Radioactive Decay
Number of nuclei remaining over time.
Formula
Variables
How it works
Radioactive decay follows an exponential law because each nucleus has the same constant probability of decaying in a given time interval, regardless of how many other nuclei are present. That means the rate of decrease is proportional to how many undecayed nuclei remain, which leads to the differential equation dN/dt = -λN and its solution N = N0 e^{-λt}. The formula tells you how many nuclei are left after time t, with larger λ values meaning faster decay.
When to use it
Use this formula when modeling the number of radioactive nuclei remaining in a sample after a known time, or when comparing how quickly different isotopes decay. It is also used in nuclear medicine, dating materials, and calculating decay-related activity in physics and chemistry problems.
Worked example, step by step
Suppose a sample starts with N0 = 8000 nuclei and has decay constant λ = 0.25 per hour. Find the number of nuclei remaining after t = 6 hours. N = N0 e^{-λt} N = 8000 e^{-0.25 \times 6} N = 8000 e^{-1.5} N ≈ 8000 \times 0.2231 N ≈ 1785 So about 1.79 × 10^3 nuclei remain after 6 hours.
Common mistakes to avoid
- Using a linear decrease instead of exponential decay
- Forgetting that λ and t must be in compatible units
- Rounding too early and losing accuracy in the final result
Did You Know?
Radioactive decay is perfectly random for a single atom yet utterly predictable for trillions of them.
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