Speculative physics: protons with double mass

Problem

Speculative physics: protons with double mass. Imagine that the proton’s mass suddenly doubled. What would change for: (a) the inertia of atoms, (b) the thermal speed of a gas, (c) hydrogen’s Bohr orbit (which depends on the reduced mass $\mu = m_e m_p/(m_e+m_p)$)? Would chemistry be turned upside down?

Solution

(a) Inertia of atoms. Almost all of an atom’s mass sits in the nucleus (protons and neutrons weigh $\sim 1836$ times the electron). Doubling $m_p$, the mass of every nucleus — and hence of every atom — roughly doubles. Atoms become more sluggish: for a given force they accelerate less.

(b) Thermal speed. The average kinetic energy of thermal agitation depends only on temperature: $\tfrac{1}{2}m\langle v^2\rangle = \tfrac{3}{2}k_B T$. Hence $v_\text{rms} = \sqrt{3k_B T/m}$. If the mass doubles at fixed $T$, the thermal speed drops by a factor of $\sqrt{2}$: $\ev{v_\text{rms} \to \frac{v_\text{rms}}{\sqrt{2}}}.$ Gases become slower: diffusion, the speed of sound and the molecular escape speed all fall by $\sqrt{2}$.

(c) Bohr orbit. The radius and energy levels of hydrogen depend on the reduced mass $\mu = m_e m_p/(m_e + m_p)$. Since $m_p \gg m_e$, we already have $\mu \approx m_e$, and doubling $m_p$ changes it only by a tiny amount: $\mu = \frac{m_e}{1 + m_e/m_p} \approx m_e\left(1 - \frac{m_e}{m_p}\right),$ with $m_e/m_p \approx 1/1836$. Doubling $m_p$ shifts the correction from $\sim 0{,}054\%$ to $\sim 0{,}027\%$: an infinitesimal shift of the energy levels and spectral lines. The Bohr radius stays practically unchanged.

Chemistry. Chemistry is governed by electronic structure, which depends on $\mu \approx m_e$: so it would not be turned upside down. Bonds, molecular geometries and reaction energies would remain almost identical. Only the dynamical aspects tied to nuclear mass would change (reaction rates, isotope effects, vibrational frequencies $\propto 1/\sqrt{m}$), exactly as happens when hydrogen is replaced by its heavy isotope, deuterium.

Links

Topics: Campo elettrico e potenziale Concepts: Energia potenziale elettrostatica · Carica elettrica Skills: Conservazione dell’energia Methods: Metodo del gradiente Objects: Atomo di idrogeno