NBPhO 2025

Difficulty medium

Prerequisites

• 1D elastic collisions (energy + momentum conservation)
• Non-relativistic kinematics, $E_k = \tfrac{1}{2} m v^2$
• Ideal gas law, $pV = nRT$
• Equipartition theorem / Maxwell–Boltzmann distribution
• Boyle's law and Dalton's law

Learning objectives

• Justify the non-relativistic limit via the small parameter $E_k / m c^2$
• Apply energy + momentum conservation to derive the 1D elastic-collision identity $v_1 + v_1' = v_2 + v_2'$
• Identify the moderator mass that maximises neutron energy transfer ($M = m_n$) and explain why hydrogen is the gold-standard moderator
• Model repeated head-on elastic collisions as geometric decay of speed, $v_N = v_0\,\alpha^N$
• Distinguish the mode ($E = k_B T$) and mean ($E = \tfrac{3}{2} k_B T$) of a Maxwell–Boltzmann distribution
• Combine Boyle's law and Dalton's law to analyse a gas mixture at constant temperature

Watch out for

• Mode-vs-mean confusion in Maxwell–Boltzmann. The oft-quoted $0.025\ \text{eV}$ is $k_B T$ at room temperature (the most-probable energy), not $\tfrac{3}{2} k_B T$ (the average). Reading the problem literally requires $\tfrac{3}{2} k_B T$ and gives $T_f \approx 193\ \text{K}$, not $290\ \text{K}$.
• Forgetting the volume change in part (iv). The rod's free volume drops from $V_0$ to $V$ due to pellet swelling; the helium partial pressure in the final state must be computed with the new volume via Boyle's law before applying Dalton's law.
• Treating all elastic collisions as head-on. The problem explicitly asks for head-on collisions, but the realistic "logarithmic energy decrement" formula gives roughly twice as many collisions to thermalise, which is often quoted in reactor physics literature.
• Dropping the sign in $\alpha = (m_n - M)/(m_n + M)$. For $M > m_n$, $\alpha$ is negative; solving $v_N = v_0\,\alpha^N$ for $N$ requires taking $|\alpha|$ so that the logarithm is real.