How To Calculate Radioactivity

Like a seemingly boundless number of chemistry and physics terms, the word "radioactive" has been co-opted by the general public to mean something other than what physical scientists mean. In everyday English, to describe something as radioactive is to imply that getting near it is a bad idea because whatever you're talking about has been irreversibly stricken a contaminating force.

In reality, radioactivity can indeed be dangerous to living things in certain forms, and it probably can't be helped that many people reflexively associate the term with unwanted images of atomic bombs and "leaky" nuclear power plants. But the term encompasses a host of physical events, many of them agonizingly slow to unfold but also vital to scientists in a number of ways.

Radioactivity, which is not a "thing" but a group of related processes, refers to changes within the nuclei of atoms that result in the emission of particles. (Contrast this with ordinary chemical reactions, in which electrons of atoms interact, but the atomic nuclei remain unchanged.) Because the processes occur in different atoms in a given sample of material at different times, calculations involving radioactivity focus on these samples, not on the behavior of individual atoms.

Radioactivity is a term referring to the decay of a radionuclide. As you will see, this "decay" is unlike that related to biological matter, in the sense that it obeys strict mathematical rules, but it nevertheless describes the reduction in mass of a substance over time, with the resultant accumulation of a different substance or substances (in accordance with the law of conservation of mass).

The activity of a radioactive sample results from the tension between the strong nuclear force, the strongest force in nature and the "glue" that binds protons and neutrons in the nucleus, and the electrostatic force, the second-strongest force and one that tends to push the protons in atomic nuclei apart. This continual "battle" results in the occasional spontaneous reformation of nuclei and the discharge of discrete particles from them.

"Radiation" is the name of these particles, which are the result of radioactivity. The three most common types of radiation (or decay) are alpha (α), beta (β) and gamma (γ) radiation, described in detail below.

• Alpha radiation consists of two protons and two neutrons, equivalent to the nucleus of a helium (He) atom, that is, helium without its two electrons. Because of the combination of this particle's sizable mass (about 7,000 times that of a beta particle, below) and +2 electrical charge, these particles do not move very far from the nuclei that emit them. They interact strongly with most matter and can do serious biological damage if ingested (swallowed).

• Beta radiation is the emission of a negatively charged electron along with a subatomic particle called an electron antineutrino. It can also refer to the emission of a positron, which has the mass of an electron (about 9.9 × 10^–31 kg) but a positive charge. Being smaller, these particles are more penetrating than alpha radiation but also do the bulk of their health harm if swallowed.

• Gamma radiation is the emission of electromagnetic energy from the nucleus rather than particles with even negligible mass. These emissions are similar to X-rays, except that the latter do not originate in nuclei. This radiation is useful in medical applications for the same reason it can be highly dangerous: It penetrates deep into biological (and sometimes far denser) matter.

The radioactive decay law, which you will be formally introduced to shortly, relates the number of decayed nuclei at two different time points to a parameter called the decay constant λ (the Greek letter lambda). This constant is derived from the half-life of a particular radionuclide.

• Think of a radionuclide as similar to an isotope, except that it emphasizes a specific proton and neutron number, e.g., carbon-14 is a carbon nucleus with six protons and eight neutrons. Neutron number is unimportant in chemical reactions but vital in radioactivity. This is why isotopes can all be grouped with the same element on the periodic table, as this emphasizes chemical behavior over physical behavior.

The half-life of a substance is the time it takes for the amount of a substance present at time t = 0 to be cut in half. Critically, this property is independent of absolute amounts at any point. This time period is designated **t_1/2 **and varies spectacularly between atomic species.

The activity of a sample is the number of decays per unit time, making it a rate. Think of the difference between the total number of decays and activity as analogous to the difference between position and velocity, or between energy and power: The latter is just the former divided by a unit of time (typically seconds, the SI unit of time across the sciences).

The basic radioactivity formula with which you should become familiar has been established as a law, meaning that nowhere under any conditions is it believed to be violable. It takes the form:

\(N = N_{0}e^{–λt}\)

Here, N₀ is the number of nuclei present at time t = 0, and N is the number remaining at time t. The e is a constant known as the base of the natural logarithm and has a value of approximately 2.71828. The λ is, as mentioned, the decay constant, which represents the fraction (not number) of nuclei that decay per unit time.

