A nuclear reactor starts producing a radioactive nuclide X from t = 0 , at a constant rate of per second. Each decay of X produces energy E₀ , which is utilized to heat a liquid of mass m and specific heat s . Assuming no heat loss from the liquid and taking as the decay constant of X , the rate of increase in the temperature of the liquid is:

Step-by-step solution

Let N be the number of active nuclei at time t . The rate of change of the number of nuclei is given by dN dt = - N Integrating with the initial condition N = 0 at t = 0 : ₀^ N dN - N = ₀^ t dt - 1 ( - N ) = t N = (1 - e^ - t ) The rate of decay of the nuclide at time t is A = N = (1 - e^ - t ) Since each decay produces energy E₀ , the rate of energy production is dE dt = A E₀ = E₀ (1 - e^ - t ) This energy is used to heat the liquid. The rate of heat absorption is dQ dt = m s dT dt Equating the rate of energy prod