Radiometric Dating

Radiometric Dating

Radioactive decay provides a natural clock. If we know the initial amount $N_0$ of a radioactive isotope and measure the fraction $N/N_0$ remaining today, we can calculate the age of the sample.

Starting from $N(t) = N_0 e^{-\lambda t}$, solving for $t$:

$t = -\frac{\ln(N/N_0)}{\lambda} = \frac{t_{1/2} \cdot \ln(N_0/N)}{\ln 2}$

Common Radiometric Clocks

Carbon-14 Dating

Carbon-14 is produced continuously in the upper atmosphere by cosmic rays. Living organisms exchange carbon with the environment, maintaining a constant $^{14}$C/$^{12}$C ratio. When an organism dies, the exchange stops and $^{14}$C decays with $t_{1/2} = 5730$ years.

Uranium-238 Dating

With $t_{1/2} = 4.468 \times 10^9$ years, U-238 is used to date the oldest rocks and meteorites. The decay chain ends at stable Pb-206.

Your Task

Implement three functions. All constants (half-lives) must be defined inside each function body.

• age_from_fraction(fraction_remaining, half_life_years) — general formula
• carbon14_age(fraction_remaining) — uses $t_{1/2} = 5730$ years internally
• uranium238_age(fraction_remaining) — uses $t_{1/2} = 4.468 \times 10^9$ years internally