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Math Help - Uranium Diffusion and Decay

  1. #1
    Junior Member
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    Uranium Diffusion and Decay

    Hi, I have the following problem and I am stuck in question 2. I would really appreciate it if you could check that the answer to question 1 is OK, and if you could also give me a hand with question 2.

    "A tank of volume 2 V, filled with water, is separated into two compartments A and B of identical volume by a semi-permeable membrane. Uranium is introduced into compartment A at a constant mass rate S. Uranium is exchanged between compartments A and B through the membrane at a rate given by p(C_A-C_B), where C_A and C_B are the concentrations of Uranium in each compartment, and p>0 is a constant. Because Uranium is radioactive, its mass in each compartment also decays, at a rate proportional to mass, with proportionality constant q>0.

    Let M_A(t) and M_B(t) be the masses of Uranium in compartments A and B. Assume that there is no Uranium in the tank at t = 0, that is, M_A(0) = M_B(0) = 0, and that the water in each compartment is well mixed.

    1.- Write down the differential equations for M_A(t) and M_B(t) describing their evolution.
    2.- Find the masses at equilibrium."

    So for the first one I have:

    dM_A(t)/dt=S-p(C_A-C_B)-qM_A
    dM_B(t)/dt=p(C_A-C_B)-qM_B

    Is it correct? What about the second question, how do I work that out? Thanks!
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  2. #2
    A Plied Mathematician
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    Note that

    C_{A}:=\dfrac{M_{A}(t)}{V}, and

    C_{B}:=\dfrac{M_{B}(t)}{V}.

    You need to plug this information into your DE's, which are correct as far as they go. I'm assuming the units compatibility is taken care of by the constants p and q.
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