Note from the radioactivity formula that the time it takes for the size of the sample to be halved, or reduced to the value (1/2)N₀, is represented by the equation (1/2)N₀ = N₀e^–λt. This equation reduces easily to (1/2) = e^–λt. Taking the natural logarithm (ln on a calculator) of each side, and replacing t with the specific value t_1/2, transforms this expression into ln (1/2) = –λt_1/2, or –(ln 2) = –λt_1/2. Solving for lambda gives:

λ = ln 2/t_1/2 = ~0.693/t_1/2

• The ~, or tilde, represents "approximately" in mathematics when appended to the front of a number.

This means that if you know the rate constant for a decay process, you can determine the half-life and conversely. One important type of calculation involves figuring out how much time has passed since a specimen was "complete" based on the fraction N/N₀ of nuclei remaining. An example of such a computation as well as a radioactive decay calculator are included later in the article.

Many students find the radioactive decay definition with its concept of half-life somewhat frustrating or at least foreign at first. If you are the person who shops for fruit juice in your home, and you notice that the number of cans has dropped from 48 to 24 over the past week, then you can probably determine without doing any formal math that you will have to pick up more fruit juice in exactly a week. In the real world, "decay" processes are linear; they occur at a fixed rate no matter how much substance is present.

• Certain drugs obey a half-life pattern of metabolism in the body. Others, such as ethanol, disappear at a fixed rate, e.g., about one alcoholic beverage per hour.

The fact that some radionuclide decay processes occur at such a slow rate, with correspondingly enormous half-lives, makes certain kinds of radioisotope dating methods invaluable in various sciences, among them archaeology and history. Just how long do some of these half-lives stretch out?

The radioactivity formula says nothing about individual atoms If you stared at a single atomic nucleus with a known half-life, even a fairly short one (say 60 minutes), you would have to guess to know whether this radionuclide will decay, or disintegrate, over the next 15, 30 or 60 minutes. But if you have a sizable sample, you can use statistical principles to determine what fraction will be converted in a given time frame; you just won't be able to pick out in advance which ones.

• The SI unit of activity is known as the becquerel, or Bq, which represents one decay per second. A nonstandard unit called the curie (Ci) equals 3.7 × 10¹⁰ Bq.

Note that, unlike the decay constant, activity changes over time. You should expect this from the graph of a substance undergoing radioactive decay; as the number of nuclei drops from N₀ to (N₀/2) to (N₀/4) to (N₀/8) and so on over successive half-lives, the curved graph flattens out; it is as if the substance is happy to disappear, but it just wants to linger and linger some more, never quite getting all the way out the door. For this to be the case, the rate of change of nuclei (equal to the calculus expression –dN/dt) must be decreasing over time (that is, the slope of the graph becomes less negative over time).

Many earnest people often use the term carbon dating incorrectly. This practice refers to a general process known as radioisotope (or radionuclide) dating. When something dies, the carbon-14 it contains begins to decay, but its far more stable carbon-12 nuclides do not. Over time, this drops the ratio of carbon-14 to carbon-12 progressively down from 1:1.

The half-life of carbon-14 is about 5,730 years. This is a long time compared to a chemistry course, but a mere wink compared to geological time since the Earth is 4.4 to 4.5 billion years old. But this can be useful for determining the ages of artifacts of antiquity on a human scale.

Example: The ratio of carbon-14 to carbon-12 in a well-preserved sweat stain on an old book cover is 0.88. How old is the book?

Note that you do not need to know how the exact values of N₀ or N; having their ratio is sufficient. You must also compute the decay constant λ from the half-life of carbon-14: λ = 0.693/5,730 = 1.21 × 10^–4 decays/yr. (This means that the probability of any one nucleus decaying in a 1-second period is about 1 in 12,100.)

The radioactive decay law equation for this problem gives:

(0.88)N₀ = N₀e^{– λt}

0.88 = e^–λt

ln 0.88 = –λt

–1.2783 = –(1.21 × 10^–4)t

t = 10,564 years.

This value is imprecise and would be rounded off to 10,560 or even 10,600 years depending on the number of tests run and other factors.

For much older specimens such as fossils, other radionuclides with far longer half-lives must be used. Potassium-40, for example, has a half-life of about 1.27 billion (1 × 10⁹) years.

In the Resources, you'll find a tool that allows you to play around with hundreds of different nuclei with a vast range of half-lives, and determine the fraction of it remaining given an initial date, or use the remaining amount to back-date the appearance of the specimen (or at least the approximate date on which biological activity concerning the specimen stopped).