 Solution

Hydrostatic equilibrium balances the vertical pressure gradient against weight, $dP/dz=-\rho g$. For a dry ideal gas parcel displaced slowly enough to maintain pressure balance but rapidly enough to exchange negligible heat, the first law givesCombining the two equations yields the dry adiabatic lapse rateusing $c_p\simeq 1004\mathrm{J}\mathrm{k}{\mathrm{g}}^{-1}{\mathrm{K}}^{-1}$. Here temperature in the thermodynamic differential is absolute, $g$ and $c_p$ are treated as constant, composition is fixed, and condensation is excluded. Hydrostatic balance alone does not force the atmosphere to be adiabatic; the parcel assumption supplies the extra relation.

Water condensation releases latent heat, reducing the cooling of a rising saturated parcel and giving a smaller moist adiabatic lapse rate. Radiation, mixing, large-scale motion and incomplete convective adjustment also shape the environmental lapse rate. Thus an observed $6$--$7\mathrm{K}\mathrm{k}{\mathrm{m}}^{-1}$ need not equal the dry parcel value and is broadly consistent with a moist, radiatively influenced atmosphere.