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Chapter 21: Problem 44

Iodine-131 is used as a nuclear medicine to treat hyperthyroidism. The half- life of \({ }^{131} \mathrm{I}\) is 8.04 days. How long will it take for a $500 \mathrm{mg}\( sample of \){ }^{131} \mathrm{I}\( to decay into \)1 \%$ of its original mass?

Short Answer

Expert verified
It will take approximately \(53.65\) days for a \(500 \mathrm{mg}\) sample of Iodine-131 to decay into \(1 \%\) of its original mass.

Step by step solution

01

Determining the decay constant

Since we are given the half-life of Iodine-131, we can use the following relationship between the decay constant \(\lambda\) and the half-life \(t_{1/2}\): \[\lambda = \frac{\ln(2)}{t_{1/2}}\] Plugging in the given value of the half-life, which is 8.04 days, we get: \[\lambda = \frac{\ln(2)}{8.04}\] Calculate the decay constant: \[\lambda \approx 0.0863 \text{ per day}\]
02

Setting up the radioactive decay equation

Now that we have the decay constant, we can set up the radioactive decay equation with the given initial mass (\(500 \mathrm{mg}\)) and desired final mass (\(1 \% \text{ of the initial mass}\)): \[N(t) = N_0 e^{-\lambda t}\] Where \(N(t) = 500 \times 0.01 = 5 \mathrm{mg}\) and \(N_0 = 500 \mathrm{mg}\). Plug in these values into the equation: \[5 = 500 e^{-0.0863 t}\]
03

Solve for time

To solve for \(t\), we'll first need to divide by \(500\) on both sides of the equation: \[\frac{5}{500} = e^{-0.0863 t}\] Then take the natural logarithm of both sides: \[\ln{\frac{1}{100}} = -0.0863 t\] Now, divide by \(-0.0863\) to isolate \(t\): \[t = \frac{\ln{\frac{1}{100}}}{-0.0863}\] Calculate the time it takes to decay to \(1 \%\) of the initial mass: \[t \approx 53.65 \text{ days}\] It will take approximately \(53.65\) days for a \(500 \mathrm{mg}\) sample of Iodine-131 to decay into \(1 \%\) of its original mass.

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Most popular questions from this chapter

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