The Mathematics of Radioactive Decay
The Heart of the Matter: The Exponential Decay Law
At its core, radioactive decay is a random process. We never know exactly which individual atom will decay next. However, when we look at a large group of identical radioactive atoms, a very clear and predictable pattern emerges. This pattern is described by the Exponential Decay Law.
$N(t) = N_0 e^{-\lambda t}$
Where:
• $N(t)$ is the number of undecayed nuclei at time $t$.
• $N_0$ is the initial number of undecayed nuclei (at $t = 0$).
• $\lambda$ (the Greek letter lambda) is the decay constant.
• $e$ is Euler's number (approximately 2.71828), the base of natural logarithms.
• $t$ is the time elapsed.
Think of it like a large bag of popping candy. You can't predict which piece will pop next, but you can be sure that over a minute, a certain percentage of the candy will have popped. The exponential decay law gives us the mathematical tool to make that prediction precise for atoms.
The formula shows that the number of remaining nuclei decreases exponentially. This means it doesn't decrease at a constant rate (like a car losing 10 miles per hour every second). Instead, it decreases by a constant percentage or fraction over equal time intervals. For example, a substance might lose half of its remaining atoms every 5 years, not a fixed number like 1000 atoms every year.
Understanding the Decay Constant and Half-Life
The decay constant ($\lambda$) is a probability. It represents the chance that any single nucleus will decay in the next second. A larger $\lambda$ means the substance is more unstable and decays faster.
A more intuitive and commonly used concept is the Half-Life, symbolized as $T_{1/2}$. The half-life is the time required for half of the radioactive nuclei in a sample to decay.
$T_{1/2} = \frac{\ln(2)}{\lambda}$
Since $\ln(2)$ is approximately 0.693, this is often written as:
$T_{1/2} \approx \frac{0.693}{\lambda}$
This relationship is powerful because it links the probabilistic decay constant ($\lambda$) to a measurable, concrete time ($T_{1/2}$). If you know one, you can always find the other. Different radioactive isotopes have vastly different half-lives, as shown in the table below.
| Isotope | Symbol | Half-Life | Common Use |
|---|---|---|---|
| Carbon-14 | $^{14}C$ | 5,730 years | Radiocarbon dating |
| Iodine-131 | $^{131}I$ | 8.02 days | Medical treatment |
| Uranium-238 | $^{238}U$ | 4.47 billion years | Dating the Earth |
| Polonium-214 | $^{214}Po$ | 0.000164 seconds | Smoke detectors |
Putting the Equations to Work: A Step-by-Step Example
Let's imagine a scientist has 1.0 gram of the radioactive isotope Iodine-131, which has a half-life ($T_{1/2}$) of 8.02 days. She needs to know how much will remain after 30 days to plan a medical treatment.
Step 1: Find the Decay Constant ($\lambda$)
We use the half-life formula: $T_{1/2} = \frac{\ln(2)}{\lambda}$.
Rearranging for $\lambda$: $\lambda = \frac{\ln(2)}{T_{1/2}} = \frac{0.693}{8.02 \text{ days}} \approx 0.0864 \text{ per day}$.
Step 2: Use the Exponential Decay Law
We know $N_0$ (the initial amount) is proportional to our 1.0 gram. We can use the mass directly in the formula since the fraction remaining will be the same as the fraction of nuclei remaining.
$N(t) = N_0 e^{-\lambda t}$
$N(30) = (1.0 \text{ g}) \times e^{-(0.0864 \text{ / day}) \times (30 \text{ days})}$
First, calculate the exponent: $-(0.0864) \times (30) = -2.592$.
Then, calculate $e^{-2.592}$. You can use a scientific calculator for this, which gives a value of approximately 0.075.
So, $N(30) = 1.0 \times 0.075 = 0.075$ grams.
Conclusion: After 30 days, only about 0.075 grams (or 7.5%) of the original Iodine-131 sample remains.
Common Mistakes and Important Questions
Q: Does the half-life mean that after two half-lives, all the radioactive material is gone?
A: This is a very common mistake. No, it does not. After one half-life, 50% remains. After a second half-life, you lose half of what was left, not half of the original. So, you are left with 25% of the original amount. After three half-lives, 12.5% remains, and so on. Theoretically, the amount never quite reaches zero, but it becomes immeasurably small.
Q: Can anything change the half-life of a radioactive element?
A: For all practical purposes, no. The half-life of a given radioactive isotope is a fundamental physical constant. It is not affected by external factors such as temperature, pressure, chemical bonding, or magnetic fields. The decay process happens within the nucleus itself, and these external conditions do not influence the strong nuclear forces involved.
Q: What is the difference between the decay constant ($\lambda$) and half-life ($T_{1/2}$)?
A: They are two sides of the same coin, but they represent the idea differently. The decay constant ($\lambda$) is a probability rate—it tells you the likelihood a single nucleus will decay in the next instant. The half-life ($T_{1/2}$) is a tangible measure of time—it tells you how long it takes for a large collection of nuclei to reduce by half. A substance with a large $\lambda$ has a short $T_{1/2}$, and vice-versa.
The equations of radioactive decay provide a powerful and elegant mathematical description of a seemingly random natural process. The exponential decay law, $N(t) = N_0 e^{-\lambda t}$, along with the concept of half-life, allows us to make precise predictions about how matter transforms over time. From carbon-dating ancient artifacts to ensuring the correct dosage in medical treatments, these equations are not just theoretical concepts but vital tools in science and technology. Understanding that the decay rate is constant and independent of external conditions is key to applying these formulas correctly. While the process is random for a single atom, for a large group, it is beautifully predictable.
Footnote
1 Isotope: Atoms of the same element that have the same number of protons but different numbers of neutrons. For example, Carbon-12 and Carbon-14 are isotopes of carbon.
2 Nucleus (plural: Nuclei): The small, dense, positively charged center of an atom, made up of protons and neutrons. Radioactive decay is a process that originates in the nucleus.
3 Exponential Decay: A process where a quantity decreases at a rate proportional to its current value. It is characterized by a constant half-life and produces a curved, constantly decreasing graph.
4 Euler's Number (e): An important mathematical constant, approximately equal to 2.71828, which is the base of the natural logarithm. It appears frequently in models of natural growth and decay.