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Radioactive Decay — Definition, Formula & Examples

Radioactive decay is the process in which unstable atoms break down over time, so the amount of a radioactive substance keeps shrinking. During each half-life, half of whatever is left decays.

Radioactive decay is a first-order process. The rate at which nuclei decay is directly proportional to the number NN of undecayed nuclei present, so dNdt=−λN\frac{dN}{dt} = -\lambda N for a constant λ>0\lambda > 0. Its solution is the exponential decay function N(t)=N0e−λtN(t) = N_0 e^{-\lambda t}. The half-life T1/2T_{1/2} is the constant time it takes NN to fall to half its value, and it is related to the decay constant by λ=ln⁡2T1/2\lambda = \frac{\ln 2}{T_{1/2}}.

Key Formula

N(t)=N0(12)t/T1/2=N0 e−λt,λ=ln⁡2T1/2N(t) = N_0 \left(\frac{1}{2}\right)^{t/T_{1/2}} = N_0\, e^{-\lambda t}, \qquad \lambda = \frac{\ln 2}{T_{1/2}}
Where:
  • N(t)N(t) = amount remaining after time t (in grams, atoms, or percent)
  • N0N_0 = initial amount at t = 0
  • tt = elapsed time
  • T1/2T_{1/2} = half-life, in the same time units as t
  • λ\lambda = decay constant, the fraction-per-unit-time rate of decay

How It Works

You use radioactive decay models to predict how much of a substance remains after a given time, or to work out how much time has passed from the amount that remains. The half-life form is usually easiest. Divide the elapsed time by the half-life to count how many half-lives have passed, then multiply the starting amount by 12\frac{1}{2} raised to that power. To solve for time, set up an exponential equation and take a logarithm of both sides. The half-life never changes, so a sample loses the same fraction of what remains in every equal time interval, not the same fixed amount.

Worked Example

Problem: A lab has 80 grams of a radioactive isotope with a half-life of 5 years. How much of the isotope remains after 15 years?
Step 1: Find the number of half-lives that have passed by dividing the elapsed time by the half-life.
tT1/2=155=3\frac{t}{T_{1/2}} = \frac{15}{5} = 3
Step 2: Raise one-half to that power to get the fraction of the sample that remains.
(12)3=18\left(\frac{1}{2}\right)^3 = \frac{1}{8}
Step 3: Multiply the starting amount by this fraction.
N(15)=80⋅18=10N(15) = 80 \cdot \frac{1}{8} = 10
Answer: 10 grams of the isotope remain after 15 years.

Another Example

Problem: Carbon-14 has a half-life of about 5,730 years. A fossil contains 25% of the carbon-14 it had when the organism died. About how old is the fossil?
Step 1: Write the remaining fraction as an exponential equation in t.
0.25=(12)t/57300.25 = \left(\frac{1}{2}\right)^{t/5730}
Step 2: Take the natural log of both sides and solve for the exponent.
t5730=ln⁡0.25ln⁡0.5=2\frac{t}{5730} = \frac{\ln 0.25}{\ln 0.5} = 2
Step 3: Multiply by the half-life to find t.
t=5730⋅2=11,460t = 5730 \cdot 2 = 11{,}460
Answer: The fossil is about 11,460 years old, which is two half-lives.

Visualization

Why It Matters

Radiocarbon dating uses carbon-14 decay to date bones, wood, and artifacts up to roughly 50,000 years old, and geologists use uranium isotopes to date rocks billions of years old. In medicine, doctors choose tracers such as technetium-99m because their short half-lives clear the body quickly. You will use these models in Algebra 2, Precalculus, and Chemistry, and again in Calculus when you solve the differential equation dNdt=−λN\frac{dN}{dt} = -\lambda N.

Common Mistakes

Mistake: Subtracting the same amount each half-life, as in 80 g, 40 g, 0 g, as if the decay were linear.
Correction: Each half-life removes half of what is currently left, not half of the original amount. The sequence is 80, 40, 20, 10, 5, and so on. The decrease gets smaller every time, which is what makes the decay exponential.
Mistake: Using the elapsed time itself as the exponent, for example computing (12)15\left(\frac{1}{2}\right)^{15} for 15 years.
Correction: The exponent is the number of half-lives, tT1/2\frac{t}{T_{1/2}}. With a 5-year half-life, 15 years is 155=3\frac{15}{5} = 3 half-lives, so use (12)3\left(\frac{1}{2}\right)^3. Make sure tt and T1/2T_{1/2} are in the same units first.

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