In a sulfuric acid plant, sulfur is burned in the presence of excess oxygen to produce sulfur dioxide which in turn is further reacted in the next step with oxygen in a converter to produce sulfur trioxide.

In the plant $S{O}_2$ along with $10\%$ excess air is fed into the converter which operates at $1500C$ and $1$atm. The per pass conversion of $S{O}_2$ is $75\%$ and overall conversion is $100\%.$ If ${10}^6\frac{m^3}{h}r$ of $S{O}_3$ at $1100C$ and $1$atm is fed to the converter, calculate the:

$($a$)$ flow rate of the product stream $P$ in $\frac{m^3}{h}r$ at $1500C$ and $1$atm and its composition in mole percent;

$($b$)$ flow rate of the recycle stream $R$ in $\frac{m^3}{h}r$ at $1500C$ and $1$atm.