Answer to Question #4113 Submitted to "Ask the Experts"

I am learning about radiation and have come to a disagreement with a fellow student. The question is "what mass, in grams, of pure phosphorus-32 has an activity of 100 mCi?" My answer is 5.0 x 10^-6 grams; hers is 3.5 x 10^-7 grams. Who is correct and how do you calculate the answer?

I hope you didn’t have too large a wager riding on this one, Kevin. Your fellow student has it right. The solution is as follows:

By definition, the activity A = λN, where λ is the radioactive decay constant for the radionuclide of interest and is expressed in terms of the half-life, T_1/2, by λ = (ln 2)/T_1/2 = 0.693/T_1/2, and N is the number of radioactive atoms of the radionuclide. The activity must have the same time dimension as does λ.

For your problem: A = (100 mCi)(3.7 x 10⁷ atoms/s/mCi) = 3.7 x 10⁹ atoms/s; this represents the rate at which ³²P atoms are decaying. For ³²P, with a 14.3 day half-life, λ = 0.693/((14.3day)(24 hr/day)(3600 s/hr)) = 5.61 x 10^-7/s.

This represents the instantaneous fraction of ³²P atoms decaying per unit time. We can rearrange the equation for A to obtain N = A/λ, and using the values here we obtain N = (3.7 x 10⁹ atoms/s)/5.61 x 10^-7/s = 6.60 x 10¹⁵ atoms. We are now left with simply having to convert this number of atoms to a mass, m, by noting that one gram-atomic-weight of ³²P (or of any other atomic species) contains Avogadro’s number of atoms, namely, 6.023 x 10²³. Thus, m = (6.60 x 10¹⁵ atoms/6.023 x 10²³ atoms/g-atomic-weight)(32 grams/g-atomic-weight) = 3.51 x 10^-7 grams. I notice that your answer, Kevin, is greater than your fellow student’s by a factor of 14.3, the numerical value of the ³²P half-life in days. Did you somehow divide by the 14.3 days twice when you calculated λ?

I’m pleased to see that your class is learning about radiation and more pleased to see that you had the interest and volition to seek help with an answer to your question. Good luck to you and your fellow students in your future studies.

George Chabot, PhD, CHP