Radioactivity · Thermal Heating Rate

Build up radioactive nuclei and watch the heating rate approach steady state

A reactor produces nuclide X at constant rate α from t = 0. The number of X nuclei first builds up, then approaches equilibrium as production and decay balance. Since each decay releases usable energy E₀, the liquid heating rate follows the decay rate.

2026 Paper 2 Q2 · production-decay heating
Production is constant · decay rate grows as radioactive nuclei accumulate
Observation stage
Time scale 1.0×
Nuclide build-up
The reactor adds α nuclei per second, while existing nuclei decay at rate λN. The population approaches α/λ.
ExpressionN = (α/λ)(1 − e⁻λᵗ)
Meaningproduction balances decay
Production αconstant source from the reactor
Decay λNgrows as X nuclei accumulate
Heating ratedecay energy delivered to the liquid

Derivation path

01 / BALANCE

Production competes with decay

The number of radioactive nuclei changes by source minus decay.

dN/dt = α − λN
02 / BUILD-UP

Population approaches α/λ

Starting from zero, the solution is an exponential approach to equilibrium.

N(t) = (α/λ)(1 − e−λt)
03 / HEATING

Each decay heats the liquid

Power is decay rate times energy per decay, and ms converts heat rate into temperature rate.

dT/dt = (λN E₀)/(ms)
Rate of increase in temperature
dT/dt = (αE₀/ms)(1 − e−λt)
Correct option: (A)