Assume that two radioactive substances A and B disintegrate as:
`(-d[A])/(dt) = k_(A), t_(1//2)(A) = 0.693/(K_(A))`
`(-d[B])/(dt) = k_(B)[B]^(2), t_(1//2)(B) =1/(k_(B)[B]_(0)]`
If both the half life periods and initial concentrations are equal, what will be the ratio of the rates of the two substances at the start of the reaction?

Explanation

According to available data, `t_(1//2)(A) = t_(1//2)(B)`
`0.693/k_(A) = 1/(k_(B)[B]_(0))` or `k_(A)/k_(B)= 0.693[B]_(0)= 0.693[A]_(0)`
At the start of the reaction
`r_(A) = k_(A)[A]_(0)`, `r_(B) = k_(B)[B]_(0)^(2) = k_(B)[A]_(0)^(2)` ……......`[therefore[A]_(0) = [B]_(0)]`
`therefore r_(A)/r_(B) = (k_(A)[A]_(0))/(k_(B)[A]_(0)^(2))` or `r_(A)/r_(B) = 0.693 [A]_(0) xx ([A]_(0))/([A]_(0^(2))) = 0.693`