= Solution
Substituting the <chemical potentials> into the equilibrium relation gives
$$
2\ln(n_c\lambda_c^3)-\beta\epsilon
=2\ln(n_{\rm CO}\lambda_{\rm CO}^3)+\ln(n_{\rm O_2}\lambda_{\rm O_2}^3).
$$
Exponentiation and the positive square root give the <law of mass action> in number-density form:
$$
\boxed{n_c=n_{\rm CO}\sqrt{n_{\rm O_2}}\,\frac{\lambda_{\rm CO}^3\lambda_{\rm O_2}^{3/2}}{\lambda_c^3}\,e^{\epsilon/(2k_BT)}.}
$$
All three <thermal de Broglie wavelengths> have the same $T^{-1/2}$ dependence. Define the temperature-independent mass factor
$$
B=\left(\frac{m_c}{m_{\rm CO}}\right)^{3/2}\left(\frac{m_c}{m_{\rm O_2}}\right)^{3/4},\qquad
\frac{\lambda_{\rm CO}^3\lambda_{\rm O_2}^{3/2}}{\lambda_c^3}=B\lambda_c^{3/2}.
$$
If $f_\alpha=n_\alpha/n$, this exact equilibrium relation can be written
$$
n_c=Bf_{\rm CO}\sqrt{f_{\rm O_2}}\,n^{3/2}\lambda_c^{3/2}e^{\epsilon/(2k_BT)}.
$$
It 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 $n_{\rm CO}=2n_{\rm O_2}$ throughout. Hence
$$
n_{\rm CO}=\frac23(n-n_c),\qquad n_{\rm O_2}=\frac13(n-n_c),
$$
and the exact relation within this ideal-mixture model is
$$
n_c=A_0(n-n_c)^{3/2}\lambda_c^{3/2}e^{\epsilon/(2k_BT)},\qquad A_0=\frac{2B}{3\sqrt3}.
$$
At high <temperature> with $k_BT\gg\epsilon$ and in the dilute <ideal gas> regime, $n\lambda_c^3\ll1$, this equation is self-consistently dissociation dominated:
$$
\frac{n_c}{n}\simeq A_0\sqrt{n\lambda_c^3}\,e^{\epsilon/(2k_BT)}\ll1.
$$
Thus $n-n_c\simeq n$, and to leading order
$$
\boxed{n_{\rm CO_2}\simeq A\,n^{3/2}\lambda_{\rm CO_2}^{3/2}e^{\epsilon/(2k_BT)},\quad
A=\frac{2}{3\sqrt3}\left(\frac{m_{\rm CO_2}}{m_{\rm CO}}\right)^{3/2}\left(\frac{m_{\rm CO_2}}{m_{\rm O_2}}\right)^{3/4}.}
$$
Using nominal molecular masses $28,32,44$ in common mass units gives $A\simeq0.963$, 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-$\mathrm{CO_2}$ assumption, the universal result is the boxed mass-action relation, and $A=Bf_{\rm CO}\sqrt{f_{\rm O_2}}$ 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, $k_BT\gg\epsilon$ alone does not justify replacing $n-n_c$ by $n$ at arbitrarily high density; the dilute classical regime is part of the model.
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