At the * Protons for Breakfast* session the other night, someone asked me ‘

*What is the ‘source’ of internal heat in the Earth?*‘. I knew that William Thomson (Lord Kelvin) had made estimates of the age of Earth based on its current temperature and arrived at an estimate well short of what we now believe. I had heard the difference had been ‘explained’ by radioactive decay but I had never looked into problem. Well I looked around on the web, found the appropriate Wikipedia page, and sketched in the answer. The answer turns out be that indeed the weak radioactivity of the rocks within the Earth is responsible. I wrote:

The internal heat is partly left over from the Earth’s creation, but there is a substantial contribution from the decay of radioactive elements. There are many long-lived radioactive elements that heat each kilogram of rock only minutely. For example,

Wikipediatells me that even though Uranium 238 is present only as an impurity in most rocks, the heating from it is estimated to be 0.000 000 000 003 Watt per kg (3 x 10^{-12}W/kg) of rock. Not much! But there is a lot of rock! The mass of the Earth is 6 x 10^{24}kg so that amounts to a whopping 18 x 10^{12 }Watts! If we divide this by the surface area of the Earth (5.1 x 10^{14}metres squared) we arrive at 0.035 Watts per square metre. Since there are three or four other significant long-lived isotopes with similar heating effects, this in line with the figure of around 0.1 Watts per square metre I mentioned in the talk.

Well I was pleased that these numbers sort of tied together. But I wondered if the heating really could be that great. So I made some further order of magnitude calculations and actually it does sort of tie up. Let me explain what I did. I calculated three quantities and then multiplied them together to get a fourth.

- The first quantity I calculated was the probability per second that a nucleus of an atom of uranium would decay.
- The second was the number of Uranium atoms in each kilogram of rock.
- The third was the energy released when each nucleus of uranium decays
- I then multiplied them together to work out the average amount of energy realsed into each kilogram of rock per second.

Lets look at each of these in turn

### Decay Probability

The half life of Uranium 238 – the most common isotope of Uranium is an astonishing 4.468 billion years! This is close to our estimate of the age of the Earth, and so very roughly only about half of the Uranium 238 atoms that originally existed on Earth have decayed. Converting to seconds by multiplying by the number of seconds in a minute, the numbe rof minutes in an hour, the number of hours in a day and the number of days in a year I find the following stupendous number 4.468 x 10^9 x 60 x 60 x 24 x 365 = 1.41 x 10^17 seconds. The probility of decaying in that many seconds is 0.5, so very roughly the probability of decay per second is 0.5÷1.41 x 10^17 or 3.57 x 10^-18 per second.

### Number of uranium atoms in each kilogram of rock

Wikipedia tells me that Uranium is present as impurity at a level somewhere between 2 parts per million and 4 parts per million in the Earth’s crust. If we assume that concentration is typical of the whole Earth, then we just need to work out how many atoms there are in a kilogram of rock, and we will then know roughly how many Uranium atoms there are. Well the interior of teh Earth has a complex structure and contains many different types of rocks. The most common rocks are silicates (SiO2) of magnesium, iron, and aluminium. Calculating the average molecular weight of this mixture is 40 and of 1 mole ‘average Earth’ of atoms would have a mass of 40 grams. So in 1 kg of ‘average Earth material’ there are roughly 25 moles of atoms. Now 1 mole of a substance 6.023 x 10^23 of its microscopic constituents, in thsi case atoms and so 1 kg of ‘Average Earth material’ contains roughly 25 x 6.023 x 10^23 = 1.5 x 10^25 atoms. And we know that approximately 2 out of every million of these atoms will be uranium. So the number of uranium atoms per kilogram of ‘average Earth rock’ is roughly 3 x 10^19 per kilogram.

### Energy release

Each radioactive decay of a Uranium 238 nucleus (about 99.3% of naturally occurring Uranium) emits an alpha particle with an energy of 4.27 MeV (Million electron volts). To convert this to joules of energy I need to multiply by the number of joules per electron volt of energy which is 1.6 x 10^-19 joules. Thus we find that each decay releases 4.27 x 10^6 x 1.6 x 10^-19 = 6.83 x 10^-13 joules. This is energy of the alpha particle emitted from the nucleus of uranium 238. The alpha particle bashes into the neighbouring atoms and sets them moving – i.e. it heats them.

### Bringing it all together

If we multiply (the probability per second that a uranium nucleus will decay) by (the number of such nuclei in a kilogram of average Earth rock) we find out the average number of radioactive decays per second per kilogram of ‘Average Earth rock. The sum is (3.57 x 10^-18 per second) x (3 x 10^19 per kilogram) = 107 decays per second. This seems reasonable, but perhaps a bit on the high side (based on my experience of measuring radioactivity of rocks – I genrally just get a few counts per second). If we now multiply this number by the amount of energy released in each decay we find the energy released into the rock per second is (107 per second) x (6.83 x 10^-13 joules) = 7.3 x 10^-11 joules per second (Watts). Looking back up the page I see that originally I took my estimate of heating by Uranium to be 3 x 10^-12 Watts per kg of rock and this estimate is around 20 times bigger.

So What? Well I just wanted to show that starting with estimates of the half life of Uranium and the energy released in each decay, I could sort of explain the amount of heat produced per second. The big uncertainty here is in the actual concentration of Uranium in the Earth. Given this uncertainty, I feel that my calculations do sort of tie up, and I feel more confident now in repeating what I have heard, that the reason the Earth is hot inside is because of the radioactive decay of long lived isotopes.

### One more thing…

It is hard to imagine how this tiny heating – a few millionths of a millionth of a watt per kilogram – could ever get anything hot! Indeed the heating rate would be infinitesimally slow – roughly 10^-13 of a degree per second. At that rate it would have taken about 1.5 billion years to have heated up the interior of Earth to our current estimate of around 5000 °C. However the heating rate is sufficient to slow down the cooling of a previously hot body, which is what we believe has happened. OK. I feel I have done this problem to death now! Time to move on!

Tags: Geothermal Gradient, Heating of the Earth, Natural Radioactivity, Uranium

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