Question 14.4.7: Use half-life to calculate mass remaining for radioactive de......
Use half-life to calculate mass remaining for radioactive decay.
The radioactive isotope ^{32}P has a half-life of 14.3 days. If a sample contains 0.884 g of ^{32}P, what mass of the isotope will remain after 22 days?
You are asked to determine the mass of a radioactive isotope remaining after a given period of time.
You are given the mass of isotope originally present, the half-life of the isotope, and the period of time that passes.
First, use the half-life to calculate the rate constant for this radioactive decay.
t_{1/2} = \frac{0.693}{k}
k = \frac{0.693}{t_{1/2}} = \frac{0.693}{14.3\text{ d}} = 0.0485 d^{-1}
Next, use the integrated first-order rate law to calculate the mass of ^{32}P remaining after 22 days.
ln\frac{N_{t}}{N_{o}} = -kt
ln\frac{N_{t}}{0.884\text{ g}} = -(0.0485 d^{-1}) (22.0 d)
ln(N_{t}) − ln(0.884 g) = −1.07
ln(N_{t}) = −1.19
N_{t} = e^{−1.19} = 0.304 g