Past exam of the thermal and statistical physics course of the University of Cambridge / 2021 / Section B / 4 / e / Solution

Substituting the chemical potentials into the equilibrium relation givesExponentiation and the positive square root give the law of mass action in number-density form:All three thermal de Broglie wavelengths have the same $T^{-1/2}$ dependence. Define the temperature-independent mass factorIf $f_{\alpha }=n_{\alpha }/n$, this exact equilibrium relation can be writtenIt already has the requested dimensional structure: $n^{3/2}{\lambda }_c^{3/2}$ has units ${\mathrm{m}}^{-3}$. Its coefficient, however, depends on the conserved elemental composition through the species fractions.

For the intended case of a gas obtained by dissociating initially pure carbon dioxide, conservation of the elemental ratio gives $n_{\mathrm{CO}}=2n_{{\mathrm{O}}_2}$ throughout. Henceand the exact relation within this ideal-mixture model isAt high temperature with $k_{B}T\gg ϵ$ and in the dilute ideal gas regime, $n{\lambda }_c^3\ll 1$, this equation is self-consistently dissociation dominated:Thus $n-n_c\simeq n$, and to leading orderUsing nominal molecular masses $28,32,44$ in common mass units gives $A\simeq 0.963$, confirming that it is of order unity. The exponential has been retained as the leading binding correction, although it tends to one as $T\to \infty$.

The printed question does not explicitly fix the elemental composition. Without the initially-pure-$\mathrm{C}{\mathrm{O}}_2$ assumption, the universal result is the boxed mass-action relation, and $A=B{f}_{\mathrm{CO}}\sqrt{f_{{\mathrm{O}}_2}}$ approaches a composition-dependent constant when both dissociation products have nonzero high-temperature fractions. It need not be of order unity for an arbitrarily unbalanced mixture. Also, $k_{B}T\gg ϵ$ alone does not justify replacing $n-n_c$ by $n$ at arbitrarily high density; the dilute classical regime is part of the model.