A radioactive isotope X has a half life of 3 seconds. At t=0, a given sample of this isotope contains 8000 atom. Calculate (i) its decay constant (ii) average life (iii) the time `t_1`, when 1000 atoms of the isotope X remain in the sample (iv) number of decay/sec in the sample at `t=t_1sec.`

Explanation

`N_0 =8000`
`T_(1//2) =3 s`
(i) `lambda=(0.6931)/(T(1//2))=(0.69931)/(3)=0.231 s^(-1)`
(ii) `T_(av) =1/lambda=(1/(0.231))=4.33 s`
(iii) `N=1000`
`N/(N_0)=(1/2)^n`
`((1000)/(8000))=(1/2)^n`
`rArr (1/2)^3=(1/2)^n`
`rArr n=3`
`t_1=nT_(1//2)=3xx3 s=9s `
(iv) `((dN)/(dt))_(t=t_1)=lambdaN=0.231 xx1000 =231 s^(-1)`