Question

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 1

Correct Answer - A::B::D
Let N be the number of radio nuclei at any time t. Then, net rate of formation of nuclei at time t is
`(dN)/(dt)=alpha-lambdaN`
or `int_0^N(dN)/(alpha-lambdaN)=int_0^tdt`
or `N=(alpha)/(lambda)(1-e^(-lambdat))`

Rate of formation `=alpha` Rate of decay `=lambdaN`
Number of nuclei formed in time `t=alphat`
and number of nuclei left after time
`t=alpha/lambda(1-e^(-lambdat))`
Therefore, number of nuclei disintegrated in time
`t=alphat-alpha/lambda(1-e^(-lambdat))`
`:.` Energy released till time,
`t=E_0[alphat-alpha/lambda(1-e^(-lambdat))]`
But only 20% of it is used in raising the temperature of water.
So, `0.2E_0[alphat-alpha/lambda(1-e^(-lambdat)]=Q`
where, `Q=msDeltatheta`
`:.` `Deltatheta=` increase in temperature of water `=(Q)/(ms)`
`:.` `Deltatheta=(0.2E_0[alphat-alpha/lambda(1-e^(-lambdat))])/(ms)`