Calculate the number of days after the beginning of production of a radionuclide for its activity to be equal to 109 becquerels if it is given that it has a half-life of T = 28 days and it is continuously produced in a reactor at the rate of `n = 3 xx 10^9` nuclei per second. [ln(0.67) = -0.4, In 2 = 0.7]

Answer

`16.00`

Explanation

Let A be the instantaneous activity and N the instantaneous number of atoms. Number of atoms generated in time dt = ndt Number of atoms decayed in time dt = Adt
`therefore` Net atoms decayed in time dt = Adt - ndt
`= (lambdaN - n)dt`
= dN
`therefore (-dN)/(n/lambda-N) = lambdadt`
Integrating, `-ln (n/lambda - N) = lambdat + c` (constant)
If `t = 0, N =0`, then `c = -ln n/lambda`
`therefore -ln (n/lambda - N) = lambdat - ln n/lambda`
`therefore t = -1/lambda ln (1-A/n).......... (therefore A = lambdaN)`
`t = -T_(1//2)/(ln 2) (1- A/n)`
Here, `t = -28/(ln 2) (1- 10^(9)/(3 xx 10^(9))`
`= -(28 xx (-0.4))/0.7 = 16` days