Calculate the amount of americium-241 consumed in a smoke detector

Question

Americium-241 is widely used in smoke detectors. Calculate the amount of americium-241 consumed in a smoke detector (initially containing 5.00 µg of Am), in micrograms, that has been used for one year if the half-life is 433 years.

Answer 1

Radioactive decay is a first order reaction, and so we can use the following expression:

FR = 0.5ⁿ

FR = fraction remaining = ?

n = number of half lives that have elapsed = 1 yr x 1 half life/433 yrs = 0.0023 half lives elapsed

FR = 0.5^0.0023

FR = 0.998 (this is the fraction remaining, which means only 0.002 was used)

Mass used = 0.002 x 5.00 ug = 0.01 ug of the Am-241 has been consumed in 1 year