= Solution
\b[Take dry <air> as unity, and compare each <gas> at the same <temperature> and <pressure>.] In the <ideal gas> approximation, $\rho=pM/(RT)$, so the <relative density> is
$$
\boxed{d_{\rm air}=\frac{\rho_{\rm gas}}{\rho_{\rm air}}\simeq\frac{M_{\rm gas}}{28.97\ \mathrm{g\,mol^{-1}}}.}
$$
Here $M$ is <molar mass>. The following values use modern approximate <molar masses>; rounding and historical <atomic weight> conventions can shift the last figures.
|| Historical name and modern identification
|| Formula
|| $M/(\mathrm{g\,mol^{-1}})$
|| <Relative density>, <air> $=1$
| Carbonic acid: <carbon dioxide> <gas>
| $\mathrm{CO}_2$
| 44.01
| \b[1.519]
| Carbonic oxide: <carbon monoxide>
| $\mathrm{CO}$
| 28.01
| \b[0.967]
| <Nitrous oxide>
| $\mathrm{N}_2\mathrm{O}$
| 44.01
| \b[1.519]
| Dry <air>
| Mixture
| 28.97
| \b[1.000]
| <Ammonia>
| $\mathrm{NH}_3$
| 17.03
| \b[0.588]
| Olefiant <gas>: <Ethylene>
| $\mathrm{C}_2\mathrm{H}_4$
| 28.05
| \b[0.968]
| Alcohol vapour: <ethanol>
| $\mathrm{C}_2\mathrm{H}_6\mathrm{O}$
| 46.07
| \b[1.590]
| Ether vapour: <diethyl ether>
| $\mathrm{C}_4\mathrm{H}_{10}\mathrm{O}$
| 74.12
| \b[2.559]
“Olefiant <gas>” is confirmed as a historical synonym of <Ethylene> in the <NIST Chemistry WebBook, SRD 69>. In this <gas> list, “carbonic acid” means <carbon dioxide>, not an isolated vapour of aqueous $\mathrm{H}_2\mathrm{CO}_3$. “Alcohol” and “ether” refer to <ethanol> and <diethyl ether>, rather than their entire modern chemical classes.<NIST Chemistry WebBook, SRD 69>[Ethylene entry, Formula, Molecular weight and Other names; <Ammonia> entry, Formula and Molecular weight]{ref}
The order is therefore <Ammonia>, then <carbon monoxide> and <Ethylene> nearly tied, then <air>, then <carbon dioxide> and <nitrous oxide> nearly tied, then <ethanol> vapour, then <diethyl ether> vapour. Nearly equal <molar masses> do not imply equal chemical behaviour.
If instead <molecular hydrogen> is chosen as unity, divide each <molar mass> by about $2.016\ \mathrm{g\,mol^{-1}}$ and state that convention. In the older nominal $\mathrm H_2=2$ approximation, the corresponding values are about $22,14,22,14.5,8.5,14,23,37$ in the source order. These are a different reference scale, not different substances.
The vapour comparisons are not comparisons of the <liquid> <densities>. Pure <ethanol> and <diethyl ether> must be considered at conditions where they remain gaseous, or by a low-<pressure> <gas> approximation. Their <liquids> can be lighter than <water> while their vapours are heavier than <air>.
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