Radiometric Dating/Example/Radium in Lead/100 Years

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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:

\(\ds \dfrac x {x_0}\) \(=\) \(\ds e^{100 \paren {\ln 2 / 1600} }\)
\(\ds \) \(=\) \(\ds e^{\ln 2 / 16}\)
\(\ds \) \(=\) \(\ds 0.9576\)


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

$\blacksquare$


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