Heat capacities for real gases

A similar expression is also obtained from the Gibbs free energy equation when pressure approaches zero. If the partial differentiation inside the integral is determined from an equation of state (EOS), then t/real can be calculated. For example, (777) from the van der Waals EOS is 

The second term on the right side of Eq. (1.130) represents the energy of molecular interactions per unit volume. As the volume increases, the interactions get smaller, as is the case for gases at low density. If we use the Berthelot equation 

Equation (1.130) also yields a relation for the heat capacity of a real gas at constant volume 

If the second partial derivative inside the integral is determined from an EOS, then the heat capacity of a real gas at constant volume can be calculated. For example, the integral in Eq. (1.134) vanishes for the van der Waals equation, and as Eq. (1.129) shows, pressure is a linear function of temperature. However, by using the Berthelot EOS, (Eq. 131), the heat capacity is obtained from Eq. (1.134). 

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