(Solved): a) For a binary mixture of species 1 and 2 at constant temperature and pressure, the summability r ...

a) For a binary mixture of species 1 and 2 at constant temperature and pressure, the summability relation of total properties from partial properties becomes: $$M=x_1\overline{M_1}+x_2\overline{M_2}$$ From this relationship, show that $$\overline{{\mathrm{M}}_1}=\mathrm{M}+{\mathrm{x}}_2\frac{\mathrm{dM}}{\mathrm{d}{\mathrm{x}}_1}$$ and $$\overline{{\mathrm{M}}_2}=\mathrm{M}?{\mathrm{x}}_1\frac{\mathrm{dM}}{\mathrm{d}{\mathrm{x}}_1}$$ (Hint: Start by taking a differential of both sides of the summability relation) b) The molar volume $$\left({\mathrm{cm}}^3/\mathrm{mol}\right)$$ of a binary liquid mixture at temperature $$\mathrm{T}$$ and pressure $$\mathrm{P}$$ is given by: $$\mathrm{V}=7{\mathrm{x}}_1{}^2?16{\mathrm{x}}_2{}^2+2{\mathrm{x}}_1+30{\mathrm{x}}_2$$ Find expressions for the partial molar volumes of species 1 and 2 in terms of $$x_1$$ at $$T$$ and $$P$$ using the equations from a). c) Show that these expressions satisfy the Gibbs-Duhem equation.