Sometimes a radioactive nucleus decays into a nucleus which itself is radioactive . An example is
`""^(38)"Sulphur"underset(=2.48h)overset("half -life")(to) ""^(38)Cl underset(=0.62 h)overset("half-life")(to) ""^(38)Ar(" stable")`
Assume that we start with `1000 ""^(38)S` nuclei at time t=0. the number of `""^(38)Cl` of count zero at t=0 and will again be zero at `t= infty.`At what value of t, would the number of counts be a maximum?

Explanation

consider the chain of two decays
`.^(38)S underset(2.48n)(to)^(38)CI underset(0.62h)(to)^(38)Ar`
At time Let `.^(38)S` have `N,(t)` active nuclei and `.^(38)CI` have `N_(2)(t)` active nuclei
`(dN_(1))/(dt)= - lambda_(1)N_(1) = "rate of formation of "CI^(38)`
`"Also"" "(dN_(2))/(dt) =- lambda_(1)N_(2)+ lambda_(1)N_(1)`
`"But"" " N_(1) =N_(0)e^(-lambda1t)`
`(dN_(2))/(dt) =lambda_(1)N_(0)e^(-lambdat)-lambda_(2)N_(2)" ".......(i)`
Multiplying by `e^(lambda_(2^(t)))` dt and rearrange
`e^(lambda_(2^(t)))dN_(2)+lambda_(2)N_(2)e^(lambda_(2^(t)))dt=lambda_(1)N_(0)e^((lambda_(2)-lambda_(1))^(t)))dt`
Integratin both sides
`N_(2)e^(lambda_(2^(t)))=(N_(0)lambda_(1))/(lambda_(2)-lambda_(1)) e^((lambda_(2)-lambda_(1))^(t))+C`
`"Since"" ""at " t=0,N_(2) =0,C =-(N_(0)lambda_(1))/(lambda_(2)-lambda_(1))`
`:." "N_(2)e^(lambda_(2^(t)))=(N_(0)lambda_(1))/(lambda_(2) -lambda_(1)) "("e^((lambda_(2)-lambda_(1))^(t))-1")"" "......(ii)`
`N_(2) =(N_(0)lambda_(1))/(lambda_(2)-lambda_(1))(e^(-lambda_(1^(t)))-e^(-lambda_(2^(t))))`
For maximum count `(dN_(2))/(dt)=0`
`lambda_(1)N_(0)e^(-lambda_(1^(t)))-lambda_(2)N_(2)=0" ""[From Eq.(i)]"`
`rArr" "(N_(0))/(N_(2)) =(lambda_(2))/(lambda_(1))e^(lambda_(1^(t)))" ""[From Eq".(ii)]`
`e^(lambda_(2^(t)))-(lambda_(2))/(lambda_(1)).(lambda_(1))/((lambda_(1)-lambda_(1)))e^(lambda_(1^(t)))[e^((lambda_(2)-lambda_(1))^(t))-1]=0`
`"or "" " e^(lambda_(2^(t)))-(lamda_(2))/((lambda_(2)-lambda_(1)))e^(lamda_(2^(t)))+(lambda_(2))/((lambda_(2)-lambda_(1)))e^(lamda_(1^(t)))=0`
`1-(lambda_(2))/((lambda_(2)-lambda_(1)))+(lambda_(2))/((lambda_(2)-lambda_(1)))e^((lambda_(1)-lambda_(2))^(t))=0`
`(lambda_(2))/((lambda_(2)-lambda_(1))) e^((lamda_(1)-lambda_(2))^(t))=(lambda_(2))/((lambda_(2)-lambda_(1)))-1`
`e^((lambda_(1)-lambda_(2))^(t))=(lambda_(2))/(lambda_(1))`
` t=("log"_(e)(lambda_(1))/(lambda_(2)))//(lambda_(1)-lambda_(2))`
`=("log"_(e)((2.48)/(0.62)))/(2.48-0.62)`
`=("log"_(e)4)/(1.86) =(2.303 xx 2 xx 0.3010)/(1.86)" " (":."lambda =(0.693)/(T_(1//2)))`
`=0.745 s`
Note Do not apply directly the formula of radioactive .Apply formulae related to chain decay.