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Vanderwall’s Gas in Joule Thomson Expansion-Thermodynamics

Let us suppose one gram mole of a real gas allowed to expand through a porous plug from a pressure P1 and volume V1 to pressure P2 and volume V2.During this process suppose that temperature changes from T1 to T2.
The gas will have to do external as well as internal work. 
External work done on the gas = P1V1
External work done by gas         = P2V2

Therefore net external work done
by the gas =   P2V2 – P1V1……………(i)

Now an internal work is also done by gas in overcoming forces of molecular attractions 
Let, the gas expands from volume V1 to volume V2 ,then internal work done in gas is given by :-

Let the temperature change is very small,  then T1 & T2 are nearly equal ,then we can write :

Since the gas is thermally insulated. Hence resulting in a fall in temperature by dT(T1-T2).
If Cv is the specific heat at constant volume then heat lost by gas in doing work is equal to CvdT.
Here three cases arises which are as follows :


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