Radiometric Dating/Example/Radium in Lead/1000 Years

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Example of Radiometric Dating

Let $Q$ be a sample of lead.

Let it be established that $10 \%$ of the radium decays in $200$ years.


After $1000$ years, there will be approximately $59.05 \%$ of the original amount of radium in $Q$.


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.

We are given that when $t = 200$, $x = 0.9 \times x_0$.

Hence:

\(\ds 0.9 x_0\) \(=\) \(\ds x_0 e^{-200 k}\)
\(\ds \leadsto \ \ \) \(\ds 0.9\) \(=\) \(\ds e^{-200 k}\)
\(\ds \leadsto \ \ \) \(\ds -200 k\) \(=\) \(\ds \ln 0.9\)
\(\ds \leadsto \ \ \) \(\ds k\) \(=\) \(\ds -\dfrac {\ln 0.9} {200}\)

So after $1000$ years, we have:

\(\ds \dfrac x {x_0}\) \(=\) \(\ds e^{-1000 \times \paren {-\ln 0.9 / 200} }\)
\(\ds \) \(=\) \(\ds e^{5 \times \ln 0.9}\)
\(\ds \) \(\approx\) \(\ds 0.59045\)

$\blacksquare$


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