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 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.

Answer :

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_{P}dT - nC_{v}dT = nRdT

which gives

(dQ)_{P} - (dQ)_{v} = nRdT …(ii)

Since (dQ)_{p} = nC_{p}dT and (dQ)_{v} = nC_{v}dT … (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.

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