`""^(227)Ac` has a half-life of 22 years with respect to radioactive decay. The decay follows two prallel paths : `""^(227)Ac to ""^(227)Th " and "^(227)Ac to ""^(223)Fr.` If the percentage of the two daughter nuclides are 2.0 and 98.0, respectively, the decay constant (in `"year"^(-1)`) for `""^(227)Ac to ""^(227)Th` path is closest to

Explanation

`t""_(1//2)" of "Ac""^(227)=21.8yr.`
`lambda=" Disintegration constant "=(0.693)/(t""_(1//2)`

`lambda=(0.693)/(21.8)yr""^(-1)`
`"Let "lambda""_(1)and lambda""_(2)" are decay constant for " Th""^(227) and`
`Fe""^(223) " respectively"`
`lambda=lambda""_(1)(Th)+lambda""_(2)(Fr)`
`lambda""_(1)=%of" "Th xx lambda=(2)/(100)xx(0.693)/(21.8)`
`=(0.02xx0.693)/(21.8)yr""^(-1)`
`lambda""_(2)=%"of " Fr xx lambda=(98)/(100)xx(0.693)/(21.8)=0.98xx(0.693)/(21.8)`
`lambda=lambda""_(1)+lamda""_(2)," No = Initial conc.`
`"The rate of decay of Ace to Th"=[(dN)/(dt)]""_(Th)=lamda""_(1)xxN""_(0)`
`"The rate of decay of Ace to Fr"=[(dN)/(dt)]""_(Fr)=lamda""_(2)xxN""_(0)`
`"Since" (lamda""_(1))/(lamda""_(2))=(2)/(98)=(1)/(49)`
`lamda=lamda""_(1)+lamda""_(2),lamda""_(2)=49lamda""_(1)`
`(0.693)/(21.8)=lamda""_(1)+49lamda""_(1)=50lamda""_(1)`
`lamda""_(1)=(0.693)/(21.8xx50)=6.3xx10""^(-4)year""^(-1)`
`lamda""_(2)=49lamda""_(1)=49xx6.3xx10""^(-4)`
`=3.087xx10""^(-2)year""^(-1)`