- The time taken to reach one-fourth of an initial radioactive amount is \(t = 2T\), where \(T\) is the material's half-life.
- Comparing the half-lives directly, the material with the shortest half-life will decay to one-fourth the fastest.
- Bismuth-212 has the shortest half-life (60.5 seconds) compared to Sodium-24 (15 hours), Iodine-131 (8.07 days), and Radium-226 (1,600 years).
- Therefore, Bismuth-212 decays the fastest, with a time of \(\boxed{\text{Bismuth-212}}\).
Explanation
- Analyzing the Decay Process
The rate at which a radioactive material decays depends on its half-life, which is the time required for half of the initial quantity of the substance to decay. For a radioactive material with an initial amount \(N_0\) and a half-life \(T\), the amount remaining at time \(t\) is given by the formula \(N(t) = N_0 \left( \frac{1}{2} \right)^{\frac{t}{T}}\). We want to find the time \(t\) it takes for the substance to decay to one-fourth (\(1/4\)) of its original amount. We set \(N(t) = \frac{N_0}{4}\) and solve for \(t\):
\[ \frac{N_0}{4} = N_0 \left( \frac{1}{2} \right)^{\frac{t}{T}} \]
\[ \frac{1}{4} = \left( \frac{1}{2} \right)^{\frac{t}{T}} \]
Since \(\frac{1}{4} = \left( \frac{1}{2} \right)^2\), the equation becomes:
\[ \left( \frac{1}{2} \right)^2 = \left( \frac{1}{2} \right)^{\frac{t}{T}} \]
Equating the exponents, we get:
\[ 2 = \frac{t}{T} \implies t = 2T \]
This shows that the time taken to reach one-fourth of the initial amount is exactly twice the half-life of the material. Therefore, the material with the shortest half-life will reach one-fourth of its original amount in the shortest time.
- Comparing Half-Lives and Determining the Fastest Decay
Comparing the half-lives provided in the table:
- Bismuth-212: 60.5 seconds
- Sodium-24: 15 hours
- Iodine-131: 8.07 days
- Radium-226: 1,600 years
Clearly, 60.5 seconds is the smallest duration among the listed values. Since Bismuth-212 has the smallest half-life, it will take the least amount of time to reach one-fourth of its initial amount. Specifically, the time required is \(t = 2 \times 60.5 \text{ seconds} = 121 \text{ seconds}\). All other materials require significantly longer periods (measured in hours, days, or years). Thus, Bismuth-212 decays the fastest.
- Conclusion
The analysis confirms that the radioactive material that decays the fastest to one-fourth of its original amount is Bismuth-212. The final answer is \(\boxed{\text{Bismuth-212}}\).
Examples
Understanding the decay rates of radioactive materials is crucial in various fields, such as medicine and geology. For instance, in nuclear medicine, clinicians use isotopes with short half-lives, like Technetium-99m, for diagnostic imaging because they decay quickly, minimizing the patient's exposure to radiation. Similarly, in archaeology, researchers use carbon-14, which has a much longer half-life, to date organic materials over thousands of years. Knowing how these decay processes work helps scientists select the right isotope for their specific needs, ensuring both safety and accuracy in their work.