Consider the decay of a radioactive element A to a daughter B and granddaughter

Question

Consider the decay of a radioactive element A to a daughter B and granddaughter C.
           t1/2(1)                      t1/2(2)
A ---------------------> B --------------------> C
Take the half-lives to be t1/2(1) = 2200 years and t1/2(2) = 1850 years.
(a) Show equation for the relative amounts of A, B and C such that you can see the production of C to at least 75% of the original amount of A.
(b) Use a little math to find the time when the amount of B is a maximum. You can use this to check that your plot is correct.

Explanation / Answer

part a

given [C]=75% 0f [A]

      or [C]=0.75 [A]...(1)

[x]=[xo](1/2)^t/T1/2

[x]= concentration

[xo]=initial conc

now using this equation for A,B,C

[B]/[A]=(1/2)^t/2200 years ...............(2)

[C]/[B]=(1/2)^t/1850 years from equation (1)

0.75 [Ao]/[B]=(1/2)^t/1850 years.........(3)

now equation (2)*(3) gives,

0.75=(1/2) ^(t/2200+t/1850)

log 0.75=log 0.5 * t(1/2200+1/1850)

solving t=416.99 years

So the reaction continued for t=416.99 years=417 years

so from equations 2 and 3

[B]/[A]=(1/2)^417/2200

[B]/[Ao]=(1/2)^0.19

log[B]/[Ao]= 0.3010*0.19....a

similarly

[C]/[B]=(1/2)^417/1850

log[C]/[B]= 0.3010* 0.23....b

hencea-b gives the relative amounts of A,B,C

part b

[B]/[Ao]=(1/2)^t/2200   (from above)

log [B]/[A]=(t/2200) * 0.3010

t=2200/0.3010 log [B]/[A]

t=7309 log [B]/[A]

but [B] is maximum when [A]=minimum

dA/dt=-K[A]=0

    k=0

so putting this k value in ,

[B]=[A] e^ -kt

[B]=[A] e^ 0

   [B]=[A] so

1=e^ -kt

                   -

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