A radioactive element A says to B I am half of what you were when you ar one-fourth of what I was Moreover I was 1.414 times that what you were. If the half-life of A is 8 days. What is the half-life of B ?

Options

A. 2 days
B. 4 days
C. 6 days
D. 8 days

Answer

D

Explanation

Let the initial number of nuclide of x be `x_(0)`, the initial number of nuclide of Y be `y_(0)`,
the number of nuclide of X after time t be `x_(t)` and the number of nuclide of Y after time t be `Y_(t)`
`therefore x_(t) = 1//2y_(0), y_(t) = 1//4x_(0) " "x_(0) = 1.414 y_(0)`
Now, `x_(t) = x_(0)e^(-lambda_(1)t) = 1//2y_(0) " ".....(1) " "y_(t) = y_(0)e^(-lambda_(2)t) = 1//4 x_(0) " "....(2)`
Dividing equation (1) by (2)
`(2y_(0))/(x_(0)) = x_(0)e ((lambda_(2) - lambda_(1))t)/(y_(0)) rArr 2 = (x_(0)^(2))/(y_(0)^(2)) e^((lambda_(2)-lambda_(1))t) " since" x_(0) = sqrt(2) y_(0) rArr x_(0)^(2)`
`therefore 2 = 2e^(lambda_(2) - lambda_(1))t " "therefore e^((lambda_(2)-lambda_(1))t = 1` . Since t is not zero, hence `lambda_(2) = lambda_(1)` Therefore, the half-life of the other radioactive element is also 8 days.