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I asked this question in the Physics SE but it still lies unanswered. Hopefully Earth Science SE is more knowledgeable in the matter.

Wikipedia gives the following equation to calculate the moist adiabatic lapse rate $\Gamma_w$, assuming that there is only one condensible gas (water vapour) mixed in the "dry air":

$\Gamma_w = g\frac{\left(1+\frac{H_v r}{R_{sd}T} \right)}{\left(c_{pd} + \frac{H_{v}^2r}{R_{sw}T^2} \right)}$

Where:

  • $\Gamma_{w}$: moist adiabatic lapse rate [K/m]
  • $g$: gravitational acceleration [m/s2]
  • $H_{v}$: latent heat of vaporization of water [J/kg]
  • $R_{sd}$: specific gas constant of dry air [J/kg·K]
  • $R_{sw}$: specific gas constant of water vapour [J/kg·K]
  • $r={\frac {\epsilon e}{p-e}}$: mixing ratio of the mass of water vapour to the mass of dry air
  • $\epsilon = \frac{R_{sd}}{R_{sw}}$: ratio of the specific gas constant of dry air to the specific gas constant for water vapour = 0.622 [dimensionless]
  • $e$: water vapour pressure of the saturated air [Pa]
  • $p$: pressure of the saturated air [Pa]
  • $T$: temperature of the saturated air [K]
  • $c_{pd}$: specific heat of dry air at constant pressure [J/kg·K]

What's the form of this equation when the atmosphere composition differs from Earth's, thus allowing multiple condensible gases or even be entirely composed of only one condensible gas (for example 100% water vapour)?

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  • $\begingroup$ One can only hope to get an analytical form for condensble vapors that have close properties in regard to condensation. Fortunately, this is not always the case - e.g. Jupiter has few stacked cloud systems, each one related to different gas. Near each of these, the equation should be the same and you just have to put the proper numbers for the particular substance. $\endgroup$
    – fraxinus
    Jan 25, 2023 at 22:00

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