Question

Radioactive isotopes often occur together in mixtures. Suppose a $100g$sample contains $131Ba$, with a half-life of $12days$, and $47Ca$, with a half-life of localid="1650485393662" $4.5days$. If there are initially twice as many calcium atoms as there are barium atoms, what will be the ratio of calcium atoms to barium atoms localid="1650485404787" $2.5$weeks later?

Short answer

Ratio of calcium to barium atoms is, $\frac{N_{Ca}}{N_{Ba}}=0.373$

Step-by-step solution

Given information

We are given that,

Half life of Ba is, ${\left(t_{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\right)}_{Ba}=12days$

Half life of Ca is,localid="1650485477257" ${\left(t_{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\right)}_{Ca}=4.5days$

Simplify

Let time constant for calcium is,${\tau }_{Ca}$

And time constant for barium is,${\tau }_{Ba}$

Time constant,

localid="1650485605612" $$\begin{gathered} {\tau }_{Ca}=\frac{{\left(t_{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}\right)}_{Ca}}{\ln \left(2\right)} \\ =\frac{4.5days}{\ln \left(2\right)} \\ =6.492days \end{gathered}$$

And,

localid="1650485623266" $$\begin{gathered} {\tau }_{Ba}=\frac{{\left(t_{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}\right)}_{Ba}}{\ln \left(2\right)} \\ =\frac{12days}{\ln \left(2\right)} \\ =17.30days \end{gathered}$$

Now, ratio is,

localid="1650485643779" $$\begin{gathered} \frac{N_{Ca}}{N_{Ba}}=\frac{{\left(N_a\right)}_{Ca}}{{\left(N_a\right)}_{Ba}}\times \frac{e^{{\displaystyle \raisebox{1ex}{$-t$}\!\left/ \!\raisebox{-1ex}{${\tau }_{Ca}$}\right.}}}{e^{\raisebox{1ex}{$-t$}\!\left/ \!\raisebox{-1ex}{${\tau }_{Ba}$}\right.}} \\ =\frac{2\times 0.0675}{0.3639} \\ =0.373 \end{gathered}$$

Here, $N_{Ca}$is final no. of atoms of calcium

And $N_{Ba}$is final no. of atoms of barium

And ${\left(N_{\alpha }\right)}_{Ca}$is initial no. of atoms of calcium

And ${\left(N_{\alpha }\right)}_{Ba}$ is initial no. of atoms of barium

And $t$is time after which ratio of atoms is calculated