A radio nuclide with disintegration constant `lambda` is produced in a reactor at a constant rate `alpha` nuclei per second. During each decay energy `E_0` is released. `20%` of this energy is utilized in increasing the temperature of water. Find the increase in temperature of `m` mass of water in time `t`. Specific heat of water is `s`. Assume that there is no loss of energy through water surface.

Answer

`DeltaT = (0.2E_(0)[alphat-(alpha)/(lambda)(1-e^(-lambdat))])/(mS)`

Explanation

Production of radioactive nudel `=alpha//sec`
Disintegration `=lambdaN_(A)`
At any time `t=(dN_(A))/(dt)=alpha-lambdaN_(A)`
`implies underset(0)overset(N_(A))(int) (dN_(A))/(alpha-lambdaN_(A))=underset(0)overset(t)(int) dt implies N_(A)=alpha/lambda (1-e^(-lambdat))`
Nuclei disinterated `=alphat-N_(A)=alpha[t-1/lambda(1-e^(-lambdat))]`
Total energy produced `=E_(0)[alphat-alpha/lambda (1-e^(-lambdat))]`
Energy used in water heating
`=0.2E_(0)[alphat-alpha/lambda(1-e^(-lambdat))]=msDeltaT`
`DeltaT=(0.2 E_(0)[alphat-alpha//lambda(1-e^(-lambdat))])/(mS)`