The rate of decay of an isotope is proportional to the amoun

The rate of decay of an isotope is proportional to the amount of isotope present. If the half-life of Plutonium-239 is 24,110 years, and you start with 100 grams, how long will it take until the sample decays to one gram of Plutonium-239? Use the necessary integrations. First find the value of C, then knowing time, find the value of K, then fully answer the question.

Solution

-d[A]dt = k[A]

-d[A]/[A]= kt

On integrating,

Integral of -d[A]/[A] = Integral of kt

-ln[A]t = kt + C ------------- Eq (1)

At time t= 0,

C = 0+ -ln[A]o

C = -ln[A]o

Hence,

-log[A]t = kt + C ------------- Eq (1)

-log[A]t = kt - ln[A]o

kt = ln {[A]o / [A]t}

k =(1/t)  ln {[A]o / [A]t} --------Eq (2)

Calculating value of k from half life:

t = t1/2

[A]t = [Ao/2

k =(1/t1/2)  ln {[A]o / [Ao]/2}

= (1/t1/2) In 2

k = In 2/ t_1/2

Given that half life t1/2 = 24110 yrs

Then,

k = In 2/ t_1/2 = In 2/ 24110 yrs = 2.875 x 10^-5 yr^-1

calculating time required to decay sample from 100 g to 1 g :

Hence,

k = 2.875 x 10^-5 yr^-1

   [Ao] = 100 g

[At] = 1g

t = ?

Substitute these values in Eq (2),

k =(1/t)  ln {[A]o / [A]t} -------- Eq (2)

2.875 x 10^-5 yr^-1 = (1/t) In ( 100 g/ 1 g)

   2.875 x 10^-5 yr^-1 = (1/t) In ( 100)

t = In (100) / [ 2.875 x 10^-5 yr^-1 ]

= 160180 yrs

t = 160180 yrs

Therefore,

time required to decay sample from 100 g to 1 g = 160180 yrs