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    Hello!

    I need help in calculating certain values in nuclear physics. Here are the tasks:

    1.) For the radioactive decay of the element polonium, it shall be deems

    ^{210}_{\ 84}\mathrm{Po} \to{}^{206}_{\ 82}\mathrm{Pb} + {}^{4}_{2}\mathrm{He}

    The half period of Polonium is T = 138.4 days. The mass of the polonium at the beginning of the decay is M = 0.3 mg. The relative masses m of polonium and helium are Po: 210 and He: 4,003

    What is the mass M of helium after two years?

    Before I calculate the mass M of helium, I have to calculate the mass M of polonium after two years first. I thought it would be possible to determine the mass M of poloium by the law of decay:

     N = N_{0}*e^-^\lambda^*^t where  \lambda = \dfrac {ln 2}{T} , t = 780 days (= two years) and N_{0} = 0.3 mg.

    And after calculating the mass M of polonium, the mass M of helium can be calculated by:

     \dfrac{M(Po)}{m(Po)} = \dfrac{M(He)}{m(He)}

     M(He) = \dfrac{M(Po)*m(He)}{m(Po)}

    Is this solution process right?

    2.) A radioactive preparation has the half period of 10 minutes. Determine the decay konstant  \lambda , that is to say the fraction of the decaying atoms per second in percent (the percent rate of decaying atoms in a second).

    I guess it has to be:

     \lambda = \dfrac {ln2}{T} , where T = 10 minutes = 600 seconds.

    Multiplied by 100, I get the percent rate. Is that right?
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    (Original post by Kallisto)
    Hello!

    I need help in calculating certain values in nuclear physics. Here are the tasks:

    1.) For the radioactive decay of the element polonium, it shall be deems

    ^{210}_{\ 84}\mathrm{Po} \to{}^{206}_{\ 82}\mathrm{Pb} + {}^{4}_{2}\mathrm{He}

    The half period of Polonium is T = 138.4 days. The mass of the polonium at the beginning of the decay is M = 0.3 mg. The relative masses m of polonium and helium are Po: 210 and He: 4,003

    What is the mass M of helium after two years?

    Before I calculate the mass M of helium, I have to calculate the mass M of polonium after two years first. I thought it would be possible to determine the mass M of poloium by the law of decay:

     N = N_{0}*e^-^\lambda^*^t where  \lambda = \dfrac {ln 2}{T} , t = 780 days (= two years) and N_{0} = 0.3 mg.

    And after calculating the mass M of polonium, the mass M of helium can be calculated by:

     \dfrac{M(Po)}{m(Po)} = \dfrac{M(He)}{m(He)}

     M(He) = \dfrac{M(Po)*m(He)}{m(Po)}

    Is this solution process right?

    2.) A radioactive preparation has the half period of 10 minutes. Determine the decay konstant  \lambda , that is to say the fraction of the decaying atoms per second in percent (the percent rate of decaying atoms in a second).

    I guess it has to be:

     \lambda = \dfrac {ln2}{T} , where T = 10 minutes = 600 seconds.

    Multiplied by 100, I get the percent rate. Is that right?


    1. what do the upper and lower case M's represent here? - the mass of helium has to increase as the mass of polonuim decreases and the lhs and rhs of your equation look like they're going to be going in opposite directions. Also if the original mass of helium is zero it looks like the rhs is going to be a divide by zero.

    fwiw I think they've provided the mass of helium and the mass of polonium because they don't want you to say "mass decrease of polonium" = "mass increase by helium",I think they want you to calculate the number of Po nuclei decaying over 2 years and multiply that by the mass of helium.

    btw 2 years is 365*2 = 730 days btw, not 780.
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    (Original post by Joinedup)
    1. what do the upper and lower case M's represent here? (...)
    Do you mean the M's in the equation above? 'M' is just the mass of the atom, while 'm' is the relative mass

    (Original post by Joinedup)
    (...)
    fwiw I think they've provided the mass of helium and the mass of polonium because they don't want you to say "mass decrease of polonium" = "mass increase by helium",I think they want you to calculate the number of Po nuclei decaying over 2 years and multiply that by the mass of helium. (...)
    Multiply the number of polonium nuclei over two years by the mass of helium? I know how to calculate the number of Polonium nuclei, But I wonder how to calculate the mass of helium by that. I'm sure I should calculate the mass of helium after a two year decay. But what can I do with the informations above to get it?
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    Oh right - have you subtracted the mass of polonium at t=2yrs from the mass of polonium at t=0 ?
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    (Original post by Joinedup)
    Oh right - have you subtracted the mass of polonium at t=2yrs from the mass of polonium at t=0 ?
    No, I have not. Do you mean I should subtract the mass of polonium at the beginning from the mass of polonium after two years? how can I calculate the mass of polonium after two years?

    And after I have done it, the mass of helium can be calculated by the equation above, so by the masses and relative masses of polonium and helium?
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    (Original post by Kallisto)
    No, I have not. Do you mean I should subtract the mass of polonium at the beginning from the mass of polonium after two years? how can I calculate the mass of polonium after two years?

