Fuel oils contain primarily organic compounds and sulfur. The molar composition of the organic fraction of a fuel oil may be represented by the formula CpHqOr; the mass fraction of sulfur in the fuel is xS $($kg S$/$kg fuel$)$; and the percentage excess air, Pxs$,$ is defined in terms of the theoretical air required to burn only the carbon and hydrogen in the fuel.

$($a$)$ For a certain high$-$sulfur No$.6$ fuel oil, p $0$:$71,$ q $1$:$1,$ r $0$:$003,$ and xS $0$:$02.$ Calculate the composition of the stack gas on a dry basis if this fuel is burned with $18\%$ excess air, assuming complete combustion of the fuel to form CO$2,$ SO$2,$ and H$2$O and expressing the SO$2$ fraction as ppm $($mol SO$2/106$ mol dry gas$).$