Decay Constant and Half-Life

Decay Constant and Half-Life

Radioactive decay is a stochastic process — any individual nucleus may decay at any moment, but large ensembles follow a precise statistical law.

The Decay Law

The rate of decay is proportional to the number of nuclei present:

$\frac{dN}{dt} = -\lambda N$

where $\lambda$ is the decay constant (units: s⁻¹). This gives the exponential solution:

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

Half-Life

The half-life $t_{1/2}$ is the time for half the nuclei to decay:

$t_{1/2} = \frac{\ln 2}{\lambda}$

Conversely, given the half-life:

$\lambda = \frac{\ln 2}{t_{1/2}}$

Mean Lifetime

The mean lifetime $\tau$ is the average time a nucleus survives before decaying:

$\tau = \frac{1}{\lambda} = \frac{t_{1/2}}{\ln 2} \approx 1.4427 \cdot t_{1/2}$

Activity

The activity $A$ is the number of decays per second (unit: Becquerel, Bq):

$A = \lambda N$

Your Task

Implement three functions using math.log(2) for $\ln 2$. All constants must be defined inside each function.

• decay_constant(half_life_s) — returns $\lambda = \ln 2 / t_{1/2}$ in s⁻¹
• half_life(lambda_s) — returns $t_{1/2} = \ln 2 / \lambda$ in s
• mean_lifetime(half_life_s) — returns $\tau = t_{1/2} / \ln 2$ in s