    And after I have done it, the mass of helium can be calculated by the equation above, so by the masses and relative masses of polonium and helium?
    I thought you'd got it in your first post

     N = N_{0}*e^-^\lambda^*^t where  \lambda = \dfrac {ln 2}{T} , t = 780 days (= two years) and N_{0} = 0.3 mg.

    the N on the left hand side is Nt - the number of un-decayed nuclei remaining after time t.
    of course the number of nuclei is proportional to the mass of atoms, so feeding in the mass of Polonium at t0 and the correct decay constant and elapsed time would give you the mass of remaining Polonium after 2 years.

    the mass of polonium which has decayed = mass of polonum at start minus the mass of polonum left at time t
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    (Original post by Joinedup)
    x
    If I put the values above in the law of decay, I get N = 0,061 mg (rounded) as decaying mass of polonium after two years. Substract from the mass at the beginning, I get:

    0,3 mg - 0,061 mg = 0,239 mg (rounded) as mass of polonium after two years. Is this calculation right?
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    (Original post by Kallisto)
    If I put the values above in the law of decay, I get N = 0,061 mg (rounded) as decaying mass of polonium after two years. Substract from the mass at the beginning, I get:

    0,3 mg - 0,061 mg = 0,239 mg (rounded) as mass of polonium after two years. Is this calculation right?
    I don't get that - can you show your working?

    the decay equation gives you the amount remaining at time t based on the amount you had at starting time 0. we're interested in the amount that has decayed (i.e. isn't remaining) at time t because 1 Po decay produces 1 He atom

    we can multiply the mass reduction of the polonium by 4.003/210 to get the mass of the helium produced.
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    (Original post by Joinedup)
    I don't get that - can you show your working? (...)
    As you will!

    I have calculated the mass after the law of decay:  N = N_{0}*e^-^\lambda^*^t , where:

     \lambda = \dfrac{ln2}{T}

     T = 138,4

     t = 730 (= 2 years)

     N_{0} = 0.3 mg

    Put in the law of decay, I get:

    N = 0,061 mg (rounded)

    Subtracted to the mass of polonium at the beginning of the decay, I get: 0,3 mg - 0,061 mg = 0,239 mg

    Why is this result wrong? Do I have this right that the mass of helium can be calculatred by the relative masses and the mass by that:

     M(He) = \dfrac{M(Po)*m(He)}{m(Po)}

    If that is the case, the mass of Helium is:

     M(He) = \dfrac{0,239*4,003}{210} = 0,0046 mg (rounded)
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    (Original post by Kallisto)
    As you will!

    I have calculated the mass after the law of decay:  N = N_{0}*e^-^\lambda^*^t , where:

     \lambda = \dfrac{ln2}{T}

     T = 138,4

     t = 730 (= 2 years)

     N_{0} = 0.3 mg

    Put in the law of decay, I get:

    N = 0,061 mg (rounded)

    Subtracted to the mass of polonium at the beginning of the decay, I get: 0,3 mg - 0,061 mg = 0,239 mg

    Why is this result wrong? Do I have this right that the mass of helium can be calculatred by the relative masses and the mass by that:

     M(He) = \dfrac{M(Po)*m(He)}{m(Po)}

    If that is the case, the mass of Helium is:

     M(He) = \dfrac{0,239*4,003}{210} = 0,0046 mg (rounded)

    so
    T=138.4 days


    λ=ln(2)/T
    =0.6931/138.4
    0.005008

    t=730 days


    λt=0.005008x730
    =3.656

    e^-3.656 = 0.02584



    0.3x0.02584
    =0.007751 (mg)

    QUICK CHECK

    730/138.4=5.27 (number of half periods)

    so we'd expect the amount in mg to be somewhere between
    0.3/(2^5) = 0.00937 and 0.3/(2^6) = 0.00468
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    (Original post by Joinedup)
    (...)

    QUICK CHECK

    730/138.4=5.27 (number of half periods)

    so we'd expect the amount in mg to be somewhere between
    0.3/(2^5) = 0.00937 and 0.3/(2^6) = 0.00468

    That makes sense, as there are 5 up to 6 half periods in two years.



    (Original post by Joinedup)
    (...)
    0.3x0.02584
    =0.007751 (mg)
    Why do you multiply the mass 0.3 mg with 0.2584 mg instead of to substract?

    If I put your calculated value in the equation  M(He) = \dfrac{M(Po)*m(He)}{m(Po)} I get:

     M(He) = \dfrac{0.007751*4.003}{210} = 0.00015 mg (rounded)

    Your result is not between 0.00937 and 0.00468, in contrast to mine, as I got 0.0046. So I suppose that your calculation must be wrong.
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    (Original post by Kallisto)
    That makes sense, as there are 5 up to 6 half periods in two years.





    Why do you multiply the mass 0.3 mg with 0.2584 mg instead of to substract?

    If I put your calculated value in the equation  M(He) = \dfrac{M(Po)*m(He)}{m(Po)} I get:

     M(He) = \dfrac{0.007751*4.003}{210} = 0.00015 mg (rounded)

    Your result is not between 0.00937 and 0.00468, in contrast to mine, as I got 0.0046. So I suppose that your calculation must be wrong.
    my long calculation and the quick check are just for the mass of Po that decayed between t0 and t=730 days

    because every 1 atom of Po that decays produces 1 atom of He we can then take that number and work out the mass of He produced using the ratio of masses.


    I multiply 0.3mg by 0.02584 (not 0.2584) because e to the power of -λt is 0.02548 and the decay law is telling us to multiply the original quantity by e to the power of -λt to get the quantity remaining at time t
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    (Original post by Joinedup)
    (...)
    I multiply 0.3mg by 0.02584 (not 0.2584) because e to the power of -λt is 0.02548 and the decay law is telling us to multiply the original quantity by e to the power of -λt to get the quantity remaining at time t
    Of course, I see!

    I have had a closer look to your calculation and now I have no doubts. Everything is well. So is the mass of helium M = 0.00015 mg, which I have calculated above, right?
 
 
 
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