Question 43.5: A Radioactive Isotope of Iodine A sample of the isotope ^131...
A Radioactive Isotope of Iodine
A sample of the isotope {}^{131}I, which has a half-life of 8.04 days, has an activity of 5.0 mCi at the time of shipment. Upon receipt of the sample at a medical laboratory, the activity is 2.1 mCi. How much time has elapsed between the two measurements?
Conceptualize The sample is continuously decaying as it is in transit. The decrease in the activity is 58% during the time interval between shipment and receipt, so we expect the elapsed time to be greater than the half-life of 8.04 d.
Categorize The stated activity corresponds to many decays per second, so N is large and we can categorize this problem as one in which we can use our statistical analysis of radioactivity.
Analyze Solve Equation 43.7 for the ratio of the final activity to the initial activity and take the natural logarithm of both sides:
R=\left|\frac{d N}{d t}\right|=\lambda N=\lambda N_0 e^{-\lambda t}=R_0 e^{-\lambda t} (43.7)
\frac{R}{R_0}=e^{-\lambda t} \rightarrow \ln \left(\frac{R}{R_0}\right)=-\lambda tSolve for the time t and use Equation 43.8 to substitute for λ:
T_{1 / 2}=\frac{\ln 2}{\lambda}=\frac{0.693}{\lambda} (43.8)
(1) t=-\frac{1}{\lambda} \ln \left(\frac{R}{R_0}\right)=-\frac{T_{1 / 2}}{\ln 2} \ln \left(\frac{R}{R_0}\right)
Substitute numerical values:
t=-\frac{8.04 d}{0.693} \ln \left(\frac{2.1 mCi}{5.0 mCi}\right)=10 dFinalize This result is indeed greater than the half-life, as expected. This example demonstrates the difficulty in shipping radioactive samples with short half-lives. If the shipment is delayed by several days, only a small fraction of the sample might remain upon receipt. This difficulty can be addressed by shipping a combination of isotopes in which the desired isotope is the product of a decay occurring within the sample. It is possible for the desired isotope to be in equilibrium, in which case it is created at the same rate as it decays. Therefore, the amount of the desired isotope remains constant during the shipping process and subsequent storage. When needed, the desired isotope can be separated from the rest of the sample; its decay from the initial activity begins at this point rather than upon shipment.