Summary. Every second, an adult human body experiences the radioactive decay of about 4000 naturally-occurring potassium-40 atoms. Beta particles and gamma rays emitted during such decays cause cellular damage. Nonetheless, humans typically live more than 70 years. How can this be? A first step in addressing this biophysical quandary is understanding how the value of 4000 decays per second arises. This document derives that value, thoroughly explaining assumptions made and methods used. Along the way I introduce important ideas taken from probability theory, chemistry, and calculus. I write at the level of a scientifically-literate high school senior or college freshman. I hope that the reader might gain an added appreciation of quantitative scientific reasoning.
The fundamental probability model
Let's begin with two fundamental assumptions. First, potassium-40 (or, K-40, as "K" is the symbol for potassium in the periodic table) atoms are all identical in their likelihood of decay. By definition, in any large collection of K-40 atoms, half will decay within some specific duration of time called the half life (abbreviated thalf). thalf is an empirical property measured for many radioactive isotopes, including K-40. Because decay is random, we cannot know in advance which particular atoms will constitute the decaying half. But we do know that half of the any sample will decay over a time interval of thalf. Second, each atom decays independently of all others. When a particular atom decays is completely unaffected, for example, by the behavior of its neighbor. Taken together, these two assumptions describe a probability model of statistically identical and independent particles (here, particles mean K-40 atoms).
I assert that, for radioactive atoms, these two assumptions are fully warranted. Nothing I know of in chemistry and physics suggests otherwise. Of course, I dearly hope that these assumptions are valid because, if they are, they make the math wonderfully simple, namely
Ndecays = N * P
where
Ndecays is the expected number of K-40 decays in a one second time period,
N is the number of K-40 atoms in an adult human body, and
P is the probability that any single K-40 atom will decay during any one second time period.
Our model is intuitive and simple. To find our answer, Ndecays, just multiply the number (N) of K-40 atoms by their probability (P) that any single atom will decay with a specified time period (here, one second). If Ndecays is calculated to be, say, 4330 decays per second, then, if you could reliably measure the decays in a human over one second periods, you would get individual measurements sometimes higher, sometimes lower, than 4330. After all, radioactive decay is individually random but well-defined when averaged over many measurements. It is the average of your individual measurements that would tend to Ndecays.
Two crucial assumptions
Our first big idea is that an appropriately chosen model (here, it is a probability model) can greatly simplify a daunting problem but at the cost of adopting simplifying assumptions. Those assumptions "cost" because they reduce the generality of the model and, more importantly, they could be wrong.
We assume K-40 atoms, in their decay, are statistically identical and independent. These two assumptions are essential to this derivation. So let's consider a counterexample where both assumptions fail to apply. Counterexamples are important to keep in mind. You begin to worry if your model resembles the counterexample.
Consider the spread of infectious disease (e.g., influenza) by casual contact. In this case, people are analogous to K-40 atoms and falling ill with the flu is analogous to decay. We might be interested in predicting the number of people falling ill each hour or day. The assumption of identical particles fails because people vary in susceptibility, those with impaired immune function being more susceptible, those who have received a recent flu shot being less susceptible. The assumption of independence also fails. Your likelihood of falling ill changes (it increases) if someone with whom you live or work in close proximity
become ill.
Calculating N, the number of K-40 atoms in a typical human body
What's left to do is to determine N and P. In this section we calculate N. Our typical human (male or female does not matter) weighs, say, 150 pounds, that is, 68,100 grams (at 454 grams per pound). By mass, the human body is about 0.2 percent potassium. So our typical body has 136.2 grams potassium. Potassium atoms have, by definition, 19 protons in their nucleus. Potassium isotopes are potassium atoms varying in the number of neutrons. Most potassium atoms have either 20 or 22 neutrons and are stable. They do not decay; they do not exhibit radioactivity. However, out of every million potassium atoms, 120 have 21 neutrons. These constitute the radioactive K-40 isotope of interest to us. Therefore, our typical 150 pound human has just 0.0163 grams of K-40 widely distributed throughout his or her body. That mass of elemental potassium metal would appear under a magnifying glass as a silvery speck of soft metal.
Mass of K-40 is not what we need. To calculate N, we instead need the corresponding number of K-40 atoms. To determine that, we employ the second important concept: the mole concept in chemistry. The number of atoms in some sample is proportional to the sample's mass and to the mass of each individual atom. In short, 40 grams of K-40 is, by definition, composed of 6.022 x 10^23 number of atoms. (Incidentally, 6.022 x 10^23 is Avogrado's constant, one of the most famous and universal constants in chemistry. The ^ symbol indicates exponentiation so that, for example, 10^2 is one hundred and 10^3 is one thousand) Using simple algebra, it follows that 0.0163 grams K-40 is composed of 2.46 x 10^20 atoms. That is our value for N. It is a huge, but not infinite, number of atoms.
Calculating P, the probability that a K-40 atom will decay over any one second interval
K-40 has an extremely long half life of 1.25 billion years. If it did not, then all of it would have long since decayed away during the 4.57 billion years of earth's existence. As it is, about 7.9 percent of the K-40 originally present when the earth and solar system formed is still with us. It is the K-40 that decays within us (and within everything containing potassium). K-40 is a naturally-occurring substance. It is not the result of pollution, nuclear weapons testing, or any other human activity.
A given atom of K-40 has a 50 percent chance of decaying in the next 1.25 billion years. Remember that this number, abbreviated thalf, is not computed or derived, rather it is empirically measured in the laboratory. So the probability that a K-40 atom will decay in the next second is, well, a truly tiny number, but it is still larger than zero. Because we are working in seconds, not years, it is helpful to convert thalf to its equivalent expressed in seconds, namely,
K-40 thalf = 1.25 x 10^9 years
= 3.94 x 10^16 seconds
We need to convert thalf into a more useful quantity, a rate constant. Radioactive decay obeys an exponential decay model.
