The formula
- Amount left after a whole number of half-lives
N = N₀ · (1/2)^(t / T½)- Exponential decay law
N = N₀ · e^(-λt)- Half-life and decay constant
T½ = ln 2 / λ ≈ 0.693 / λ- Activity of the sample
A = λN
What the symbols mean
| Symbol | Meaning | Unit |
|---|---|---|
N | Number of undecayed nuclei left at time t | count (or mass, since the two are proportional) |
N₀ | Number of undecayed nuclei at the start | count |
t | Elapsed time | s (any time unit, matched to T½ and λ) |
T½ | Half-life of the isotope | s |
λ | Decay constant, the probability per unit time that a nucleus decays | s⁻¹ |
A | Activity, the number of decays per unit time | Bq (1 Bq = 1 decay per second) |
e | Base of the natural logarithm, used in the exponential form | 2.718 (dimensionless) |
ln 2 | Natural logarithm of 2, the constant linking half-life and decay constant | 0.693 (dimensionless) |
When it applies
- The sample holds a large number of nuclei, since the law is statistical and describes an average over many of them.
- The half-life belongs to the isotope, not to the sample. Doubling the amount of material does not change it.
- Keep t, T½ and λ in matched time units. A half-life in days gives a decay constant per day, so t has to be in days too.
Worked example
Problem. A 40 g sample of a radioactive isotope has a half-life of 8.0 days. How much is left after 24 days, and what is the decay constant?
- Count the half-lives: t / T½ = 24 days / 8.0 days = 3.
- Halve three times: N = 40 g × (1/2)³ = 40 / 8 = 5.0 g.
- For the decay constant, λ = 0.693 / T½ = 0.693 / 8.0 days = 0.087 per day.
- Check with the exponential form, keeping the unrounded λ = 0.693 / 8.0 = 0.086625 per day: N = 40 · e^(-0.086625 × 24) = 40 · e^(-2.079) = 40 × 0.1251 = 5.0 g, which agrees.
Answer. 5.0 g remains after 24 days, and the decay constant is 0.087 per day.
Common mistakes
- Thinking two half-lives leave nothing. Each half-life removes half of what is still there, never half of the original amount.
- Dividing the starting amount by the number of half-lives instead of halving repeatedly.
- Mixing time units between t, T½ and λ. A decay constant per day used against a time in hours puts the exponent out by a factor of 24, and the answer with it.
- Confusing half-life with average lifetime. The half-life is 0.693 times the average lifetime, not equal to it.
Related formulas
- Kinematics equations:
v = v₀ + at - Ideal gas law:
pV = nRT