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Thread: Uranium Diffusion and Decay

  1. #1
    Junior Member
    Sep 2010

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

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

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

    So for the first one I have:

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

    Is it correct? What about the second question, how do I work that out? Thanks!
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  2. #2
    A Plied Mathematician
    Jun 2010
    CT, USA
    Note that

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

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

    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 $\displaystyle p$ and $\displaystyle q.$
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