Half-Life of Radioactive Substance

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Theorem

Let a radioactive element $S$ decay with a rate constant $k$.

Then its half-life $T$ is given by:

$T = \dfrac {\ln 2} k$


HalfLife.png


Proof

Let $x_0$ be the quantity of $S$ at time $t = 0$.

At time $t = T$ the quantity of $S$ has been reduced to $x = \dfrac {x_0} 2$.

This gives:

\(\displaystyle x_0 e^{-k T}\) \(=\) \(\displaystyle \frac {x_0} 2\) First-Order Reaction
\(\displaystyle \leadsto \ \ \) \(\displaystyle e^{k T}\) \(=\) \(\displaystyle 2\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle k T\) \(=\) \(\displaystyle \ln 2\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle T\) \(=\) \(\displaystyle \frac {\ln 2} k\)

$\blacksquare$


Sources