Fu = exp( -k * t )
where
Fu the fraction of original amount still undecayed after time t has elapsed. Fu ranges from 0.0 (none undecayed, i.e., all atoms of decayed) to 1.0 (all undecayed, i.e., all atoms still present),
t is the time elapsed (in this example, one second),
k is the rate constant, a positive number specifying the rate at which decay happens. (I use a lower case k for the rate constant to distinguish it from the periodic table symbol (K) for potassium), and
exp() is the exponential function (i.e., exponentiation using Euler's constant e, 2.718, as a base).
To derive k, we substitute thalf for t and 0.5 for Fu.
Fu = exp( -k * thalf )
0.5 = exp( -k * 3.94 x 10^16 seconds )
and solve the equation for k (by taking the natural logarithm of both sides). We get
k = -Ln(0.5) / thalf
= 0.693 / (3.94 x 10^16 seconds)
= 1.76 x 10^-17 per second
where Ln(0.5) is the natural logarithm of one half, or approximately -0.693. Thus, we now have a numerical value for the desired rate constant k.
What we want probability P that a single K-40 atom will decay in a one second time interval. I assert that P (for one atom) is identical to Fd (for a collection, or sample, of many K-40 atoms), where Fd is the fraction decayed
P = Fd = 1 - Fu
= 1 - exp( -k * t )
= 1 - exp( -1.76 x 10^-17 ) for t set to 1 second
We could stop right here and attempt a direct calculation of P with a calculator or spreadsheet. However, because you would be subtracting two numbers that are very nearly equal, the answer you get may well be degraded by round-off error. For example, my HP-15c scientific calculator returns 0.0. Were that correct, then K-40 would not decay. We can however avoid this problem by replacing exp( -k * t ) by an accurate approximation.
Calculus shows that mathematical functions can be represented as a power series (specifically, a Taylor Series). This useful ability to approximate mathematically is the third important concept of this derivation. However, I leave it to college calculus textbooks to prove the following power series equation.
exp( A ) = 1 + A + A^2/2 + A^3/6 + A^4/24 + ...
where for A we could, for example, substitute -k * t. The ellipsis means that the terms of the power series continue infinitely. We approximate exp( A ) by retaining only a finite number of terms right of the equal sign and ignoring the rest. Because -k * t is so small in magnitude, we can get an accurate approximation using only the first two terms. Thus
exp( -k * t ) =~= 1 - k * t
where the =~= symbol means "approximately equal to". When I claimed that -k * t is "so small in magnitude", I meant
0 < abs( -k * t ) << 1
where abs() specifies absolute value. In plain language, 1.76 x 10^-17 is much, much closer to zero than it is to one. Our new (approximate) equation for P is
P = 1 - [ 1 - k * t ]
= k * t
= 1.76 x 10^-17
As it must be for a probability, P is a positive real number having no units with value between 0.0 and 1.0. It is also a very small number. That seems intuitively right given the extremely long half life of K-40. It was derived in a way avoiding round-off error. The key point here is that while P is indeed minuscule, it is still greater than zero. And that is what matters.
Putting it all together
Our probability model presented above is
Ndecays = N * P
where N is 2.46 x 10^20 and, for a one second, P is 1.76 x 10^-17. Therefore,
Ndecays is about 4000 (per second)
That is the number sought. One could insist on 4330 decays because that is what results from the multiplication. However, the precision of our answer is only as good as the least precisely specified parameter used in our derivation. Above, we used the fact that a human body is 0.2 percent potassium. That's not very precise (it has only one significant figure). So any answer we derive will likewise be limited in precision.
Four thousand K-40 decays per second should be regarded as a ballpark estimate. It's more than 1000 and it's less than 10,000. Eat a banana (a good dietary source of potassium) and maybe the value goes up a bit for a few hours. Urinate and it goes down.
Correctness of the result
Is Ndecays of about 4000 per second correct? To answer that, one would compare this value to experimentally-measured values. Unfortunately, I know of no such data. (If you do, please leave me a comment or email me!) Instead, all we have is consistency with a published value, that of 4300 per second in a 70 kg (154 pound) human. It remains possible that such consistency arises because both derivations are each consistently wrong!
K-40 atoms primarily decay by the emission of beta particles (high-energy electrons). Detecting such particles by sensors external to the body would be difficult because most beta particles never escape the body. Instead, they are chemically captured before reaching body surface. However, human tissue is more transparent to gamma rays, and about 11 percent of K-40 decays involve gamma ray emission. Detecting those gamma rays (and distinguishing them from other sources, such as stray cosmic rays) might allow experimental measurement of Ndecays.
Meaning of 4000 decays per second
Thousand of times per second the cells in our body are subjected to the beta particles ejected during the decay of K-40 atoms. These fast-moving electrons chemically damage nucleic acids, enzymes, membranes, and other cellular components.
Given this abuse, I remain amazed that we are not all dead from cancer by age 8 or 10 years. So why not? Here are two of my speculations. First, K-40 is naturally-occurring and has been present since earth's formation. Life has evolved in a naturally radioactive environment and so, in response, has evolved defenses such like DNA repair mechanisms. Second, K-40 is distributed throughout the human body (which makes its experimental measurement difficult). Concentration of it would accentuate its damage. Iodine, for example, is concentrated by the body at one location, the thyroid gland. Radioactive iodine -- e.g., the iodine 131 released into the environment during the Chernobyl nuclear power plant disaster of 1986 -- does cause thyroid cancer.
posted: 2019-11-22, last edited: 2019-11-27
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