A radioactive sample at any instant has its disintegration rate `5000` disintegrations per minute After `5` minutes , the rate is `1250` disintegration per minute. Then , the decay constant (per minute)

Options

A. `0.4` ln `2`
B. `0.2` ln `2`
C. `0.1` ln `2`
D. `0.8` ln `2`

Answer

A

Explanation

Rate of disintegration at any instant is directly proportional to the number of undecayed nuclei at that instant, I.e., `-(dN)/(dt)= lambdaN`
Where `lambda=` decay constant
We are given that at `t=0`
`-(dN)/(dt)=lambdaN_(0)=5000 ("disintegration")/("minute")`
at `t=5` minute
`-(dN)/(dt)=lambdaN=1250 ("disintegration")`
`N=N_(0) e^(-lambda t)`
On multiplying both sides by `lambda` we get
`(lambda N)=(lambdaN_(0))e^(-lambdat) implies 1250=5000 e^(-lambdat)implies 1/4=e^(-lambdat)`
Taking logarithm on both sides we get
`ln1-ln4=e^(-lambda(5))implies 0-ln4= -5lambdaimplies lambda=1/5 ln 2^(2)`
`implies lambda=0.4 ln2`.