Answer to Question #213412 in General Chemistry for khola2709

Question #213412

Find an expression for activity coefficient from the  Two-suffix- Margules equation for excess Gibbs energy.

The expression for Two-suffix-Magules Equation is given by :

"G^E=Ax_1x_2"


1
Expert's answer
2021-07-05T03:22:51-0400

Given expression for two-suffix-magules equation is given by-


"\\dfrac{G^{E}}{RT}=AX_1X_2"


,"where \\ RT=constant" "and\\ X_1\\ andX_2\\ are \\ the fractions\\ of \\ each \\ component."



Now portion molar excess gibbs free energy is given by the equation -


"G_i^{E}=RTln\\gamma_{i}"


The above equation can also be written as-


"\\dfrac{G_{i}}{RT}=ln\\gamma_{i}"


now by definition of extensive partial molar property -


"ln\\gamma_{i}=(\\dfrac{\\partial({\\dfrac{nG^{E}}{(RT)}})}{\\partial{n}_{i}})_T,_P,n_{j}" (partial molar excess gibbs free energy)


"ln\\gamma_{1}=(\\dfrac{\\partial({\\dfrac{nG^{E}}{(RT)}})}{\\partial{n}_{1}})_T,_P,n_{1}"



Now, by expression for - two - suffix- magules -


"\\dfrac{G^{E}}{RT}=AX_1X_2" "X_{1}=\\dfrac{n_{1}}{n_{1}+n_{2}}" "=\\dfrac{n_1}{n}"



"X_{2}=\\dfrac{n_{2}}{n_{1}+n_{2}}" "=\\dfrac{n_{2}}{n}"



Now putting "X_1,X_2\\" in the above equation we get


"\\dfrac{G^{E}}{RT}=AX_1X_2"



"\\dfrac{G^{E}}{RT}=A(\\dfrac{n_{1}}{n_{1}+n_{2}})(\\dfrac{n_{2}}{n_{1}+n_{2}})" "=A\\dfrac{n_{1}n_{2}}{(n_1+n_2)^{2}}"



now multiplying by n on both side of the , equation we get -


"n\\dfrac{G^{E}}{RT}=(n_1+n_2)A(\\dfrac{n_{1}}{n_{1}+n_{2}})(\\dfrac{n_{2}}{n_{1}+n_{2}})" "=\\dfrac{An_1n_2}{n_1+n_2}"


"ln\\gamma_{1}=(\\dfrac{\\partial({\\dfrac{nG^{E}}{(RT)}})}{\\partial{n}_{i}})_T,_P,n_{2}"



"ln\\gamma_{i}=(\\dfrac{\\partial({\\dfrac{An_1n_2}{n_1+n_2}})}{\\partial n_1})_T,_P,_{N_J}"



"ln\\gamma_{i}=An_2(\\dfrac{\\partial({\\dfrac{n_1}{n_1+n_2}})}{\\partial n_1})_{n_2}"



"ln\\gamma_{i}=A(\\dfrac{n_2^{2}}{n_1+n_2})"


"ln\\gamma_{i}=A(\\dfrac{n_2^{2}}{n})"


"ln\\gamma_{i}=A(X_2^{2})"


"ln\\gamma_{i}=A(X_1^{2})"


so this is our equation .
















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