Problem 1The excess Gibbs free energy G# for a binary mixture of two fluids (1) and (2) is given byĜE= f(x1,x2)RTa) Determine yi expressions for both Margules models given below. Show your derivations stepby step.If f(x1, x2) = A x1 X2 (Two-Suffix Margules Equation)If f (x1, x2) = x1 x2(A+ B(x1 – x2)) (Three-Suffix Margules Equation)b) Determine In y ® expressions for both models given in Part (a).

Given data: The expression of excess Gibbs energy of the fluid (1) and (2) is given.(a) The expression of excess Gibbs energy of the fluid (1) and (2) is G⏞ERT = fx1, x2 ............... (1) Where, GE represents the excess Gibbs energy. R represents the universal gas constant. T represents the temperature. x1 represents the composition of fluid (1). x2 represents the composition of fluid (2). From the Two –suffix Margules equation, fx1, x2 = Ax1x2 Therefore the equation (1) becomes, G⏞ERT = Ax1x2 ................. (2)From the excess property relation, the activity coefficient of fluid (1) is calculated as follows: ln γ1 = ∂nGERT∂niT,P,nj .................... (3) Where, γ1 represents the activity coefficient of fluid (1). n represents the total number of moles. ni represents the number of moles of ith species, The mole fraction can be written in terms of number of moles as follows: x1 =n1n1 + n2 x2 =n2n1 + n2 Therefore the equation (2) becomes, nGERT = An1n2n1 + n22 nGERT = An1n2n2 ............... (4) Differentiate equation (4) with respect to n1 at constant temperature and pressure ∂nGERT∂n1T, P ,n2 = ∂∂n1 An1n2n2From equation (3)ln γ1 = An21n1 +n22 - 2n1n1 +n23ni = nxiln γ1 = Ax22 Similarly differentiate equation (4) with respect to n2 ∂nGERT∂n2T, P ,n1 = ∂∂n2 An1n2n2ln γ2 = An11n1 +n22 - 2n2n1 +n23ni = nxiln γ2 = Ax12 Form the three-suffix Margules equation, fx1, x2 = A+Bx1 -x2x1x2 Equation (1) becomes, G⏞ERT = A+B x1 - x2x1x2 x2 can be written in terms of x1 as follows: x2 = 1 – x1 And, A +B = A21 A – B = A12 GEx1x2RT = A21x1 +A12 x2 .............. (5) The reciprocal of equation (5) is written as follows: x1x2RTGE = A' +B'x1-x2x1 +x2 = 1x1x2RTGE = A' x1 +x2+B'x1-x2x1x2RTGE = A' +B'x1 +A' -B'x2Assume,A' +B'= 1A21'A' -B'= 1A12'x1x2RTGE = x1 1A21'+x2 1A12' GEx1x2RT = A12'A21'A12'x1 + A21'x2 .................. (6) On solving equation (6) ln γ1 = A12'1+A12'x1A21'x22ln γ2 = A21'1+A21'x2A12'x12 The values of activity coefficients at infinite dilution for two-suffix and three –suffix Margules model is calculated as follow: For,ln γ1∞ means,x1 →0x2 →1Therefore,ln γ1∞ for two-suffix Margules equation becomes,ln γ1∞ = Ax22ln γ1∞ = A×12ln γ1∞= A For three- suffix Margules equation,ln γ1∞ becomes,ln γ1∞ = A12'1 + A12'x1A21'x2ln γ1∞ = A12'1 + A12'× 0A21'× 1 ln γ1∞ = A12'1 + 0ln γ1∞ = A12'