A nuclear reactor starts producing a radioactive nuclide X from t = 0, at a constant rate of α 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 λ as the decay constant of X, the rate of increase in the temperature of the liquid is:
Options

Question

A nuclear reactor starts producing a radioactive nuclide X from t = 0, at a constant rate of α 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 λ as the decay constant of X, the rate of increase in the temperature of the liquid is:

Options

G1 Physics · Accepted Answer

If N is the number of radioactive nuclei, dN/dt = α − λN with N(0)=0. Hence N = (α/λ)(1 − e⁻λᵗ). The decay rate is λN = α(1 − e⁻λᵗ). Therefore the heating rate is ms(dT/dt) = E₀λN, giving dT/dt = αE₀(ms)⁻¹(1 − e⁻λᵗ).