Answer to Question #96722 in Physics for Abhishek

Question #96722
Derive an expression for adiabatic lapse
rate.
1
Expert's answer
2019-10-28T10:55:07-0400

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=0dQ=0


and the atmosphere is in hydrostatic equilibrium.

The gravity side


dp=gρdzdp=-g\cdot\rho\cdot dz


dpρ=gdz\frac{dp}{\rho}=-g\cdot dz


The thermodynamics side


dA=dUpdV=mcVdT-dA=dU \to-pdV=mc_{V}dT


Vdpd(pV)=mcVdTVdp-d(pV)=mc_{V}dT


Vdpd(nRT)=VdpnRdT=mcVdTVdp-d(nRT)=Vdp-nRdT=mc_{V}dT


VmdpnmRdT=cVdT\frac{V}{m}dp-\frac{n}{m}RdT=c_{V}dT


1ρdpnnMRdT=cVdT\frac{1}{\rho}dp-\frac{n}{nM}RdT=c_{V}dT


1ρdp=(RM+cV)dT\frac{1}{\rho}dp=(\frac{R}{M}+c_{V})dT


dpρ=cpdT\frac{dp}{\rho}=c_{p}dT


cpdT=gdzc_{p}dT=-g\cdot dz


And finally


dTdz=gcp\frac{dT}{dz}=-\frac{g}{c_{p}}.




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