Question

Consider a nucleus with mass number $A$. Its mass, within $1\%$, is $A$atomic mass units.

The fact that $A$cancels means that all nuclei have this density. It is a staggeringly large density, roughly ${10}^{14}$times larger than the density of familiar liquids and solids.

One early objection to Rutherford’s model of a nuclear atom was that matter simply couldn’t have a density this high. Although we have no direct experience with such matter, nuclear matter really is this dense.

Short answer

The expression for the number of atoms present in a sample after some time is $N=N_0{e}^{-\lambda x}$.

Here ,$N_0$ is the initial number of atoms in the sample, $t$is the time, and $\lambda$is the decay constant.

Therefore, the ratio of calcium atoms to barium atoms after$2.5$weeks is$0.37$.

Step-by-step solution

Lifetime of Ca-47 isotope

${\tau }_{Ca-47=\raisebox{1ex}{$4.5d$}\!\left/ \!\raisebox{-1ex}{$ln2$}\right.}$

=$6.5d\left(\raisebox{1ex}{$1.0week$}\!\left/ \!\raisebox{-1ex}{$7d$}\right.\right)$

$=0.93$week

Similarly, calculate the lifetime of$Ba-131$by substituting $12d$for $t\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$.