# 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:

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

## 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

- 1972: George F. Simmons:
*Differential Equations*... (previous) ... (next): $1$: The Nature of Differential Equations: $\S 4$: Growth, Decay and Chemical Reactions