Question

medical technician is working with the four samples of radionuclides listed in the tabl first, order the samples by decreasing initial radioactivity. then calculate how long it wi f the initial amount.

sample	radionuclide	initial radioactivity	time for amount of radionuclide to decrease to 1/8 of initial amount
a	\\(\ce{^{13}_{7}n}\\)	10. minutes	select	30 minutes
b	\\(\ce{^{33}_{15}p}\\)	25. days	select	75 days
c	\\(\ce{^{90}_{39}y}\\)	64.0 hours	select	192 hours
d	\\(\ce{^{131}_{53}i}\\)	8.0 days	select	24 days

Explanation:

Step1: Calculate the number of half - lives

The formula for the amount of a radionuclide remaining is \(N = N_0(\frac{1}{2})^n\), where \(N\) is the final amount, \(N_0\) is the initial amount, and \(n\) is the number of half - lives. If \(N=\frac{1}{8}N_0\), then \(\frac{1}{8}=(\frac{1}{2})^n\). Solving for \(n\), we know that \((\frac{1}{2})^3=\frac{1}{8}\), so \(n = 3\) for all samples.

Step2: Analyze initial radioactivity

Initial radioactivity is not related to the time calculation (since the time to reach \(\frac{1}{8}\) of the initial amount is \(n = 3\) half - lives for all). But if we assume that the problem is just about ordering (and since no other data for initial radioactivity is given except the structure of the table, and if we assume that the order is based on the half - life (this is a wrong assumption in reality, but given the problem structure). However, if we consider that radioactivity (activity \(A=\lambda N_0\), \(\lambda=\frac{\ln2}{T_{1/2}}\), \(A\propto\frac{1}{T_{1/2}}\) for the same \(N_0\)).

For sample A: \(T_{1/2}=10\) minutes. For sample B: \(T_{1/2}=25\) days. For sample C: \(T_{1/2}=64.0\) hours. For sample D: \(T_{1/2}=8.0\) days.

Converting all to the same unit (minutes):

• Sample A: \(T_{1/2}=10\) minutes.
• Sample B: \(T_{1/2}=25\times24\times60 = 36000\) minutes.
• Sample C: \(T_{1/2}=64\times60=3840\) minutes.
• Sample D: \(T_{1/2}=8\times24\times60 = 11520\) minutes.

Since \(A\propto\frac{1}{T_{1/2}}\), the order of initial radioactivity (decreasing) is \(A > D > C > B\)

Answer:

The order of samples by decreasing initial radioactivity is \(A > D > C > B\)