The Universe of Discourse

It's the radioactive potassium, dude!
A correspondent has informed me that my explanation of the snow-melting properties of salt is "wrong". I am torn between paraphrasing the argument, and quoting it verbatim, which may be a copyright violation and may be impolite. But if I paraphrase, I am afraid you will inevitably conclude that I have misrepresented his thinking somehow. Well, here is a brief quotation that summarizes the heart of the matter:

Anyway the radioactive [isotope of potassium] emits energy (heat) which increases the rate of snow melt.

So what's going on here? I picture a years-long conspiracy in the MIT chemistry department, sort of a gentle kind of hazing, in which the professors avow with straight faces that the snow-melting properties of rock salt are due to radioactive potassium, and generation after generation of credulous MIT freshmen nod studiously and write it into their notes. I imagine the jokes that the grad students tell about other grad students: "Yeah, Bill here was so green when he first arrived that he still believed the thing about the radioactive potassium!" "I did not! Shut up!" I picture papers, published on April 1 of every year, investigating the phenomenon further, and discoursing on the usefulness of radioactive potassium for smelting ores and frying fish.

Is my correspondent in on the joke, trying to sucker me, so that he can have a laugh about it with his chemistry buddies? Or is he an unwitting participant? Or perhaps there is no such conspiracy, the MIT inorganic chemistry department does not play this trick on their students, and my correspondent misunderstood something. I don't know.

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You know, the great thing about this theory is that you can get the salt to melt the snow without even taking it out of the bag. When you're done, just pick up the bag again, put it back in the closet. All those people who go to the store to buy extra salt are just a bunch of fools!

Well, it's not enough just to scoff, and sometimes arguments from common sense can be mistaken. So I thought I'd do the calculation. First we need to know how much radioactive potassium is in the rock salt. I don't know what fraction of a bag of rock salt is NaCl and what fraction is KCl, but it must be less than 10% KCl, so let's use that. And it seems that about 0.012% of naturally-occurring potassium is radioactive . So in one kilogram of rock salt, we have about 100g KCl, of which about 0.012g is K⁴⁰Cl.

Now let's calculate how many K⁴⁰ atoms there are. KCl has an atomic weight around 75. (40 for the potassium, 35 for the chlorine.) Thus 6.022×10²³ atoms of potassium plus that many atoms of chlorine will weigh around 75g. So there are about 0.012 · 6.022×10²³ / 75 = 9.6×10¹⁹ K⁴⁰-Cl pairs in our 0.012g sample, and about 9.6×10¹⁹ K⁴⁰ atoms.

Now let's calculate the rate of radioactive decay of K⁴⁰. It has a half-life of 1.2×10⁹ years. This means that each atom has a (1/2)^T probability of still being around after some time interval of length t, where T is t / 1.2×10⁹ years. Let's get an hourly rate of decay by putting t = one hour, which gives T = 9.5×10^-14, and the probability of particular K⁴⁰ atom decaying in any one-hour period is 6.7 × 10^-14. Since there are 9.6×10¹⁹ K⁴⁰ atoms in our 1kg sample, around 64,000 will decay each hour.

Now let's calculate the energy released by the radioactive decay. The disintegration energy of K⁴⁰ is about 1.5 MeV. Multiplying this by 64,000 gets us an energy output of 86.4 GeV per hour.

How much is 86.4 GeV? It's about 3.4×10^-9 calories. That's for a kilogram of salt, in an hour.

How big is it? Recall that one calorie is the amount of energy required to raise one gram of water one degree Celsius. 3.4×10^-9 calorie is small. Real small.

Note that the energy generated by gravity as the kilogram of rock salt falls one meter to the ground is 9.8 m/s² · 1000g · 1m = 9.8 joules = 2.3 calories. You get on the order of 750 million times as much energy from dropping the salt as you do from an hour of radioactive decay.

But I think this theory still has some use. For the rest of the month, I'm going to explain all phenomena by reference to radioactive potassium. Why didn't the upgrade of the mail server this morning go well? Because of the radioactive potassium. Why didn't CVS check in my symbolic links correctly? It must have been the radioactive potassium. Why does my article contain arithmetic errors? It's the radioactive potassium, dude!

The nation that controls radioactive potassium controls the world!

[ An earlier version of this article said that the probability of a potassium atom decaying in 1 hour was 6.7 × 10^-12 instead of 6.7 × 10^-14. Thanks to Przemek Klosowski of the NIST center for neutron research for pointing out this error. ]

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