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Chapter 11: Problem 46

Iodine-131 is used to treat tumors in the thyroid. Its first-order half-life is \(8.1\) days. If a patient is given a sample containing \(5.00 \mathrm{mg}\) of \(\mathrm{I}-131\), how long will it take for \(25 \%\) of the isotope to remain in her system?

Short Answer

Expert verified
Answer: It takes approximately 10.63 days for only 25% of the initial amount of Iodine-131 isotope to remain.

Step by step solution

01

Find the decay constant (k)

We can find the decay constant using the half-life for Iodine-131. The relation between decay constant (k) and half-life (t_1/2) is as follows: \( t_{1/2} = \frac{ln(2)}{k} \) We are given half-life, so let's rearrange the formula to get k: \( k = \frac{ln(2)}{t_{1/2}} \) Now, plug in the half-life value for Iodine-131: \( k = \frac{ln(2)}{8.1\, \text{days}} \approx 0.0855\; \text{day}^{-1} \)
02

Determine the remaining percentage of Iodine-131 after the decay

We are given that we want 25% of the initial amount to remain. First, let's express this remaining percentage as a fraction: \( \frac{A(t)}{A_0} = \frac{25}{100} = 0.25 \) Where A(t) is the amount of Iodine-131 after time t, and A0 is the initial amount of Iodine-131 (5mg).
03

Use the integrated rate law to find the time (t)

The first-order integrated rate law equation is: \( ln\left(\frac{A(t)}{A_0}\right) = -kt \) Now, plug in the values for \(\frac{A(t)}{A_0}\) and \(k\): \( ln(0.25) = -0.0855\; t \) Calculate the value of t: \( t = \frac{ln(0.25)}{-0.0855} \approx 10.63\; \text{days} \)
04

Final Answer

The time it takes for 25% of Iodine-131 to remain in the patient's system is approximately 10.63 days.

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Most popular questions from this chapter

Hydrogen bromide is a highly reactive and corrosive gas used mainly as a catalyst for organic reactions. It is produced by reacting hydrogen and bromine gases together. $$\mathrm{H}_{2}(g)+\mathrm{Br}_{2}(g) \longrightarrow 2 \mathrm{HBr}(g)$$ The rate is followed by measuring the intensity of the orange color of the bromine gas. The following data are obtained: $$\begin{array}{cccc}\hline \text { Expt. } & {\left[\mathrm{H}_{2}\right]} & {\left[\mathrm{Br}_{2}\right]} & \text { Initial Rate }(\mathrm{mol} / \mathrm{L} \cdot \mathrm{s}) \\ \hline 1 & 0.100 & 0.100 & 4.74 \times 10^{-3} \\ 2 & 0.100 & 0.200 & 6.71 \times 10^{-3} \\ 3 & 0.250 & 0.200 & 1.68 \times 10^{-2} \\ \hline\end{array}$$(a) What is the order of the reaction with respect to hydrogen, bromine, and overall? (b) Write the rate expression for the reaction. (c) Calculate \(k\) for the reaction. What are the units for \(k ?\) (d) When \(\left[\mathrm{H}_{2}\right]=0.455 \mathrm{M}\) and \(\left[\mathrm{Br}_{2}\right]=0.215 M\), what is the rate of the reaction?

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