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)`=3s
(i)`lambda=0.6931/T_(1//2)=0.6931/3=0.231 s^(-1)`
(ii)`T_"av"=1/lambda=1/0.231=4.33s`
(iii)N=1000
`t_1`=1
`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) = 3 xx 3 s=9s`
(iv)`("dN"/"dt")_(t=t_1)=lambdaN=0.231xx1000 =231 s^(-1)`