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

Put $\beta =1/(k_{B}T)$ and define the thermal de Broglie wavelength of species $\alpha$ byIgnoring the internal degrees of freedom as stipulated, each molecule's translational canonical partition function is $V/{\lambda }_{\alpha }^3$. The two species do not interact, so their partition functions multiply; the treatment of indistinguishable particles requires a separate factorial for each species, not a factorial of the total particle number. ThusUsing $F=-k_{B}T\ln Z$ and Stirling's approximation gives the extensive Helmholtz free energyThese are ideal gas expressions, valid for $n_{\alpha }{\lambda }_{\alpha }^3\ll 1$, with $n_{\alpha }=N_{\alpha }/V$. The energy zero has been chosen so that separated CO and oxygen molecules have zero internal reference energy.

The given binding energy $ϵ$ belongs to the formation of two carbon-dioxide molecules. Relative to the reference used in (a), the internal energy offset of one carbon-dioxide molecule is therefore $-ϵ/2$, not $-ϵ$. Its mass is $m_c=m_{\mathrm{CO}}+m_{{\mathrm{O}}_2}/2$ and its thermal de Broglie wavelength is ${\lambda }_c=h/\sqrt{2\pi m_c{k}_{B}T}$, where the subscript $c$ denotes $\mathrm{C}{\mathrm{O}}_2$.

The single-molecule canonical partition function is $(V/{\lambda }_c^3)e^{\beta ϵ/2}$. For $N_c$ indistinguishable noninteracting molecules,The corresponding Helmholtz free energy, using Stirling's approximation, isThe negative sign in the last term expresses the stabilizing binding energy and ensures that low temperature favors association.

At fixed temperature, volume and the numbers of all other species, the chemical potential is ${\mu }_{\alpha }=(\partial F/\partial N_{\alpha }{)}_{T,V,N_{{\alpha }^{\prime }\ne \alpha }}$. For the combined noninteracting mixture, add the free energies from (a) and (b). Since the thermal de Broglie wavelengths are independent of particle number at fixed temperature, differentiation givesEach logarithm has a dimensionless argument, and each chemical potential has units of energy per molecule. The binding offset must be kept consistently when comparing species.

Introduce an extent of reaction $\xi$ increasing in the dissociation direction. The allowed particle-number changes conserve both carbon and oxygen atoms:At fixed temperature and volume, the resulting Helmholtz free energy change isAt an interior chemical equilibrium, both forward and backward changes are allowed, so the coefficient must vanish:Equality of the three individual chemical potentials would be wrong: a reaction exchanges species in its stoichiometric proportions rather than molecule for molecule. Stability is also explicit here, sincewhen all three species are present. Thus the equilibrium composition minimizes the free energy along the conserved-composition constraint.

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.