To determine the accurate conclusion based on the given data, we need to calculate the half-lives of both materials. The details for each material’s mass decrement were given as follows:
- Material 1: Original mass = 100 grams, Remaining mass after 21.6 seconds = 12.5 grams
- Material 2: Original mass = 200 grams, Remaining mass after 21.6 seconds = 25 grams
### Step-by-Step Solution:
1. Determine the ratio of remaining mass to original mass for each material:
[tex]\[
\text{Ratio for Material 1} = \frac{\text{Remaining mass}}{\text{Original mass}} = \frac{12.5}{100} = 0.125
\][/tex]
[tex]\[
\text{Ratio for Material 2} = \frac{\text{Remaining mass}}{\text{Original mass}} = \frac{25}{200} = 0.125
\][/tex]
2. Use the formula for radioactive decay to calculate the half-life:
The formula that relates the remaining mass to the original mass involving the half-life (T) and elapsed time (t) is:
[tex]\[
\text{Remaining mass} = \text{Original mass} \times \left( \frac{1}{2} \right)^{\frac{t}{T}}
\][/tex]
Simplifying:
[tex]\[
\frac{\text{Remaining mass}}{\text{Original mass}} = \left( \frac{1}{2} \right)^{\frac{t}{T}}
\][/tex]
Taking the logarithm base 2 of both sides:
[tex]\[
\log_2 \left( \frac{\text{Remaining mass}}{\text{Original mass}} \right) = \frac{t}{T} \log_2 \left( \frac{1}{2} \right)
\][/tex]
[tex]\[
\log_2 \left( \frac{Remaining mass}{Original mass} \right) = - \frac{t}{T}
\][/tex]
Therefore:
[tex]\[
T = \frac{t}{\log_2 \left( \frac{1}{0.125} \right)}
\][/tex]
3. Calculate the half-life for each material:
For both materials, since their ratios are the same:
[tex]\[
\log_2 \left( \frac{1}{0.125} \right) = \log_2 8 = 3
\][/tex]
For Material 1:
[tex]\[
T_1 = \frac{21.6}{3} = 7.2 \text{ seconds}
\][/tex]
For Material 2:
[tex]\[
T_2 = \frac{21.6}{3} = 7.2 \text{ seconds}
\][/tex]
Both materials have a half-life of 7.2 seconds.
4. Conclusion:
Since both materials share the same half-life:
[tex]\[
T_1 = T_2 = 7.2 \text{ seconds}
\][/tex]
Thus, the correct conclusion is:
- The half-life of Material 1 and Material 2 are equal.
Hence, the most likely correct conclusion is:
The half-life of Material 1 and Material 2 are equal.
