Question

a medical technician is working with the four samples of radionuclides listed in the table below. initially, each sample contains 5.00 μmol of the radionuclide. first, order the samples by decreasing initial radioactivity. then calculate how long it will take for the amount of radionuclide in each sample to decrease to 1/32 of the initial amount. table with columns: sample, radionuclide (symbol, half - life), initial radioactivity, time for amount of radionuclide to decrease to 1/32 of initial amount; rows: a (⁶⁷₃₁ga, 78.0 hours, select, blank hours), b (¹⁴⁵₆₂sm, 340.0 days, select, blank days), c (²¹²₈₃bi, 1. hour, select, blank hours), d (¹⁵²₆₃eu, 13. years, select, blank years)

Explanation:

Step1: Relationship between half - life and initial radioactivity

The shorter the half - life of a radionuclide, the higher its initial radioactivity.
Half - life of sample A (\(^{67}_{31}Ga\)): \(t_{1/2,A}=78.0\) hours.
Half - life of sample B (\(^{145}_{62}Sm\)): \(t_{1/2,B}=340.0\) days.
Half - life of sample C (\(^{212}_{83}Bi\)): \(t_{1/2,C}=1\) hour.
Half - life of sample D (\(^{152}_{63}Eu\)): \(t_{1/2,D}=13\) years.
Since \(1\) hour\(<78.0\) hours\(<340.0\) days\(<13\) years, the order of initial radioactivity (decreasing) is \(C > A> B > D\).

Step2: Formula for radioactive decay

The formula for radioactive decay 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.
We want \(N=\frac{1}{32}N_0\). So, \(\frac{1}{32}=(\frac{1}{2})^n\).
Since \(\frac{1}{32}=\frac{1}{2^5}\), then \(n = 5\).

Step3: Calculate the time for each sample

For sample A: \(t=n\times t_{1/2}\). Given \(n = 5\) and \(t_{1/2,A}=78.0\) hours, \(t_A=5\times78.0\) hours \( = 390\) hours.
For sample B: Given \(n = 5\) and \(t_{1/2,B}=340.0\) days, \(t_B=5\times340.0\) days \(=1700\) days.
For sample C: Given \(n = 5\) and \(t_{1/2,C}=1\) hour, \(t_C=5\times1\) hour \(=5\) hours.
For sample D: Given \(n = 5\) and \(t_{1/2,D}=13\) years, \(t_D=5\times13\) years \(=65\) years.

Answer:

• Order of initial radioactivity (decreasing): \(C > A> B > D\)
• Time for sample A: \(390\) hours
• Time for sample B: \(1700\) days
• Time for sample C: \(5\) hours
• Time for sample D: \(65\) years