In the context of radioactive decay, method 3 focuses on calculating the fraction or percentage of a radioisotope remaining after a certain period, utilizing both the radioactive half-life equation and the integrated rate law. The fraction of a radioisotope can be expressed as the ratio of the final concentration to the initial concentration. To convert this fraction into a percentage, one can multiply by 100.

For instance, consider iodine-131, which has a half-life of 8.021 days and is used in thyrotherapy. If a sample is estimated to be 6.25 months old, we first need to determine the decay constant (k). The relationship between half-life (t_1/2) and the decay constant is given by the equation:

$$ t_{1/2} = \frac{\ln(2)}{k} $$

Rearranging this equation allows us to solve for k:

$$ k = \frac{\ln(2)}{t_{1/2}} $$

Substituting the half-life value, we find:

$$ k = \frac{\ln(2)}{8.021 \text{ days}} \approx 0.08642 \text{ days}^{-1} $$

Next, we need to convert the age of the sample from months to days. Assuming an average month is about 30 days, 6.25 months translates to:

$$ 6.25 \text{ months} \times 30 \text{ days/month} = 187.5 \text{ days} $$

Now, we can apply the integrated rate law, which can be rearranged to find the fraction remaining:

$$ \ln\left(\frac{[A_f]}{[A_i]}\right) = -kt $$

Substituting the known values:

$$ \ln\left(\frac{[A_f]}{[A_i]}\right) = - (0.08642 \text{ days}^{-1})(187.5 \text{ days}) $$

This results in:

$$ \ln\left(\frac{[A_f]}{[A_i]}\right) \approx -16.20375 $$

To isolate the fraction, we exponentiate both sides:

$$ \frac{[A_f]}{[A_i]} = e^{-16.20375} $$

Calculating this gives a fraction of approximately:

$$ \frac{[A_f]}{[A_i]} \approx 9.18 \times 10^{-8} $$

This value represents the fraction of iodine-131 remaining in the sample after 6.25 months, demonstrating the application of radioactive decay principles in practical scenarios.

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