Question

The molar excess entropy for a real gas is defined as s_ex(T, P) = s(T, P) - s_IG(T, P) where s_IG is the entropy of an ideal gas at the same temperature and pressure. Show that the excess entropy is related to the fugacity by: S_ex(T, P) = -(d(k_s T ln f/p)/dT)_p

Answer 1

The molar excess entropy of a real gas is related to its fugacity by the derived equation S_ex(T, P) = -(d(k_B T ln f/p)/dT)_P, which involves the temperature derivative of the Gibbs free energy with respect to temperature at constant pressure.

Step-by-step explanation:

The question asks to show the relationship between the molar excess entropy (Sex) of a real gas and its fugacity (f). According to the question, molar excess entropy is defined as Sex(T, P) = s(T, P) - sIG(T, P), where s represents the entropy of the real gas and sIG is the entropy of an ideal gas at the same temperature (T) and pressure (P). To derive the relation Sex(T, P) = -(d(kB T ln f/p)/dT)P, we use the properties of entropy and the Gibbs free energy (G).

For a real gas, the Gibbs free energy is related to its fugacity by G = GIG + RT ln(f/P). Taking the temperature derivative of this expression at constant pressure allows us to express the change in molar excess entropy with respect to temperature.

The relationship between Gibbs free energy and entropy is G = H - TS, such that by differentiating G with respect to temperature at constant pressure, we find -S = (dG/dT)P. This derivation is grounded in thermodynamic principles and is a representation of how real gas behavior deviates from ideal gas law predictions based on fugacity and molar excess entropy.