$($a$)$ Making use of Eqs. $(10\mathrm{.}15)$ and $(10\mathrm{.}16),$ written for excess properties, show for a binary system that:

$(M(bar){)}_1^E=x_2^2(X+x_1\frac{dX}{d{x}_1}),$ and $(M(bar){)}_2^E=x_1^2(X-x_2\frac{dX}{d{x}_1})$

where

$X=\frac{M^E}{x_1{x}_2}$

$[10\mathrm{.}15]$ : $M(bar{)}_1=M+x_2\frac{dM}{d{x}_1}$

$[10\mathrm{.}16]$ : $M(bar{)}_2=M-x_1\frac{dM}{d{x}_1}$

$($b$)$ Plot on a single graph the values of $\frac{H^E}{x_1{x}_2},(H(bar){)}_1^E,$ and $(H(bar){)}_2^E$ determined from the following heat$-$of$-$mixing data for the $H_{2}S{O}_4\frac{1}{H_2}O(2)$ system at $25C$ :

$---------------------------------$

$x_1-$delta$\frac{H}{k\frac{j}{k}g}$

$0\mathrm{.}1073\mathrm{.}27$

$0\mathrm{.}20144\mathrm{.}21$

$0\mathrm{.}30208\mathrm{.}64$

$0\mathrm{.}40262\mathrm{.}83$

$0\mathrm{.}50302\mathrm{.}84$

$0\mathrm{.}60323\mathrm{.}31$

$0\mathrm{.}70320\mathrm{.}98$

$0\mathrm{.}80279\mathrm{.}58$

$0\mathrm{.}85237\mathrm{.}25$

$0\mathrm{.}90178\mathrm{.}87$

$0\mathrm{.}95100\mathrm{.}71$

$---------------------------------$

Explain with reference to these plots why sulfuric acid is diluted by adding acid to water rather than water to acid.