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 C_P – C_V = R, find whether C_P – C_v will be more than R, less than R or equal to R for a real gas.

We know that, for an ideal gas,

C_P - C_V = R … (i)

where

C_p = specific heat constant at constant pressure

C_v = specific heat constant at constant volume

R = universal gas constant

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

nC_PdT - nC_vdT = nRdT

which gives

(dQ)_P - (dQ)_v = nRdT …(ii)

Since (dQ)_p = nC_pdT and (dQ)_v = nC_vdT … (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,

C_p = specific heat constant at constant pressure

C_v = specific heat constant at constant volume

R = universal gas constant

n = number of moles

dT = change in temperature

Hence, from (v), we get C_p - C_v > R.

We conclude that for a real gas, C_p - C_v > R.