For a binary mixture, theta_bar = 6x_1, x_2, where theta_bar is some molar property of the mixture and x_1 is the mole fraction of component i. Derive an expression for theta^bar _1 the partial molar property of component 1. The Gibbs-Duhem equation can be expressed for any partial molar property theta^bra_i at constant T and P as sigma_i = 1^C x_i d theta^bar_i = 0. For a binary mixture, prove that: theta^bra_1 = theta_bar + x_2 d theta_bat/dx_1 and theta^bar_2 = theta_bar + x_1 dtheta_bar/dx_2. For a ternary solution at constant T and P, the molar property of mixture theta_bar can be expressed in terms of the molar fractions and pure species properties of each component as theta_bar = x_1theta_bar1 + x_2 theta_bar_2 + x_3 theta_bar_3 + x_1x_2x_3 C, where C is a constant. Develop expressions for the partial molar properties, theta^bar_1, theta^bar_2 and tehta^bar_3 in terms of molar fractions and pure species properties. What is the partial molar property when component 1 is infinitely dilute? In a chemical plant for formaldehyde production, you require 40% methanol solution in water. You attempt to make 1000.0 L of the solution by exactly mixing 400.0 L of methanol and 600.0 L of water. Using the graph below, what is the volume of the resulting mixture? (Liquid A is water and Liquid B is methanol). As the exact volumes are important for chemical reactions and their yield, how much methanol and water should you mix to achieve a methanol solution that is the same molar concentration of 40.0%?