An atom of n radioactive substance typically decays into an atom of some other r

Question

An atom of n radioactive substance typically decays into an atom of some other radioactive substance with its own decay constant. Suppose we have 50 grams of pure Uranium-238 with a decay constant m. Each atom of Uranium-238 decays to a single atom of Thorium-234 which has a decay constant n. We have no Thorium-234 initially. Write down the differential equations and initial conditions for the radioactive decay of Uranium 238 (U(t)) and production and decay of Thorium-234 (T(t)). Solve the differential equations and initial conditions to find the amount of Thorium-234 after time t in terms of m and n. You should find that there are two cases that must be treated separately, m notequalto n and m = n. (Optional - mostly to think about later on your own) Notice that the solution in the case of n = m (T_n = m(t)) does not have the same form as the solution in the case of n notequlato m (T_n notequalto m (t)). Show that as m approaches n, the function T_n notequalto m(t) approaches T_n = m (t).

Explanation / Answer

The number of atoms decaying at any given time is proportional to the number of radiactive atoms present

dU(t)/dt = -nU(t)

n is the U-238 decay constant and U(t) is the number of U-238 atoms present at time t. The -ve sign is due to thefact that the rate of chage is decresing with time.

solving the above DE we get

ln(U(t)) = -nt + c

U(t) = exp(-nt + c)

let U(0) be the inital number of U-238 atoms at t=0 to start with the

U(0) = exp(c)

U(t) = U(0) exp(-nt)

U-238 decays to Th-234

The rate at which U-238 decays is the rate of production of Th-234

i.e.    nU(t)

out the above Th-234 decays with its own decay constant m

Th-234 atoms created per unit time is given by

dT(t)/dt = nU(t) - mT(t)

        = nU(0) exp(-nt) - mT(t)    - replace U(t) with the previous equation

dT(t)/dt +mT(t) = nU(0) exp(-nt)

multiply both sides with exp(mt)

exp(mt)dT(t) + mexp(mt) T(t) = nU(0) exp(-nt+mt) dt

d{exp(+mt) T(t)} = nU(0) exp(-nt+mt)dt   -----------(1)

integrating both side from 0 to t we get

exp(+mt)T(t) - T(0) = nU(0)/(m-n) {exp(-nt+mt) -1}

T(0) = 0 as there are no Th-234 atoms to start with

T(t) = nU(0)/(m-n) {e-nt - e-mt } , where m not.eq n

in the limit where m approaches n (1) reduces to

d{exp(+mt) T(t)} = nU(0) dt

integrating both sides we have

T(t) = nU(0) t exp(-mt)

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