Past exam of the chemistry course of the year 4 of the natural sciences course of the University of Cambridge / 2022 / Interdisciplinary paper IDP1 / 2 / a

Take a fixed amount of dry air, with no phase change, moving quasistatically so that its pressure matches the surrounding atmosphere. From the first law of thermodynamics and $H=U+pV$,For adiabatic motion $dq=0$. Treating the parcel as an ideal gas of fixed composition makes its enthalpy depend only on temperature, so for $n$ moles $dH=n{C}_{p}dT$ even though the pressure changes. Hence $n{C}_{p}dT=Vdp$. The environment is in hydrostatic equilibrium, $dp/dz=-\rho g$, and the parcel is taken to have the ambient density at the displacement point. Using $\rho =nM/V$ givesHere $C_p$ is molar and $c_p=C_p/M$ is mass-specific heat capacity. Positive $z$ points upward; the commonly quoted positive dry adiabatic lapse rate is ${\Gamma }_d=-dT/dz$. Approximate constancy of $g$ and heat capacity gives a constant gradient. Condensation, external heating, entrainment or composition changes invalidate this dry-parcel relation.