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## Homework Statement

The hydrostatic equation expresses the change in pressure dp due to a layer of

atmosphere of thickness dz as

constant volume.

##dp = −\rho g dz ##

Using this expression, show that the change in temperature with height for a parcel of air that rises adiabatically in the atmosphere can be expressed as

##-\frac{\gamma-1}{\gamma} \frac{mg}{K_B}##

## Homework Equations

## The Attempt at a Solution

So I think we're trying to find ##\frac{\partial T}{\partial z}_S ## as this seems like a reversible process

starting off with ##dU=TdS-pdV##

## \frac{\partial U}{\partial T}_z dT +\frac{\partial U}{\partial z}_T dz = TdS-pdV##

## \frac{3NK_B}{2}\frac{\partial T}{\partial z}_S +\frac{\partial U}{\partial z}_T =-p\frac{\partial V}{\partial z}_S##

The fact that i can't find ## \frac{\partial V}{\partial z}_S## makes me think I've gone wrong somewhere

Many thanks