Use the GibbsDuhem equation to show that the partial molar volume (or any partial molar

Chapter 5, Problem 5.19

(choose chapter or problem)

Use the Gibbs-Duhem equation to show that the partial molar volume (or any partial molar property) of a component \(\mathrm{B}\) can be obtained if the partial molar volume (or other property) of A is known for all compositions up to the one of interest. Do this by proving that

\(V_{\mathrm{B}}=V_{\mathrm{B}}^*-\int_{V_{\mathrm{A}}^*}^{V_{\mathrm{A}}} \frac{x_{\mathrm{A}}}{1-x_{\mathrm{A}}} \mathrm{d} V_{\mathrm{A}}\)

Use the following data (which are for \(298 \mathrm{~K}\) to evaluate the integral graphically to find the partial molar volume of acetone at x=0.500.

\(\begin{array}{llllllll}x\left(\mathrm{CHCl}_3\right) & 0 & 0.194 & 0.385 & 0.559 & 0.788 & 0.889 & 1.000 \\ V_{\mathrm{m}} /\left(\mathrm{cm}^3 \mathrm{~mol}^{-1}\right) & 73.99 & 75.29 & 76.50 & 77.55 & 79.08 & 79.82 & 80.67\end{array}\)