A `""^118Cd` radio nuclide goes through the transformation chain.
`""^118Cd underset(30min) rarr^(118)In underset(45min)rarr^(118)Sn ("stable")`
The half-lives are written below the respective arrows. At time `t=0` only Cd was present. Find the fraction of nuclei transformed into stable over `60` minutes.

Answer

A, C

Explanation

At time `t=t`, `N_1=N_0e^(-lambda_1t)` and `N_2=(N_0lambda_1)/(lambda_2-lambda_1)(e^(-lambda_1t)-e^(-lambda_2t))`
`:.` `N_3=N_0-N_1-N_2`
`=N_0[1-e^(-lambda_1t)-(lambda_1)/(lambda_2-lambda_1)(e^(-lambda_1t)-e^(-lambda_2t))]`
`:.` `N_3/N_0=1-e^(-lambda_1t)-(lambda_1)/(lambda_2-lambda_1)(e^(-lambda_1t)-e^(-lambda_2t))`
`lambda_1=0.693/30=0.0231 mi n^(-1)`
`lambda_2=0.693/45=0.0154mi n^-1`
and `t=60min`
`:.` `N_3/N_0=1-e^(-0.0231xx60)-(0.0231)/(0.0154-0.0231)(e^(-0.0231xx60)-e^(-0.0154xx60))`
`=1-0.25+3(0.25-0.4)`
`=0.31`