For water $(1)+$ ethanol $(2)$ system, its molar volume change of mixing $({}_{\text{m}\text{i}\text{x}}$ V$)$ at $300K$ can be expressed by the following equation :

${}_{\text{m}\text{i}\text{x}}{V}_{c\frac{m^3}{m}o{l}_1}=-x_1{x}_2[5+0\mathrm{.}5(x_1-x_2)]$

where $x_1$ and $x_2$ are the mole fractrons of water and ethand, respectrvely. At $300k,$ the density of pure Now, we try to prepare $100c{m}^3$ of water $+$ ethanol solution with the molar ratio of water: ethanol $=1$:$1$ at $300K.$

$($a$)$ Calculate the density of this agueous solution.

$($b$)$ How many grams of water and ethanol are needed, respectrvely, to prepare the solution?

$($c$)$ Calculate the partial molar volume of ethand for this solutron.

$($d$)$ Calculate the partial molar volume of ethanol at infinile drlutron.