$(25$ pts$)$ Suppose you prepare a solution that is $30\mathrm{.}0wt\%H_{2}S{O}_4$ in water and intend to confirm the concentration by comparing the specific gravity of the solution to a value obtained from the literature. You carefully add $30\mathrm{.}0g$ of $H_{2}S{O}_4$ to $70\mathrm{.}0g$ of water $($you know that rapidly adding acid to water or water to concentrated acid can lead to dangerous splatters of liquid$)$ and allow the resulting mixture to equilibrate at $20C.$

$($a$)$ Estimate the volumes of water and sulfuric acid that have been blended, and then estimate the density of the $30\mathrm{.}0wt\%$ sulfuric acid solution, assuming the volumes are additive. $SG$ of Sulfuric Acid $=1\mathrm{.}834$;$,SG$ of $H_{2}O=1\mathrm{.}0$;${}_{H_{2}O}=1\mathrm{.}0\frac{g}{ml}$

$($b$)$ Perry'sChemicalEngineers'Handbook $($p$\mathrm{.}2-112)$ gives the specific gravity of sulfuric acid as a function of both acid concentration and temperature. The value for $30\mathrm{.}0wt\%$ sulfuric acid at $20C$ is given as $1\mathrm{.}2185.$ Estimate the volume of the $30\mathrm{.}0wt\%$ solution prepared as described by the above process.

$($c$)($bonus $10pts$ What volume of $40\mathrm{.}0wt\%$ sulfuric acid at $)=(1\mathrm{.}2991$ must be added to $100\mathrm{.}0g$ of the $30\mathrm{.}0-wt\%$ solution to produce a solution that is $35\mathrm{.}0wt\%$ sulfuric acid?