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Found in: Page 1307

### Fundamentals Of Physics

Book edition 10th Edition
Author(s) David Halliday
Pages 1328 pages
ISBN 9781118230718

# After a brief neutron irradiation of silver, two isotopes are present: ${}^{{\mathbf{108}}}{\mathbf{Ag}}{\mathbf{\left(}}{{\mathbf{T}}}_{\mathbf{1}\mathbf{/}\mathbf{2}}{\mathbf{=}}{\mathbf{2}}{\mathbf{.}}{\mathbf{42}}{\mathbf{min}}{\mathbf{\right)}}$ with an initial decay rate of ${\mathbf{3}}{\mathbf{.}}{\mathbf{1}}{\mathbf{×}}{{\mathbf{10}}}^{{\mathbf{5}}}{\mathbf{/}}{\mathbit{s}}$,and role="math" localid="1661598035621" ${}^{{\mathbf{110}}}{\mathbf{Ag}}{\mathbf{\left(}}{{\mathbf{T}}}_{\mathbf{1}\mathbf{/}\mathbf{2}}{\mathbf{=}}{\mathbf{24}}{\mathbf{.}}{\mathbf{6}}{\mathbf{s}}{\mathbf{\right)}}$ with an initial decay rate of. Make a semilog plot similar to Fig. 42-9 showing the total combined decay rate of the two isotopes as a function of time t = 0 from until t = 10min .We used Fig. 42-9 to illustrate the extraction of the half-life for simple (one isotope) decays. Given only your plot of total decay rate for the two-isotope system here, suggest a way to analyze it in order to find the half-lives of both isotopes.

The plot similar to the total combined decay rate of the two isotopes as a function of time from t = 0 to t = 10min is plotted.

See the step by step solution

## Step 1: The given data

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

## Step 2: Determine the concept of combined decay

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

$InR=In\left({R}_{0}{e}^{-\lambda t}+{R}_{0}\text{'}{e}^{-\lambda t}\right)$ …… (i)

The disintegration constant is as follows:

…… (ii)

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

## Step 3: Plot the decay graph of the combined isotope model

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:

$\mathrm{\lambda }=\frac{\mathrm{In}2}{145.2\mathrm{s}}\phantom{\rule{0ex}{0ex}}\mathrm{\lambda }=\frac{\mathrm{In}2}{24.6\mathrm{s}}$.

The plot is given below:

Note that the magnitude of the slope for small is $\lambda \text{'}$(the disintegration constant for ${}^{110}\mathrm{Ag}$ ) , and for large t is $\lambda$ (the disintegration constant for role="math" localid="1661598893886" ${}^{108}\mathrm{Ag}$ ).

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