Radiometric Dating/Example/Radium in Lead/100 Years

Example of Radiometric Dating
Let $Q$ be a sample of lead.

It is established that the half-life of radium-226 is $1600$ years.

So, after $100$ years, the amount of radium-226 that has decayed will be $4.2 \%$.

Proof
From First-Order Reaction, we have:


 * $x = x_0 e^{-k t}$

where:
 * $x$ is the quantity of radium at time $t$
 * $x_0$ is the quantity of radium at time $t = 0$
 * $k$ is a positive number.

By definition of half-life, when $x = \dfrac {x_0} 2$, we have $t = 1600$.

So:
 * $e^{-1600 k} = \dfrac 1 2$

So:
 * $k = \dfrac {\ln 0.5} {-1600} = \dfrac {\ln 2} {1600}$

After $100$ years:

So there is $95.76\%$ remaining, and so $4.2\%$ will be lost.