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.
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.