# differential equation class radioactive decay, assignment help

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In the radioactive decay series of uranium 238U, an isotope of lead β-decays with a half-life of 26.8 92

minutes into bismuth, which in turn β-decays with a half-life 19.9 minutes into polonium:

Its is my first time see this kind of chemistry math question, could someone solve this for me? I ask all my friends no one can do this

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Math 2A Nuclear Decay Morriss __________________________________________________________________________________ In the radioactive decay series of uranium 238 92 U , an isotope of lead β-decays with a half-life of 26.8 minutes into bismuth, which in turn β-decays with a half-life 19.9 minutes into polonium: 214 82 Pb → 214 83 Bi → 214 84 Po Each isotope's β-decay rate (rate of change of mass with respect to time) is proportional to the amount of isotope present. Consider a given 100-mg sample of pure 214 82 Pb . __________________________________________________________________________________ a) Discuss how you expect the mass of lead present to change over time. Do the same for bismuth. In particular, do you expect the masses of lead and bismuth to ever be equal? b) Set up a mathematical model for the mass of lead in the sample. Identify independent and dependent variables, and specify units. c) Compose an initial value problem describing the mass of lead in the sample as a function of time. d) Solve the IVP to express the amount of 214 82 Pb present in the sample as a function of time. e) Set up a mathematical model for the mass of bismuth in the sample. Identify independent and dependent variables, and specify units. f) Compose another IVP describing the mass of bismuth in the sample as a function of time. g) Solve the second IVP to express the mass of 214 83 Bi present in the sample as a function of time. h) On one set of axes, plot the amounts of each isotope present for an appropriate time period. i) Discuss your graphical results in light of your expectation from part a). __________________________________________________________________________________ Work it out, write it up. You may consult with others, but prepare your own paper. Papers are due Tuesday, October 11 at 7:30AM
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(a) Discuss how you expect the mass of lead present to change over time. Do the same
for bismuth. In particular, do you expect the masses of lead and bismuth to ever be
equal?
The masses of lead and bismuth are expected to reduce with time. Since their half-lives
are different, it will not be possible for the masses to ever be equal. Only masses of
isotopes with same half-lives tend to be equal during their decay period.
(b) Set up a mathematical model for the mass of lead in the sample. Identify
independent and dependent variables, and specify units.
1

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Generally decay is given by; 𝐴 = 𝐴0 . 2ℎ A i...

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