[RamSose]

There are two homework problems I need help with.  They are problems # 10 and 11 from Chapter 3 of Modern Nuclear Chemistry by Loveland, Morrissey, and Seaborg, Copyright 2005.

#10: "Consider a reactor in which the production rate of ²³⁹U via the ²³⁸U(n,gamma) ²³⁹U reaction is 10⁵ atoms/s.  Calculate the activity of ²³⁹Pu after an irradiation of (a) 1 day (b) 1 month, and (c) 1 year."

An earlier problem gives the following decay chain, which I assume is the correct one to use for this problem: ²³⁹U > ²³⁹Np > ²³⁹Pu > further decay products. 

The half-lives of these nuclides are as follows:
²³⁹U - 23.45 months
²³⁹Np - 2.356 days
²³⁹Pu - 24110 years

Next, I wrote out the pertinent differential equations:

dN_U/dt = 10⁵t - lamda_UN_U

dN_Np/dt = lamda_UN_U - lamda_NpN_Np

dN_Pu/dt = lamda_NpN_Np - lamda_PuN_Pu = lamda_NpN_Np  (I assumed here that since the half-life of ²³⁹Pu is very long compared to the half-life of the others and to the time scales given in the problem, the decay of Pu is negligible)

lamda = mean lifetime = ln2/t_1/2 ; N = number of given nuclide

This is as far as I got.  I tried working with the differential equations, but to no avail.  Am I on the right track?  Where do I go from here?  HELP


#11: "What is the probability of a ²²²Rn atom decaying in our lungs?  The atmospheric concentration of ²²²Rn may be assumed to be 1 pCi/L.  In an average breath we inhale 0.5L of air and exhale it 3.5s later."

The half-life of ²²²Rn is 3.8235 days.

From the information given in the problem, I can calculate such quantities as ²²²Rn atoms/breath (about 8800) and average decays/breath (about 0.065).  However, I can't figure out how to relate these quantities to a probability.  Any suggestions?

[toluene]

I think the half-life for U-239 is in minutes, not months. Anyway, I'm stuck on the same problem. It looks similar to the secular equilibrium discussed on pg. 72 - 73 of Loveland because uranium-239 is constantly being produced, but that would mean that A₁ = A₂ for all three times in the problem and that doesn't seem right.

[Mitch]

Take the probability for #11 to be: average decays/breath x 100%

[ethylenediamine]

Take the probability for #11 to be: average decays/breath x 100%

How would average decay be calculated? Should we integrate dN/dt = -lambda N for the time interval to find the change in number of radioactive nuclei? dN/N = -lamda dt .. ln(N/N₀) = -lambda t
N(t)  = N₀e^{-lambda t}

Then average decays would be N(t) - N₀

Divide that by the time interval?


For problem 1, do we need to account for the simulatenous decay and production of U-239? I ended up getting a long equation:

let lamda U = A, lamda NP = B, lamda Pu = C (all in day^-1)
N_Pu=(7.9 E -8 day^-1)(1/C - exp(-Ct)/C + Aexp(-Bt)/[(C - B)(B-A)] - Aexp(-Ct)/[(C-B)(B-A)] + B(exp(-Ct) - exp (-At))/[(C-A)(B-A)])

Don't know if I have the right equation, but if I do, I hope everyone else enjoys all the work involved in solving it...

EDIT: For problem 11, wouldn't finding the percent of nuclei decayed be more helpful for finding the probability? If 0.06 "nuclei" decay  of 8800 possible nuclei, wouldn't the resulting ratio be more helpful? I guess the 0.06 nuclei is just a conceptual number, since a non-integer amount of nuclei can't decay