Question

Construct a mathematical model for a radioactive series of four elements W, X, Y, and Z, where Z is a stable element.

Short answer

So, a mathematical model for a radioactive series of four elements , and, where is a stable element is $\frac{dx}{dt}={\lambda }_{1}w-{\lambda }_{2}x$.

Step-by-step solution

Definition of radioactive series

Radioactive series are any of four independent sets of unstable heavy atomic nuclei that decay through a sequence of alpha and beta decays until a stable nucleus is achieved.

Make a model for radioactive series

Suppose that the series is described schematically by $W\stackrel{-{\lambda }_1}{\to }X\stackrel{-{\lambda }_2}{\to }Y\stackrel{-{\lambda }_3}{\to }Z$.

Where $-{\lambda }_1,{\lambda }_2$and$-{\lambda }_3$ are the decay constants for And respectively and is a stable element.

Let $w\left(t\right),x\left(t\right),y\left(t\right)$and $z\left(t\right)$denote the amounts of substances and respectively then,

The decay of element W is described by

$\frac{dw}{dt}=-{\lambda }_{1}w$

Now, X is gaining atoms from the decay of W and at the same time losing atoms because of its own decay.

So, the rate at which X decays is described by $\frac{dx}{dt}={\lambda }_{1}w-{\lambda }_{2}x$.