Final answer:
To find the remaining mass of lead-210 after 13.7 years, we use the formula for exponential decay with the initial mass of 5.15 g and a half-life of 22.3 years, resulting in approximately 3.94 grams of lead-210 remaining.
Step-by-step explanation:
The question is asking to determine the remaining mass of lead-210 after 13.7 years given that lead-210 has a half-life of 22.3 years and the initial mass of 5.15 grams. To calculate the remaining mass, we can use the formula for exponential decay, which describes how the quantity of a radioactive substance decreases over time. The formula is:
N = N_0 (1/2)^(t/T)
Where:
- N is the final amount of the substance
- N_0 is the initial amount of the substance
- t is the time that has passed
- T is the half-life of the substance
Using the given values, N_0 = 5.15 g, t = 13.7 years, and T = 22.3 years, we substitute these into the equation to calculate N:
N = 5.15 g (1/2)^(13.7 years / 22.3 years)
Calculating this gives us the remaining mass of lead-210:
N ≈ 5.15 g (1/2)^(0.6143)
N ≈ 5.15 g (0.765)
N ≈ 3.94 g
Therefore, after 13.7 years, approximately 3.94 grams of lead-210 will be present in the sample.