We will now derive an expression for the rate of change of temperature with height of a parcel of dry air as it moves about in the Earth’s atmosphere. Since the air parcel undergoes only adiabatic transformations
dQ=0
and the atmosphere is in hydrostatic equilibrium.
The gravity side
dp=−g⋅ρ⋅dz
ρdp=−g⋅dz
The thermodynamics side
−dA=dU→−pdV=mcVdT
Vdp−d(pV)=mcVdT
Vdp−d(nRT)=Vdp−nRdT=mcVdT
mVdp−mnRdT=cVdT
ρ1dp−nMnRdT=cVdT
ρ1dp=(MR+cV)dT
ρdp=cpdT
cpdT=−g⋅dz
And finally
dzdT=−cpg.
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