Book edition 10th Edition

Author(s) David Halliday

Pages 1328 pages

ISBN 9781118230718

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Short Answer

After a brief neutron irradiation of silver, two isotopes are present: $^{108} Ag (T_{1 / 2} = 2.42 \min)$ with an initial decay rate of $3.1 \times 10^{5} / s$ ,and role="math" localid="1661598035621" $^{110} Ag (T_{1 / 2} = 24.6 s)$ 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.

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: The given data

The half life of $^{108} Ag, {(T_{1 / 2})}_{108} = 2.42 \min or 145.2 s$
Decay rate of $^{108} Ag, {(R)}_{108} = 3.1 \times 10^{5} / s$
The half life of $^{110} Ag, {(T_{1 / 2})}_{110} = 24.6 s$
Decay rate of $^{110} Ag, {(R)}_{110} = 4.1 \times 10^{6} s$

Step 2: Determine the concept of combined decay

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

$I n R = I n (R_{0} e^{- λ t} + R_{0}' e^{- λ t})$ …… (i)

The disintegration constant is as follows:

$λ = \frac{In 2}{T_{\frac{1}{2}}}$ …… (ii)

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

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

From the given data $R_{0} = 3.1 \times 10^{5} / s$ and $R_{0} = 4.1 \times 10^{6} / s$ equation (i), the combined decay rate of the isotopes, the plot is made accordingly for disintegration constants using equation (ii) as:

$λ = \frac{In 2}{145.2 s} λ = \frac{In 2}{24.6 s}$ .

The plot is given below:

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

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Fundamentals Of Physics

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Short Answer

Step by Step Solution

Step 1: The given data

Step 2: Determine the concept of combined decay

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

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Fundamentals Of Physics

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