula for this is actually quite simple: for every mole of a material that undergoes a change of one electronvolt, 100,000 Joules are produced. (Actually, the real number is 96,485 Joules, but in my conceptual layout, I will use the more round number to simplify the arithmetic.)4 Thus, if a mole of material undergoes a change of one million electronvolts, as it does in nuclear decay, the total amount of energy produced will be 1,000,000 * 100,000, or 100,000,000,000 Joules. This is one hundred billion joules. A ton of oil has 40 billion joules in it, so a single mole of radioactive material undergoing decay will have the energy equivalent of two and a half tons of oil.
However, there is more to the story than this, because this decay takes place over time, often a great deal of time. Thus, we have to take the half-life into account. Therefore, our new expression comes out like this:
(electronvolts per decay)*100,000
time of half-life
So, for example, with Potassium-40, there is a decay energy of 1,300,000 electronvolts, multipled by 100,000, which gives us 130,000,000,000. However, this is then divided by the half-life of 1.25 billion years to give us about 100 Joules per year. 100 Joules per year is not, by most standards, a large amount of energy, which will be explained below.
Fun Interlude! So far I have been using the Joule, which is the SI unit of energy. One of the more commonly used units of energy in everyday life is the Food Calorie, which is equivalent to 4,000 Joules. Thus, it takes 40 years for a two mole quantity of Potassium-40 (about 3 ounces) to create one Food Calorie of energy. In other words, it pumps out a Tic Tac's worth of energy a century. After learning the basic formula, it is easy enough to convert Joules into Food Calories, and thus into fun units like "Tic Tac Centuries" or "Snickers Minutes" to compute radioactive energies.
Not-so-fun interlude: In order to have one mole worth of decay, you must have two moles worth of material to start with. And the average rate of energy production is an average, with more energy being produced towards the start, and less towards the end. To specify more accurately, calculus would be needed, but for our purposes, that is not necessary.
Having put out the basic formula, I can now show some applications of it:
- One of the first is how much energy was and is produced inside of the earth. The earth has a molten core, and has plate tectonics and volcanoes, due to the melting of the inside by radioactive decay. How much energy was produced inside of the earth, to still be driving volcanoes billions of years later? For example, if we take Plutonium-244, an element with a half-life of 82 million years, its entire decay chain, down to Thorium-232 (which is also radioactive, but with a very long half-life) produces 15.5 million electronvolts, over a period of 105 million years. (15,500,000 * 100,000)/105,000,000=15,000 Joules per year. This is a little bit better than our Potassium-40, but this is still only the equivalent of burning several Tic-Tacs a year. Of course, it wasn't only Plutonium-244, but a group of five long-lived isotopes that added most of the heat to the earth. To fully get an answer to this, we would have to know how quickly heat radiates through and away from rock (which involves calculus and geology) and how much of the different radioactive isotopes were (and are) present in the earth, which involves making stuff up. However, this method does provide some guidelines for how quickly the earth formed, and how much radioactive material must have been present.
- This formula can also be used to calculate the energy put off by isotopes created through technology. For example, one of the main isotopes released by Fukushima Daiichi is Iodine-131, which has a decay energy of 1 MeV, and a half life of 8 days. This gives us an energy of 100,000,000,000 Joules, divided by 8 days, or 125,000,000 Joules per day. Dividing this further, we get 5,000,000 Joules per hour. Once again, we can turn to our comparison to food calories. The heat energy produced by two moles of Iodine-131, or about 8 ounces, is equivalent to a large slice of cheesecake every hour, about 1,000 Food Calories. Although this is a large amount of energy, it is somewhat more manageable when imagined in those terms.
- The energy needed for self-dispersal. If a radioactive material is decaying too quickly, it will not be able to disperse that heat through radiative heating, or through convection. And thus, it will continue to heat up until it boils away, thus achieving self-dispersal. This is influenced by many things, such as the physical structure of the element, the environment it finds itself in, and the amount of energy put out by its decay. However, since most of these things are usually fairly similar, the biggest factor in whether a material can radiate out the energy created by nuclear decay is its half-life. Above, I said that 8 ounces, or 262 grams of Iodine-131 produces about 1000 food calories of heat per hour. A single food calorie would raise the temperature of a kilogram of water by 1 degree. 1000 food calories an hour is therefore enough energy to raise the temperature of 262 grams of Iodine-131 by about 4000 degrees in an hour. Obviously, under normal conditions, it could not radiate or convect away this much heat, especially given Iodine's relatively low melting point. Cesium-134, another common fission byproduct, has a half life of 2 years. Leaving aside that it has a different amount of energy put out, and different physical properties, the fact that its half life is 100 times longer means that its temperature would be gaining about 4 degrees in an hour. This is an amount of heat that could easily be radiated away. Therefore, the half-life needed for an isotope to self-disperse through boiling away is (depending on its physical properties and environment) somewhere between several weeks and several months.
Having explained the general formula, and shown some applications of it, I have a few caveats to throw out. First, in describing these terms, I have used spherical cows, to explain the general concept of connecting radiation to heat. There are many subtleties I have left out, such as the fact that some elements have branching decay chains, and also the fact that I have used rounded units. Therefore, anyone who wants to apply this in a more rigorous settings should use better arithmetic. Second, although the amount of energy created by radiation is more manageable when thought of in terms of heat, it does not mean that it is not dangerous. While Iodine-131 might only have the energy of eating a slice of cheesecake every hour for 8 days, that energy is in a lethal form. (Of course, that might also be the end result of eating a slice of cheese cake every hour for eight days, but the lethality with the cheesecake would be much more pleasant). The fact that nuclear energy is just another form of energy does not mean it is not a very specific, dangerous form of energy.
I hope that reading this has enabled a better understanding of the scale of nuclear energy, and I welcome any corrections to mathematical errors I may have made.
1. The full definition of electronvolt would need more explanation than is needed for this article. I have defined it in an inverse way to how it is usually defined: an electronvolt is the amount of energy an electron picks up if it is attracted by a volt. Further explanations of what an electronvolt is can be found on this site and others.
2. This mixed metaphor led to more responses than any other part of this writeup. The current mentioned here is not the electrical current, but the metaphorical current of the water. Again, the full explanation of what an electronvolt is can be found elsewhere, and I selected this metaphor under the (possibly mistaken) belief that it might allow a good conceptualization.
3. A million electron volts is the order of magnitude, with almost all decay energies lying on a bell curve from 100,000 electron volts to ten million electron volts. But a million electron volts is a good generalization.
4. 1.602 176 487 x 10^-19 J multiplied by 6.022 141 79 x 10^23 equals 96485. The first number is the tiny amount of joules that each electronvolt represents, the second number is the gigantic number of atoms in a mole. By multiplying them together, we get the joules per mole.