Atoms of X decay to atoms of Y with a half-life of 500,000 y. If there were 50,0

Question

Atoms of X decay to atoms of Y with a half-life of 500,000 y. If there were 50,000 atoms of X present at the start and 0 atoms of Y, how long will it take for there to be 3,125 atoms of X?

A fossil that formed with 10,000 atoms of a radioactive parent element now contains only 1,250 atoms of that element. If the half-life of the element was 1,000,000 y, how old is the fossil?

The age of a rock was was estimated using 238U radiometric dating. The ratio of 238U to 206Pb (238U decays to 206Pb) was approximately equal to 1. If the half-life of 238U is 4.5 billion y, and if it is assumed there was no 206Pb initially, how old is the rock sample?

Answer vary it could be 2 million yrs, 3 million yrs,2.25 billion yrs, 3 billion yrs, 4.5 billio
n yrs or 5 million yrs

Explanation / Answer

decay is a first order reaction.
if we have initiallay N atoms of X then after time t no. of atoms of X left would be N*2^(-t/T) where T is the half life of X.
in the first part N=50,000 and left are 3125 and T = 500,000 y = 5*10^6 y

3125=50000*2^(-t/T)

==> 2^(-t/T) = 3125/50000

                    = 1/16

                     = 2^-4

==> t/T = 4

t=4*5*10^6= 2*10^7 y

time taken = 2*10^7 y

in the second part of the question,

N= 10,000 and let it be formed t time before. now the amount present is 1250 and T = 10^6 y

1250 = 10000*2^(-t/T)

2^(-t/T)= 1/8

             = 2^-3

==> t/T = 3

      t= 3T = 3*10^6 y

the age of the fossil is 3*10^6 y = 3 million y

now coming to the last part of the question,

given that after time t of the formation ratio of U and Pb is 1 and T= 4.5 billion years

let initial no. of atoms of U be N.

then after time t remaining are N*2^(-t/T)

as 1 atom of U gives 1 atom of Pb, no. of atoms of Pb formed are no. of atoms of U disintegrated,

no. of atoms of Pb = N- N*2^(-t/T)

                              = N*(1-2^(-t/T))

now ratio = 1

N*2^(-t/T)/N*(1-2^(-t/T)) = 1

2*(-t/T) = 1- 2*(-t/T)

2*2*(-t/T)=1

2*(-t/T)= 2^-1

t=T

     =4.5 billion years.

hence the rock is 4.5 billion years old

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