= Solution
The three parameter groups are \b[local temperature, local pressure, and elemental composition]. The elemental composition includes both <atmospheric metallicity of a giant planet> and <atmospheric carbon-to-oxygen ratio>; these are independent compositional controls, so a specification by $T$ and $P$ alone is incomplete. Given these inputs, <thermochemical equilibrium> minimizes <Gibbs free energy> with element conservation.
Three useful chemical observables in a hydrogen-rich <hot Jupiter> are:
* The <carbon monoxide>-to-<methane> partition. For $\mathrm{CO}+3\mathrm H_2\rightleftharpoons\mathrm{CH}_4+\mathrm H_2\mathrm O$, cooler gas and, at fixed mole fractions, larger <pressure> favour the side with fewer molecules, whereas hotter gas generally favours CO. Thus relatively hot photospheres commonly show CO near $2.3$ and $4.6\,\mu\mathrm m$, while cooler conditions favour <methane> bands near $3.3\,\mu\mathrm m$. The exact boundary depends on composition and <pressure>; <carbon monoxide–methane quenching> can invalidate an equilibrium diagnosis.
* <Water> absorption, for example around $1.4$ and $2.7\,\mu\mathrm m$, traces the oxygen left after CO forms. In oxygen-rich gas with $\mathrm{C/O}<1$, water is often prominent; carbon-rich gas can have much less water and relatively more <methane> or <hydrogen cyanide>. Condensation and high-temperature dissociation are exceptions to a simple elemental bookkeeping argument.
* <Carbon dioxide>, with a strong band near $4.3\,\mu\mathrm m$, is sensitive to enrichment and oxidation chemistry. In a hydrogen-dominated trace-species regime, $\mathrm{CO}+\mathrm H_2\mathrm O\rightleftharpoons\mathrm{CO}_2+\mathrm H_2$ gives $x_{\rm CO_2}\propto x_{\rm CO}x_{\rm H_2O}/x_{\rm H_2}$ at fixed $T$, so CO2 can grow roughly quadratically with heavy-element enrichment before the trace approximation fails.
\b[CO/CH4, H2O, and CO2 bands provide complementary temperature and elemental-abundance diagnostics], but their observed strengths also depend on the <atmospheric pressure-temperature profile>, <exoplanet clouds> and observing geometry. A spectral feature is not by itself a direct equilibrium abundance.
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