= Solution
Interpret the requested <mass path> as the vapour column; no condensed-water loading is prescribed. <Hydrostatic equilibrium> gives $dp=-\rho g\,dz$, so if $q_B=\rho_B/\rho$ is vapour <mass fraction>,
$$
W_B=\int_0^\infty\rho_B\,dz=\frac1g\int_0^{p_s}q_B(p)\,dp.
$$
For the mixture $q_B=\epsilon e_s/[p+(\epsilon-1)e_s]$. Under the dilute conditions of part (a), $q_B\simeq\epsilon e_s/p$. Introduce $\zeta=-\ln(p/p_s)$, so $T=T_se^{-x\zeta}$ and $-dp=p\,d\zeta$. Integrating the constant-$L$ <Clausius-Clapeyron equation> relative to the surface gives
$$
e_s(T)=e_s(T_s)\exp\left[-\frac{L}{R_B}\left(\frac1T-\frac1{T_s}\right)\right].
$$
Therefore
$$
\boxed{W_B\simeq\frac{\epsilon e_s(T_s)}g\int_0^\infty\exp[-\beta_s(e^{x\zeta}-1)]\,d\zeta,\qquad \beta_s=\frac{L}{R_BT_s}.}
$$
The explicit $p_s$ cancels: decreasing vapour <mass fraction> in a denser dry column compensates its greater total <mass>. This cancellation belongs to the dilute, dry-adiabatic approximation, and is not a general property of moist atmospheres. With suspended condensate, an additional and unspecified condensate <mass path> would have to be added.
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