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A curve of growth describes the equivalent width $E(m)=\int (1-e^{-m{\kappa }_{\nu }})d\nu$ as a function of absorber mass path or column density. Its weak-line limit is $E\simeq mS$. Saturation makes growth sublinear: a pure Gaussian core gives $E\sim 2{\Delta }_D\sqrt{\log {\tau }_0}$, while saturated Lorentzian line shape wings give $E\sim 2\sqrt{mS{\Delta }_L}$. The spectral coordinate and coefficient convention must be consistent.

The mass mixing ratio used in a dilute radiative transfer model is absorber mass density divided by total atmospheric mass density. A hydrostatic pressure interval contributes absorber mass path $qdp/g$. Some meteorological conventions divide by dry-air mass instead; the convention must be stated. A volume or mole mixing ratio instead contributes absorber moles $xdp/(gM)$, with background molar mass $M$.

Let $m$ be the absorber mass path along a ray. If the spectral line profile is independent of position, Beer-Lambert law gives ${\mathfrak{T}}_{\nu }(m)=e^{-m{\kappa }_{\nu }}$. A finite-band mean and the line's equivalent width are respectivelyWhen the band contains all significant absorption, ${\int }_{\mathcal{B}}{\mathfrak{T}}_{\nu }d\nu \simeq \Delta {\nu }_{\mathcal{B}}-E$. An unqualified integral of transmission over the entire infinite line domain diverges because transmission tends to one far from line centre; $E$ is the finite and physically useful line deficit.

This is nonexponential band transmission: each monochromatic transmission is exponential, but their average generally is not. Differentiating with respect to mass path givesMore decisively, define the absorption coefficient averaged over the transmitted spectrum byThen $\boxed{a_{\mathrm{eff}}^{\prime }(m)=-{\mathrm{Var}}_m(\kappa )\le 0}$. The core saturates first and the surviving radiation increasingly samples weakly absorbing wings. The effective attenuation per additional mass therefore decreases. Jensen's inequality also gives $\overline{\mathfrak{T}}\ge e^{-m⟨\kappa ⟩}$, with equality for a spectrally constant coefficient. Thus the band transmission decays more slowly than the exponential with its initial mean attenuation.

The corresponding curve of growth has $E(m)\simeq mS$ for an optically weak line, and sublinear growth after saturation. For the Gaussian profile, with ${\tau }_0=mS/(\sqrt{\pi }{\Delta }_D)\gg 1$, absorption extends to $|\delta \nu |\simeq {\Delta }_D\sqrt{\log {\tau }_0}$, so $E\sim 2{\Delta }_D\sqrt{\log {\tau }_0}$. For a strongly saturated pure Lorentz profile the wings instead give $E\sim 2\sqrt{mS{\Delta }_L}$. These asymptotes apply while the line remains isolated and its important wings remain within the range where the assumed profile and nearly constant Planck function are appropriate. The integrated absorption is not an exponential either: $E^{\prime }>0$ and $E^{\prime \prime }\le 0$.

Interpret the requested mass path as the vapour column; no condensed-water loading is prescribed. Hydrostatic equilibrium gives $dp=-\rho gdz$, so if $q_B={\rho }_B/\rho$ is vapour mass fraction,For the mixture $q_B=ϵe_s/[p+(ϵ-1)e_s]$. Under the dilute conditions of part (a), $q_B\simeq ϵe_s/p$. Introduce $\zeta =-\ln (p/p_s)$, so $T=T_s{e}^{-x\zeta }$ and $-dp=pd\zeta$. Integrating the constant-$L$ Clausius-Clapeyron equation relative to the surface givesThereforeThe 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.

For a dilute saturated vapour with constant latent heat $L$ in a dry atmosphere, set $x=R_A/c_{pA}$, $ϵ=R_A/R_B$ and $\beta =L/(R_B{T}_s)$. Its mass path is $W=ϵe_s(T_s)e^{\beta }{E}_1(\beta )/(gx)$. Substituting $y=\beta e^{x\zeta }$ in the log-pressure column integral proves this identity. For $\beta \gg 1$, $W\sim c_{pA}{T}_s{e}_s(T_s)/(gL)$ with relative correction $-1/\beta +O({\beta }^{-2})$.