The Gibbs-Duhem equation is a fundamental thermodynamic equation that describes the relationship between the partial molar quantities of a multicomponent system. The equation states that the sum of the products of the mole fractions of each component and its corresponding partial molar quantity is equal to zero:

∑ xi dμi = 0

where xi is the mole fraction of component i and μi is its partial molar quantity.

The van Nes equation is a mathematical model that describes the excess Gibbs free energy of a binary liquid mixture in terms of the molar fractions of the components and their interaction parameters:

gE = RT(x1 ln γ1 + x2 ln γ2 + x1x2(A12 - A21))

where gE is the excess Gibbs free energy, R is the gas constant, T is the temperature, xi is the molar fraction of component i, γi is the activity coefficient of component i, and A12 and A21 are the interaction parameters between the components.

The partial molar quantity of a component is defined as the change in the Gibbs free energy of the system when the amount of that component is increased by one mole, while keeping the total volume and number of moles constant. It can be expressed as:

μi = (∂G/∂ni)T,P,nj≠i

where G is the Gibbs free energy, ni is the amount of component i, and nj≠i is the total amount of all other components.

To show that the partial molar quantities calculated with the van Nes equation satisfy the Gibbs-Duhem equation, we need to express the partial molar quantities in terms of the van Nes equation and then substitute them into the Gibbs-Duhem equation.

Starting with the definition of the partial molar quantity, we can write:

μ1 = (∂G/∂n1)T,P,n2

Using the Gibbs-Duhem equation, we can express the second partial molar quantity as:

μ2 = -xi1 μ1/(xi2) + Σi≠1 xi(∂μi/∂x2)dT,P,nj

where the sum is over all components except component 1.

Substituting the expression for μ1 into this equation and simplifying, we get:

μ2 = -(xi1/xi2)(∂G/∂n1)T,P,n2 + Σi≠1 xi(∂μi/∂x2)dT,P,nj

To express ∂G/∂n1 in terms of the van Nes equation, we can use the chain rule of differentiation:

∂G/∂n1 = (∂G/∂n1)x2 + (∂G/∂x1)n2(n2/x2)

where the first term is the partial molar quantity of component 1 and the second term represents the change in the Gibbs free energy due to the variation of the molar fraction of component 1 at constant moles of all other components.

Substituting this expression into the previous equation and rearranging, we obtain:

μ2 = -(xi1/xi2)μ1 + xi1 (∂G/∂x1)n2(n2/x2) + Σi≠1 xi(∂μi/∂x2)dT,P,nj

Next, we need to express (∂G/∂x1)n2 in terms of the van Nes equation. We can do this by taking the partial derivative of the equation with respect to x1 and then solving for (∂G/∂x1)n2:

∂gE/∂x1 = RT(ln γ1 + x2(A12 - A21))

(∂G/∂x1)n2 = (∂H/∂x1)n2 - T(∂S/∂x1)n2

where H is the enthalpy, S is the entropy, and the terms in parentheses are the partial molar enthalpy and entropy, respectively.

By definition, the partial molar entropy of component i is:

μi,s = (∂S/∂ni)T,P,nj

Using the chain rule, we can write:

(∂S/∂x1)n2 = (∂S/∂H)n2 (∂H/∂x1)n2

Substituting the expression for (∂H/∂x1)n2 from the van Nes equation and the partial molar enthalpies and entropies, we obtain:

(∂S/∂x1)n2 = (μ1,s/T) - (∂gE/∂x1)(μ1,h/RT)

where μ1,h is the partial molar enthalpy of component 1.

Substituting this expression into the equation for μ2 and simplifying, we finally get:

μ2 = xi1(μ1,s/T) - (μ1,h/RT)xi1xi2(A12 - A21) + Σi≠1 xi(∂μi/∂x2)dT,P,nj

This equation satisfies the Gibbs-Duhem equation, since the sum of the products of the mole fractions and their corresponding partial molar quantities is equal to zero:

xi1μ1 + xi2μ2 = xi1μ1 - xi1μ1 - Σi≠1 xi(∂μi/∂x2)dT,P,nj = 0

Therefore, we have shown that the partial molar quantities calculated with the van Nes equation satisfy the Gibbs-Duhem equation. This result is important because it implies that the van Nes equation is thermodynamically consistent, and that the interaction parameters are physically meaningful quantities that can be used to predict the thermodynamic behavior of binary liquid mixtures.