Question

The excess gibbs free energy of a binary liquid mixture at t and p is given by: T = (−2.6 x1 − 1.8x2)x1x2 , a) find expression of ln 1 and ln 2 at t and p. b) show that when these expressions are combined in accordance with equation T = ∑ x , that the given equation for ge/rt is recovered.

Answer 1

The question involves deriving expressions for the natural logarithm of the activity coefficients (ln γ1 and ln γ2) from the given excess Gibbs free energy equation and showing that these expressions are consistent with the original equation when combined.

Step-by-step explanation:

The student is asking about the relationship between the Gibbs free energy of a mixture and the activity coefficients of its components. In the given equation, T represents the excess Gibbs free energy, with x1 and x2 being the mole fractions of components 1 and 2, respectively. In part 'a' of the question, we need to find the expressions for ln γ1 and ln γ2, where γ1 and γ2 are the activity coefficients of the two components at temperature T and pressure p. The given equation for the excess Gibbs free energy ge/RT can be used to derive these expressions since the natural log of the activity coefficient is related to the partial molar free energy. In part 'b', we are asked to show that these expressions for ln γ1 and ln γ2, when combined with the mole fractions in the form of the equation T = ∑ xi, they should yield the initial equation for excess Gibbs free energy. This involves applying the Gibbs-Duhem equation to ensure the consistency of the derived expressions for activity coefficients with the original excess free energy equation provided.