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.
$\blacksquare$
Sources
- 1963: Morris Tenenbaum and Harry Pollard: Ordinary Differential Equations ... (previous) ... (next): Chapter $1$: Basic Concepts: Lesson $1$: How Differential Equations Originate: Exercise $1.2$