A $0\mathrm{.}100$ mol sample of propane, a gas used in portable gas cookers, was placed in a bomb calorimeter with excess oxygen and ignited. The reaction was

C$3$H$8($g$)+5$ O$2($g$)3$ CO$2($g$)+4$ H$2$O$($l$)$

The water bath surrounding the calorimeter contained $1\mathrm{.}000$ kg of water at an initial temperature of $25\mathrm{.}000$ oC$.$ The heat capacity of the bomb calorimeter was $93\mathrm{.}1$ kJ K$-1.$ The reaction raised the temperature of the water bath to $27\mathrm{.}282$ oC$.$ The specific heat of water is $4\mathrm{.}18$ J g$-1$ K$-1$

$($a$)$ How much heat was liberated in this reaction?

$($b$)$ What is the enthalpy of reaction of propane with oxygen?