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Gibbs-Duhem Equation

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In a certain binary solution, the activity of component 1 is given by

R * ln(alpha_1) = R * ln(x_1) + A(x_2)^2 + B(x_2)^3

Where x_1 and x_2 are the respective mole fractions and A and B are constants. Derive an expression for alpha_2 (the activity of component 2) given that the equation above is valid over the entire concentration range. (Your answer should be in terms of R, A, B x_1, and x_2). A useful form of the Gibbs-Duhem Equation is

X1 * d * ln(alpha_1) + (x_2) * d * ln(alpha_2) = 0

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Solution Preview

In the case of a two component system X1 + X2 = 1 such that
dx1/dx2 = d(1-x2)/dx2 = -1

now d(ln a1) = partial(lna1/wrx1) dx1 + partial( lna1)wrx2)dx2

holding x2 constant: partial(lna1/wrx1) = d/dx1(ln x1) = 1/x1
holding x1 constant partial( lna1)wrx2) = 2A/R x2 + ...

Solution Summary

This complete solution identifies steps involved and includes calculations. 189 words.