Hess's law to free energy to the equilibrium constant
SCH4U Grade 12 Chemistry · Energy Changes and Rates of Reaction
Methanol is made industrially from synthesis gas: CO(g) + 2H₂(g) → CH₃OH(g). The reaction cannot be run cleanly in a bomb calorimeter, so its enthalpy is routed through three steps that can: burning carbon monoxide, −283.0 kJ/mol; burning hydrogen to steam, −483.6 kJ/mol; and the reversed combustion of methanol, +676.5 kJ/mol. The reaction's entropy change is −219.0 J/(mol·K). Find the enthalpy of the target reaction, its free energy change in a 500 K reactor, and the equilibrium constant that implies.
Enthalpy is a state function, so any route that starts and ends where the target reaction does gives its ΔH. The third value is positive because that step is a combustion run backwards, and reversing a step flips its sign.
Carried onward at full precision, not this rounded figure.
Enthalpy alone cannot say whether the reaction runs. Three moles of gas collapse into one, so the entropy term is strongly negative — and at 500 K it is large enough to overturn the favourable enthalpy.
Carried onward at full precision, not this rounded figure.
The exchange rate between thermodynamics and equilibrium. A positive ΔG° at this temperature must give K < 1, and it does — barely one part in a hundred.
Carried onward at full precision, not this rounded figure.
Open the Gibbs Free Energy and the Equilibrium Constant solver →
Why this order
The order is forced by what each equation can see. Hess's law knows only about enthalpy and says nothing about whether anything happens; the Gibbs equation adds the entropy and the temperature and answers the spontaneity question; ΔG° = −RT ln K then converts that answer into a number an equilibrium table can be compared against. Skipping the middle step is the classic error — a strongly exothermic reaction feels like it must go, and here it does not, because turning three moles of gas into one is an entropic catastrophe that 500 K amplifies into +19.4 kJ/mol.
Watch the prefixes. Enthalpies are tabulated in kJ/mol and entropies in J/(mol·K), a factor of a thousand apart, and a student who subtracts 500 × 219 from 90.1 without converting concludes the reaction is impossibly endothermic. This site keeps molar energy canonically in J/mol and gives entropy its own unit type so the mismatch shows on screen rather than hiding in a placeholder. The chemistry is real, too: K ≈ 0.0094 at 500 K is why methanol plants run at 50 to 100 atmospheres. Le Chatelier's principle says squeezing three moles of gas into one rewards pressure, and the pressure is what makes an unfavourable equilibrium into a profitable one.
Carried values move at full precision, not the rounded figure shown — chaining rounded numbers compounds error.