The Mathematics of Radioactive Decay
What is Radioactive Decay?
Imagine you have a large bag of popcorn kernels. You start heating them, and they begin to pop. You don't know which kernel will pop next, but you know that, over time, more and more will pop until only a few unpopped kernels are left. Radioactive decay is very similar! Certain atoms, like some forms of Uranium or Carbon-14, are unstable. To become stable, they spontaneously release energy and particles. This transformation is what we call radioactive decay.
The key point is that the decay of any single atom is completely random and unpredictable. You cannot know when a specific Uranium-238 atom will decay. However, when you have a very large number of these atoms, a clear and predictable pattern emerges for the entire group. This is where mathematics becomes an incredibly powerful tool.
The Heart of the Matter: Exponential Decay
The pattern that governs the decay of a large group of radioactive atoms is known as exponential decay. This is a process where a quantity decreases at a rate proportional to its current value. In simpler terms, the more atoms you have, the faster they decay. As the number of atoms decreases, the rate of decay also slows down.
The number of atoms remaining after a certain time is given by:
$N(t) = N_0 e^{-\lambda t}$
Where:
• $N(t)$ is the number of atoms remaining at time $t$.
• $N_0$ is the initial number of atoms.
• $\lambda$ (lambda) is the decay constant.
• $t$ is the time elapsed.
• $e$ is Euler's number (approximately 2.71828), the base of natural logarithms.
Let's break this down. The decay constant $\lambda$ is a probability. It tells us the likelihood that any single atom will decay in a unit of time. A larger $\lambda$ means the substance is more radioactive and will decay faster. The negative sign in the exponent ($-\lambda t$) ensures that the number of atoms decreases over time.
A More Intuitive Concept: Half-Life
While the decay constant is fundamental, scientists often use a more intuitive measure called the half-life. The half-life, often written as $T_{1/2}$, is the time it takes for half of the radioactive atoms in a sample to decay.
| Isotope | Use | Half-Life |
|---|---|---|
| Carbon-14 | Dating organic materials | 5,730 years |
| Iodine-131 | Medical treatment | 8.02 days |
| Uranium-238 | Nuclear power, dating rocks | 4.47 billion years |
There is a direct mathematical relationship between the half-life and the decay constant. We can find it by plugging into the exponential decay formula. After one half-life, the number of atoms is half of the original: $N(T_{1/2}) = N_0 / 2$.
$N_0 / 2 = N_0 e^{-\lambda T_{1/2}}$
Divide both sides by $N_0$: $1/2 = e^{-\lambda T_{1/2}}$
Take the natural logarithm of both sides: $\ln(1/2) = -\lambda T_{1/2}$
This simplifies to: $-\ln(2) = -\lambda T_{1/2}$
Finally, we get the important formula:
$T_{1/2} = \frac{\ln(2)}{\lambda}$
Calculus: The Engine Behind the Model
Where does the exponential decay equation $N(t) = N_0 e^{-\lambda t}$ actually come from? The answer lies in calculus. Calculus is the mathematics of change, and radioactive decay is all about the change in the number of atoms over time.
The fundamental rule of radioactive decay is that the rate at which atoms decay is proportional to the number of atoms present. We can write this as a differential equation:
$\frac{dN}{dt} = -\lambda N$
This equation says "the rate of change of the number of atoms ($dN/dt$) is equal to negative lambda times the current number of atoms ($N$)." The negative sign indicates a decrease. Solving this differential equation (a process involving integration) is what gives us the familiar exponential decay formula. This shows that the exponential model isn't just a guess; it's the direct mathematical consequence of the core principle that decay rate depends on the current amount.
Putting the Math to Work: A Practical Example
Let's say a scientist has a 24.0-gram sample of the isotope Iodine-131, which has a half-life of 8.02 days. It is used to treat a patient's thyroid condition. How much Iodine-131 will remain after 30 days?
Step 1: Find the decay constant ($\lambda$).
We know $T_{1/2} = \frac{\ln(2)}{\lambda}$. So,
$\lambda = \frac{\ln(2)}{T_{1/2}} = \frac{0.693}{8.02 \text{ days}} \approx 0.0864 \text{ per day}$.
Step 2: Use the exponential decay formula.
$N(t) = N_0 e^{-\lambda t}$
Here, $N_0 = 24.0$ grams, $\lambda = 0.0864 / \text{day}$, and $t = 30$ days.
$N(30) = 24.0 \times e^{-(0.0864)(30)}$
First, calculate the exponent: $(0.0864)(30) = 2.592$
So, $N(30) = 24.0 \times e^{-2.592}$
$e^{-2.592} \approx 0.075$ (you can find this using a scientific calculator)
$N(30) = 24.0 \times 0.075 = 1.8$ grams.
So, after 30 days, only about 1.8 grams of the original 24.0-gram sample remains. This kind of calculation is crucial in medicine to ensure the correct dosage is active in a patient's body.
Common Mistakes and Important Questions
Q: After two half-lives, is the sample completely gone?
No. After one half-life, half of the sample remains. After the second half-life, half of that remaining half decays, leaving one-quarter (1/2 x 1/2 = 1/4) of the original sample. In theory, the amount never quite reaches zero, it just gets smaller and smaller.
Q: If the decay of a single atom is random, how can we predict the behavior of the whole sample so accurately?
This is the power of large numbers and probability. While you can't predict a single coin flip, you can be very confident that out of a million flips, about half will be heads. Similarly, with trillions upon trillions of atoms, the random individual decays average out to produce the smooth, predictable exponential decay curve we observe.
Q: Is the half-life the same if I have a bigger or smaller sample of the same material?
Yes! The half-life is a constant property of the radioactive isotope itself. It does not depend on the amount you have, the temperature, pressure, or chemical state. A gram of Carbon-14 and a ton of Carbon-14 both have the same half-life of 5,730 years.
The mathematics of radioactive decay provides a beautiful and powerful example of how we can use models to understand the natural world. The random, unpredictable behavior of a single atom gives way to the precise, predictable pattern of exponential decay for a large group. Through concepts like the half-life and the decay constant, and with the help of calculus, we can accurately describe this process. This understanding is not just theoretical; it allows us to date ancient artifacts, diagnose and treat diseases, and generate power, showcasing the profound connection between abstract mathematics and real-world applications.
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 both carbon, but Carbon-14 has two extra neutrons, making it unstable (radioactive).
[2] Differential Equation: An equation that involves an unknown function and its derivatives. It describes how a quantity changes. In our case, it describes how the number of radioactive atoms changes over time.
[3] Euler's Number (e): An important mathematical constant, approximately 2.71828, which is the base of the natural logarithm. It appears naturally in models of continuous growth or decay.