91Ó°ÊÓ

1. The half life of ${}^{108}\mathrm{Ag},{\left({\mathrm{T}}_{1/2}\right)}_{108}=2.42\min \mathrm{or}145.2\mathrm{s}$
2. Decay rate of${}^{108}\mathrm{Ag},{\left(\mathrm{R}\right)}_{108}=3.1×{10}^5/\mathrm{s}$
3. The half life of${}^{110}\mathrm{Ag},{\left({\mathrm{T}}_{1/2}\right)}_{110}=24.6\mathrm{s}$
4. Decay rate of${}^{110}\mathrm{Ag},{\left(R\right)}_{110}=4.1×{10}^6\mathrm{s}$

The total combined decay rate of two-isotope system is as follows:

$InR=In\left(R_0{e}^{-\mathrm{λ}t}+R_0\text{'}{e}^{-\mathrm{λ}t}\right)$ …… (i)

The disintegration constant is as follows:

$\mathrm{λ}=\frac{\mathrm{In}2}{{\mathrm{T}}_{{\displaystyle \frac{1}{2}}}}$ …… (ii)

Here, ${\mathrm{T}}_{\frac{1}{2}}$is the half-life of the substance.

From the given data ${\mathrm{R}}_0=3.1×{10}^5/\mathrm{s}$and${\mathrm{R}}_0=4.1×{10}^6/\mathrm{s}$equation (i), the combined decay rate of the isotopes, the plot is made accordingly for disintegration constants using equation (ii) as:

$$\begin{gathered} \mathrm{λ}=\frac{\mathrm{In}2}{145.2\mathrm{s}} \\ \mathrm{λ}=\frac{\mathrm{In}2}{24.6\mathrm{s}} \end{gathered}$$.

The plot is given below:

Note that the magnitude of the slope for smallis$\mathrm{λ}\text{'}$(the disintegration constant for${}^{110}\mathrm{Ag}$ ) , and for large tis $\mathrm{λ}$(the disintegration constant forrole="math" localid="1661598893886" ${}^{108}\mathrm{Ag}$ ).