= Solution
For ordinary condensibles, $L/(R_BT)$ 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 $\beta_sx\zeta=O(1)$. There $x\zeta=O(\beta_s^{-1})\ll1$, so $e^{x\zeta}-1\simeq x\zeta$ gives
$$
\boxed{W_B\simeq\frac{\epsilon e_s(T_s)}{g x\beta_s}=\frac{c_{pA}T_s}{L}\frac{e_s(T_s)}g.}
$$
This is an <Laplace's method>, rather than an assumption that vapour is uniformly mixed. The effective contributing depth in log-pressure is $\Delta\zeta\sim1/(x\beta_s)$, explaining why the near-surface atmosphere dominates.
An exact representation within the same dilute model makes the error explicit. Set $y=\beta_se^{x\zeta}$ and define the <exponential integral> $E_1(b)=\int_b^\infty e^{-y}dy/y$. Then
$$
W_B=\frac{\epsilon e_s(T_s)}{gx}e^{\beta_s}E_1(\beta_s)
\sim\frac{c_{pA}T_s e_s(T_s)}{gL}\left(1-\frac1{\beta_s}+\frac2{\beta_s^2}-\cdots\right).
$$
The <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.
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