Radioactive Decay Law

Radioactive Decay Law

In the previous lesson we derived the decay constant $\lambda$. Now we use it to track how a population of nuclei evolves over time.

Number of Nuclei Over Time

Solving $dN/dt = -\lambda N$ gives:

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

where $N_0$ is the initial number of nuclei. Equivalently, using the half-life $t_{1/2}$:

$N(t) = N_0 \left(\frac{1}{2}\right)^{t/t_{1/2}}$

Fraction Remaining

The fraction of original nuclei still present:

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

After exactly one half-life: $e^{-\lambda \cdot t_{1/2}} = e^{-\ln 2} = 0.5$

After $n$ half-lives: $\left(\frac{1}{2}\right)^n$

Activity Over Time

The activity (decay rate) also follows exponential decay:

$A(t) = \lambda N(t) = \lambda N_0 \, e^{-\lambda t} = A_0 \, e^{-\lambda t}$

where $A_0 = \lambda N_0$ is the initial activity in Becquerel (Bq).

Example: Carbon-14 Dating

C-14 has a half-life of 5730 years. A living organism maintains a constant C-14/C-12 ratio. After death, C-14 decays with no replenishment. Measuring the fraction remaining tells us the age:

$t = -\frac{1}{\lambda} \ln\left(\frac{N}{N_0}\right)$

After one half-life (5730 yr), exactly half the C-14 remains. After two half-lives (11460 yr), one quarter remains.

Your Task

Implement three functions. All constants must be defined inside each function.

• nuclei_remaining(N0, lambda_s, t) — returns $N_0 \, e^{-\lambda t}$
• fraction_remaining(lambda_s, t) — returns $e^{-\lambda t}$
• activity(N0, lambda_s, t) — returns $\lambda N_0 \, e^{-\lambda t}$ in Bq