Given \(C_{(graphite)} +O_2(g) \rightarrow CO_2(g)\,;\) \(\Delta _rH^o=-395.5 \, kJ\,mol^{-1}\) \(H_2 (g) + \frac{1}{2} O_2 (g) \rightarrow H_2O(l)\,;\)   \(\Delta _rH^o  =-285.8\, kJ\, mol^{-1}\) \(CO_2(g) + 2H_2O(l) \rightarrow CH_4(g) + 2O_2(g)\,;\) \(\Delta _rH^o  = + 890.3\, kJ\, mol^{-1}\) Based on the above thermochemical equations, the value of, \(\Delta _rH^o \) at \(298\, K\) for the reaction \(C_{(graphite)} + 2H_2(g) \rightarrow CH_4(g) \) will be for \(\Delta_{r} H^{\circ}\) ........... \(kJ \,mol^{-1}\)

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.