The vapour pressure of water at $50\mathrm{.}0C$ is $92\mathrm{.}50$ torr. When $12\mathrm{.}6g$ of glycerol are dissolved in $494g$ of water, the vapour pressure is lowered to $92\mathrm{.}04$ torr.

$($a$)$ What is the molar mass of glycerol?

$($b$)$ What is the molecular formula of glycerol $($a compound containing $C,H,$ and $O)$ from the following combustion analysis: when $2\mathrm{.}00g$ glycerol are combusted, $2\mathrm{.}865gC{O}_2$ and $1\mathrm{.}564g{H}_{2}O$ are recovered.