 Solution

For ordinary condensibles, $L/(R_{B}T)$ is typically of order ten or more near low atmospheric temperatures. At the approximate triple-point temperatures, the supplied substances give $\beta$ about $20$ for water and $10$–$12$ for carbon dioxide, methane and nitrogen. This is consistent with the supplied large ratios of latent heat to sensible heating: for molecular gases $c_p/R$ is only a few. The relevant $x=R_A/c_{pA}$ is typically a few tenths.

The integral is concentrated where ${\beta }_{s}x\zeta =O(1)$. There $x\zeta =O({\beta }_s^{-1})\ll 1$, so $e^{x\zeta }-1\simeq x\zeta$ givesThis is an Laplace's method, rather than an assumption that vapour is uniformly mixed. The effective contributing depth in log-pressure is $\Delta \zeta \sim 1/(x{\beta }_s)$, explaining why the near-surface atmosphere dominates.

An exact representation within the same dilute model makes the error explicit. Set $y={\beta }_s{e}^{x\zeta }$ and define the exponential integral $E_1(b)={\int }_b^{\infty }{e}^{-y}dy/y$. ThenThe saturated vapor column on a dry adiabat estimate overstates the exact dilute-model column by roughly $1/{\beta }_s$, about five to ten per cent for those illustrative values. It is not accurate for a warm, nondilute steam atmosphere.