Radioactive Decay — Question 1

Question 1

Constant decay probability produces exponential survival

Each undecayed nucleus has probability $\lambda\,dt$ of decaying in a short interval dt, independent of its age. A sample initially contains $N_0$ nuclei.

1. Derive the population law and half-life.

2. Find the fraction that decays between one and three half-lives, and the time required for 99 percent to decay.

Question 1 - Solution

Constant decay probability produces exponential survival

(a) Method and calculation

$$dN=-\lambda Ndt\Rightarrow\int_{N_0}^N\frac{dN\prime}{N\prime}=-\lambda t\Rightarrow\boxed{N=N_0e^{-\lambda t}}.$$$$N(T_{1/2})=N_0/2\Rightarrow\boxed{T_{1/2}=\ln2/\lambda}.$$

(b) Interpretation and further calculation

$$\boxed{[N(T_{1/2})-N(3T_{1/2})]/N_0=1/2-1/8=3/8}.$$$$\boxed{t_{99}=\ln100/\lambda=6.644T_{1/2}}.$$

Physical check

The probability per unit time is constant, but the number of decays per unit time decreases because fewer parent nuclei remain.