##### 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.

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