Answer
C$H_{4}$ + 2$O_{2}$ < C$O_{2}$ + 2$H_{2}$O
$\Delta$G$^{\circ}$= -817 kJ/$mol_{rxn}$
Work Step by Step
Given:
2$H_{2}$ + C < C$H_{4}$/ $\Delta$G$^{\circ}$= -51 kJ
2$H_{2}$ + $O_{2}$ < 2 $H_{2}$O/ $\Delta$G$^{\circ}$= -474 kJ
C + $O_{2}$ < C$O_{2}$/ $\Delta$G$^{\circ}$= -394 kJ
Algebraically Fixed Equations so Undesired Components Cancel Out (Add all $\Delta$G$^{\circ}$ values to get the final $\Delta$G$^{\circ}$) :
C$H_{4}$ < 2$H_{2}$ + C/ $\Delta$G$^{\circ}$= +51 kJ
2$H_{2}$ + $O_{2}$ < 2 $H_{2}$O/ $\Delta$G$^{\circ}$= -474 kJ
C + $O_{2}$ < C$O_{2}$/ $\Delta$G$^{\circ}$= -394 kJ
C$H_{4}$ + 2$O_{2}$ < C$O_{2}$ + 2$H_{2}$O
$\Delta$G$^{\circ}$= -817 kJ/$mol_{rxn}$
