What is the radioactive decay law?

N(t) = N₀ e^(−λt), where λ is the decay constant. The number of radioactive nuclei decreases exponentially.

What is the relationship between λ and half-life?

λ = ln 2 / t₁/₂. After one half-life, half of the original nuclei remain; after two, a quarter; and so on.

How is radioactive dating done?

Measure the ratio of a radioactive isotope to its decay product in a sample. Knowing the half-life, solve t = −ln(N/N₀)/λ for the age. Carbon-14 dating works for organic samples up to about 50000 years.

Does activity follow the same equation?

Yes. Activity A = λN, so A(t) = A₀ e^(−λt) with the same time constant. After n half-lives, both N and A are reduced by a factor of 2ⁿ.

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NSW · HSCModule 8

Radioactive decay calculator

N = N₀ e^(−λt) with λ = ln 2 / t₁/₂. Solve for the remaining number after time t, or the time required to reach a target number.

Inputs

Solve forInitial number N₀Half-life t₁/₂ (s (or any unit))Use any time unit; just match t below.Elapsed time t ((same unit as half-life))

Result

Number remaining N

250.0

Fraction remaining

0.2500

Number of half-lives

2.000

Decay constant λ

1.210e-4per unit time

N_t = N₀ e^(−λt), with λ = ln2 / t₁/₂. After n half-lives, the fraction remaining is (1/2)ⁿ.

How this calculator works

The calculator converts your half-life into a decay constant, then applies N(t) = N₀ e^(−λt) for the forward direction, or inverts to find t for a target N. Use any consistent time units for half-life and elapsed time.

Common questions

What is the radioactive decay law?: N(t) = N₀ e^(−λt), where λ is the decay constant. The number of radioactive nuclei decreases exponentially.
What is the relationship between λ and half-life?: λ = ln 2 / t₁/₂. After one half-life, half of the original nuclei remain; after two, a quarter; and so on.
How is radioactive dating done?: Measure the ratio of a radioactive isotope to its decay product in a sample. Knowing the half-life, solve t = −ln(N/N₀)/λ for the age. Carbon-14 dating works for organic samples up to about 50000 years.
Does activity follow the same equation?: Yes. Activity A = λN, so A(t) = A₀ e^(−λt) with the same time constant. After n half-lives, both N and A are reduced by a factor of 2ⁿ.