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The humidity ratio $w$ is the mass of water vapor divided by the mass of dry air in a sample:It is commonly expressed as kilograms of water vapor per kilogram of dry air, or grams per kilogram. The denominator excludes the water vapor itself; the related specific humidity is a mass fraction of the entire moist sample.^[ref]

Using the ideal gas law for both components giveswhere $M_v,M_d$ are molar masses, $p_v$ is the water-vapor partial pressure, and $p$ is total pressure. At $p=101.325\mathrm{kPa}$ and $p_v=1.6\mathrm{kPa}$, this yields $w\approx 0.00998$, or about $9.98g/kg$ of dry air.^[ref]

Unlike relative humidity, the humidity ratio does not compare vapor pressure with saturation vapor pressure. Warming air without adding or removing water vapor can change its relative humidity while leaving $w$ unchanged. This distinction matters when comparing incoming and outgoing air in measurements of evaporation or biological heat loss.

Let $R_A,R_B$ be the specific specific gas constants and $c_{pA},c_{pB}$ the specific heat capacities of the noncondensible gas and vapour. Write $e_s(T)$ for the saturation vapor pressure, $r$ for vapour mass per unit dry-gas mass, and $ϵ=R_A/R_B$. For a dilute ideal gas mixture,The second equation is the Clausius-Clapeyron equation, neglecting condensed-phase volume and taking constant latent heat $L$. The first law of thermodynamics for saturated ascent, with condensate removed and negligible condensate heat capacity, is to leading orderSince $dr=r[LdT/(R_B{T}^2)-d\ln p]$, the resulting moist adiabat has logarithmic slopeConsequently a sufficient dry adiabat criterion isThe first two conditions justify neglecting the vapour's effects on the gas law and sensible heat capacity; the last two suppress latent heating. With $x=R_A/c_{pA}$ and $\beta =L/(R_{B}T)$, the latent conditions are $\beta e_s/p\ll 1$ and $x{\beta }^2{e}_s/p\ll 1$. Usually $x\beta \gg 1$, so the squared-latent term gives the stronger condition. At sufficiently low surface temperature, the exponential smallness of $e_s(T_s)$ makes these conditions hold andThese are local conditions along the column. They can be checked at the surface when $x{\beta }_s>1+2x$: on this dry profile the strongest latent parameter is proportional to $\exp [(1+2x)\zeta -{\beta }_s(e^{x\zeta }-1)]$ and decreases upwards. Without that restriction a surface-only test need not bound every altitude. If suspended condensate is retained, its own heat capacity and loading must also be negligible; saturation alone does not specify those quantities.

Prescribe a dry adiabat for a background ideal gas A,where $R_A$ is its specific gas constant, $c_{pA}$ its constant specific heat capacity at fixed pressure, and $T_s,p_s$ the surface temperature and pressure. A dilute saturated vapour B has specific gas constant $R_B$, constant latent heat $L$, and saturation vapor pressure $e_s(T)$. Put $ϵ=R_A/R_B$ and $\beta =L/(R_B{T}_s)$.

The ideal-vapour Clausius-Clapeyron relation integrates toIts dilute vapour mass fraction is approximately $ϵe_s/p$. Under constant-$g$ hydrostatic balance, the vapour mass path is thereforeFor the second equality, set $\zeta =\log (p_s/p)$ and then $y=\beta e^{x\zeta }$. Thus the exponential integral follows from the combination of adiabatic cooling and rapidly falling saturation vapor pressure.

For $\beta \gg 1$, integration by parts gives $e^{\beta }{E}_1(\beta )\sim {\beta }^{-1}(1-{\beta }^{-1}+2{\beta }^{-2}-\cdots )$, henceThe column is weighted toward its warm lower part. Diluteness must hold throughout the column. This calculation prescribes the dry adiabat; if condensation heating changes that profile, a consistent moist adiabat is needed instead.