Two Stage Radioactive Decay

Theorem

Let $R_A$ be a radioactive isotope which decays into another radioactive isotope $R_B$ with the rate constant $k_1$.

Let $R_B$ decay into a third element $R_C$ with the rate constant $k_2$.

Let the initial quantity of $R_A$ be $x_0$.

Let the amounts of $R_A$ and $R_B$ present at time $t$ be $x$ and $y$ respectively.

Then:

$y = \begin{cases}

\dfrac {k_1 x_0} {k_2 - k_1} \left({e^{-k_1 t} - e^{-k_2 t} }\right) & : k_1 \ne k_2 \\ & \\ k_1 x_0 t e^{-k_1 t} & : k_1 = k_2 \end{cases}$

Proof

This theorem requires a proof. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{ProofWanted}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

Sources

• 1972: George F. Simmons: Differential Equations ... (previous) ... (next): Miscellaneous Problems for Chapter $2$: Problem $26$