For the cracking reaction, C$3$ H$8($g$)$ C$2$ H$4($g$)+$ CH$4($g$)$ the equilibrium conversion is negligible at $300$ K$,$ but it becomes appreciable at temperatures above $500$ K$.$ For a pressure of $1$ bar, determine:

$($a$)$ The fractional conversion of propane at $600$ K$.$

$($b$)$ The temperature at which the fractional conversion is $80\%.$