7.3 Consider a k-component mixture in which n represents the total number of moles of material. (a) Show that the substitution of mix=n~mix Thermodynamics of mixtures 205 into Eq. (7.1-5) gives i=~mix+n(ni~mix)T,P,nj=i (b) Use the chain rule of differentiation and express Eq. (2) as where the subscript xj=i,r indicates that mole fractions of all components other than i and r are kept constant. (c) Noting that nixr=n2nr show that Eq. (3) takes the form i=mixr=1r=ikxr(xr~mix)T,P,xj=i,r (d) For a ternary system show that 1=~mixx2(x2~mix)x3x3(x3~mix)x22=mixx1(x1mix)x3x3(x3mix)x13=mixx1(x1~mix)x2x2(x2mix)x1 (e) For a ternary system of components 1, 2, and 3, the molar volume of the mixture is given by V~mix=V~1x1+V~2x2+V~3x3+Ax1x2x3, where A is a function of temperature, and the terms V1,V2, and V3 represent the molar volumes of pure components 1, 2, and 3, respectively. Use Eqs. (6)-(8) and show that the partial molar volumes of components 1, 2, and 3 are expressed as a function of composition by the following equations: V1=V1+Ax2x3(12x1) 7.3 Consider a k-component mixture in which n represents the total number of moles of material. (a) Show that the substitution of mix=n~mix Thermodynamics of mixtures 205 into Eq. (7.1-5) gives i=~mix+n(ni~mix)T,P,nj=i (b) Use the chain rule of differentiation and express Eq. (2) as where the subscript xj=i,r indicates that mole fractions of all components other than i and r are kept constant. (c) Noting that nixr=n2nr show that Eq. (3) takes the form i=mixr=1r=ikxr(xr~mix)T,P,xj=i,r (d) For a ternary system show that 1=~mixx2(x2~mix)x3x3(x3~mix)x22=mixx1(x1mix)x3x3(x3mix)x13=mixx1(x1~mix)x2x2(x2mix)x1 (e) For a ternary system of components 1, 2, and 3, the molar volume of the mixture is given by V~mix=V~1x1+V~2x2+V~3x3+Ax1x2x3, where A is a function of temperature, and the terms V1,V2, and V3 represent the molar volumes of pure components 1, 2, and 3, respectively. Use Eqs. (6)-(8) and show that the partial molar volumes of components 1, 2, and 3 are expressed as a function of composition by the following equations: V1=V1+Ax2x3(12x1)