A 53.0-mg supply of iodine-131, used in hospitals in the treatment of hyperthyroidism, was stored for 40.11 days (d). If the half-life of iodine-131 is 8.021 days, how many mg remain? Be sure your answer has the correct number of significant figures.

Answer 1

To find the amount of iodine-131 remaining after a certain time, we can use the formula for radioactive decay based on the half-life:

$$N=N_0{\left(\frac{1}{2}\right)}^{t/T_{1/2}}$$

where:

• $N$ is the remaining quantity of substance,
• $N_0$ is the initial quantity of substance,
• $t$ is the elapsed time,
• $T_{1/2}$ is the half-life of the substance.

Given:

• $N_0=53.0,\text{mg}$
• $t=40.11,\text{days}$
• $T_{1/2}=8.021,\text{days}$

First, we calculate the number of half-lives that have elapsed during the time $t$:

$$\text{Number of half-lives}=\frac{t}{T_{1/2}}=\frac{40.11,\text{d}}{8.021,\text{d}}\approx 5.000$$

Next, we can plug this value into the decay formula:

$$N=53.0,\text{mg}{\left(\frac{1}{2}\right)}^{5.000}$$

Now we calculate ${\left(\frac{1}{2}\right)}^{5.000}$:

$${\left(\frac{1}{2}\right)}^{5.000}=\frac{1}{32}=0.03125$$

Now, we can calculate $N$:

$$N=53.0,\text{mg}\times 0.03125=1.65625,\text{mg}$$

Finally, we need to report $N$ with the correct number of significant figures. The initial amount (53.0 mg) has three significant figures. Thus, we round the final answer to three significant figures:

$$N\approx 1.66,\text{mg}$$

Therefore, the amount of iodine-131 remaining after 40.11 days is approximately 1.66 mg.