Number of nuclei of a radioactive substance at time t=0 are 2000 and 1800 at time t=2 s. number of nuclei left after t=6s is

Options

A. 800
B. 810
C. 790
D. 700

Answer

B

Explanation

From Rutherford-Soddy law, the number of atoms left after n half-lives is given by
`N=N_(0)((1)/(2))^(n)`
where, `N_(0)` is original the number of atoms.
The number of half-life
`n=("Time of decay")/("Effective half-life")`
Relation between effective disintegration constant `(lamda)` and half-life (T) is
`lamda=(ln2)/(T)`
`therefore lamda_(1)+lamda_(2)=(ln2)/(T_(1))+(ln2)/(T_(2))`
Effective half-life,
`(1)/(T)=(1)/(T_(1))+(1)/(T_(2))=(1)/(1620)+(1)/(810)`
`(1)/(T)=(1+2)/(1620) implies T=540` yr
`therefore n=(t)/(540)`
`therefore N=N_(0)((1)/(2))^(t//540)`
`implies(N)/(N_(0))=((1)/(2))^(2)=((1)/(2))^(t//540) implies (t)/(540)=2`
`implies t=2xx540=1080yr`