An isotope of postassium `K^(40)` undergoes two parallel type of decay, one by electron capture and other by `beta` decay with half lives as `1.3 xx 10^(9)` year and `(1.3)/(9) xx 10^(9)` year respectively. If in a sample of a mineral mass ratio of `K : Ar: Ca ` is 5,1.5 & 16 then calculate age of the mineral if it known that all `Ca^(40)` in the mineral is not from the radioactive decay of potassium.
[ Express your answer in terms of `10^(7)` year for eg. if you answer is `2 xx 10 ^(9)` years fill 0200 in OMR sheet]

Answer

[0026]

Explanation

Assumed that all Ar produced from K.
`{:("K","Ar","Ca",,),("t=0","a",0,0,),("t=t","a-x-y","x","y prpduced",):}`
`(x)/(y)=(lambda_(1))/(lambda_(2))=((t_(1//2))_(2))/((t_(1//2))_(1))=(1)/(9)`
`y=9x`
`x=(1.5)/(40) " " y=(9 xx1.5)/(40)`
`a-x-y=(5)/(40)`
`a=(5)/(40)+(1.5)/(40)+(9xx1.5)/(40)=(20)/(40)`
`lambda t=ln (a)/(a-x-y)`
`((ln2)/(1.3xx10^(9))+(ln2)/((1.3)/(9)xx10^(9)))t=ln.(20//40)/(5//40)`
`t=26xx10^(7)`