Q. 5

In a real gas the internal energy depends on temperature and also on volume. The energy increases when the gas expands isothermally. Looking into the derivation of CP – CV = R, find whether CP – Cv will be more than R, less than R or equal to R for a real gas.

Answer :

We know that, for an ideal gas,


CP - CV = R … (i)


where


Cp = specific heat constant at constant pressure


Cv = specific heat constant at constant volume


R = universal gas constant


Multiplying by n x dT on both sides of (i), we get


nCPdT - nCvdT = nRdT


which gives


(dQ)P - (dQ)v = nRdT …(ii)


Since (dQ)p = nCpdT and (dQ)v = nCvdT … (iii)


Where


n = number of moles


dT = change in temperature


(dQ)p = change in heat at constant pressure


(dQ)v = change in heat at constant volume


However, for a real gas, the internal energy depends on the temperature as well as the volume.


Hence, there will be an additional term on the right-hand side of (ii) which will indicate the change in the internal energy of the gas with volume at constant pressure. Let this term be u.


Hence, for a real gas, (ii) becomes :


(dQ)p - (dQ)v = nRdT + u … (iv)


Again, dividing on both sides by ndT, we get


… (v),


which is greater than R.


Here,


Cp = specific heat constant at constant pressure


Cv = specific heat constant at constant volume


R = universal gas constant


n = number of moles


dT = change in temperature


Hence, from (v), we get Cp - Cv > R.


We conclude that for a real gas, Cp - Cv > R.


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