Radioactive Decay

Radioactive nuclei spontaneously transform to lower-energy states by emitting radiation. The three main types are alpha (He-4 nucleus), beta (electron or positron), and gamma (photon) decay. All follow the exponential decay law N(t) = N₀ e^{-λt}.

Key Concepts

Decay law: N(t) = N₀ e^{-λt}
Half-life: t_{1/2} = ln2/λ = 0.693/λ
Activity: A = λN = A₀e^{-λt}
Alpha decay: A_Z X → A-4_{Z-2} Y + ⁴₂He
Beta-minus: neutron → proton + e⁻ + ν̄_e

Key Equations

Radioactive decay law

N(t) = N_0 e^{-\lambda t}

Half-life

t_{1/2} = \frac{\ln 2}{\lambda} \approx \frac{0.693}{\lambda}

Activity

A(t) = \lambda N(t) = A_0 e^{-\lambda t}

Mean lifetime

\tau = 1/\lambda = t_{1/2}/\ln 2

Worked Example

Example Problem

Problem

¹⁴C has t_{1/2} = 5730 years. Find the decay constant λ.

Solution

λ = ln2/t_{1/2} = 0.693/(5730×3.156×10⁷ s) = 3.83×10⁻¹² s⁻¹.

Practice

Exercises

7 problems

1 of 7

Find the decay constant λ of ²²⁶Ra with t_{1/2} = 1600 years in s⁻¹. (1 year = 3.156×10⁷ s)

Your answer

s⁻¹

2 of 7

A sample has N₀=10¹² atoms of ²²⁶Ra (λ=1.37×10⁻¹¹ s⁻¹). Find initial activity in Bq.

Your answer

3 of 7

After 3 half-lives, what fraction of a radioactive sample remains?

Unlock Exercise 3