After a brief neutron irradiation of silver, two isotopes...

After a brief neutron irradiation of silver, two isotopes are present: ${ }^{108} \mathrm{Ag}\left(T_{1 / 2}=2.42 \mathrm{~min}\right)$ with an initial decay rate of $3.1 \times 10^{5} / \mathrm{s},$ and ${ }^{110} \mathrm{Ag}\left(T_{1 / 2}=24.6 \mathrm{~s}\right)$ with an initial decay rate of $4.1 \times 10^{6} / \mathrm{s} .$ Make a semilog plot similar to Fig. $42-9$ showing the total combined decay rate of the two isotopes as a function of time from $t=0$ until $t=10$ min. 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.

After a brief neutron irradiation of silver, two isotopes are present: ${ }^{108} \mathrm{Ag}\left(T_{1 / 2}=2.42 \mathrm{~min}\right)$ with an initial decay rate of $3.1 \times 10^{5} / \mathrm{s},$ and ${ }^{110} \mathrm{Ag}\left(T_{1 / 2}=24.6 \mathrm{~s}\right)$ with an initial decay rate of
$4.1 \times 10^{6} / \mathrm{s} .$ Make a semilog plot similar to Fig. $42-9$ showing the total combined decay rate of the two isotopes as a function of time from $t=0$ until $t=10$ min. 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.

Key Concepts

Half-Life and Decay Constant

The half-life is the time required for half of the radioactive nuclei to decay, and it is inversely related to the decay constant through the relation T?/? = ln(2)/?. This concept is critical for quantifying how rapidly a radioactive isotope decays, making it possible to compare the stability or decay rates of different isotopes.

Semilog Plot Analysis

A semilog plot, where the logarithm of the decay rate is plotted against time, is particularly useful because exponential decay curves become straight lines on this type of graph. The slope of the line is directly related to the decay constant, which in turn provides a method to calculate the half-life. This linearity simplifies the identification of different decay rates in complex systems.

Exponential Decay

Exponential decay is the mathematical description of a quantity that decreases over time at a rate proportional to its current value. In the context of radioactive decay, the number of undecayed nuclei follows an exponential law, which is fundamental to understanding processes where the rate of decay is constant relative to the amount present.

Multi-Exponential Decay Analysis

When a system involves the sum of multiple exponential decays, such as from two different isotopes, the overall decay curve is a combination of these processes. Analysis techniques involve identifying regions where one exponent dominates – often visible as different linear segments in a semilog plot – and then isolating these segments to extract the individual decay constants and corresponding half-lives. Subtraction methods or fitting procedures can be employed to disentangle overlapping contributions.

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