Hess's Law

Key Ideas

• State-function consequence: ΔH_total = ΔH_1 + ΔH_2 + ΔH_3 + … for any valid sequence of steps.
• Reverse a reaction → flip the sign of ΔH. If A → B has ΔH = +x, then B → A has ΔH = −x.
• Multiply a reaction by a factor → multiply ΔH by the same factor. ΔH scales with the amount of substance.
• Add reactions → add their ΔH values. Intermediates that appear on both sides cancel out.
• Goal: arrange the given equations so that, after cancellation, only the target reaction's reactants and products remain.
• Common target reactions: forming CO from C and O₂ (partial oxidation), forming a compound from its elements, or an overall multi-step synthesis.

Equations and Variables

• ΔH_rxn = Σ ΔH_steps
• Reverse: ΔH → −ΔH
• Multiply by n: ΔH → n·ΔH
• Add: ΔH_sum = ΔH_A + ΔH_B

How It Works

Think of enthalpy like altitude: the change in altitude from the base of a mountain to its peak is the same whether you climb the direct trail, take a winding road, or fly a helicopter. Only the start and end points matter. Hess's law applies the same logic to chemical reactions — you may "route" a reaction through any intermediate compounds as long as you end at the correct products.

Procedure:

1. Write the target equation.
2. List the given equations with their ΔH values.
3. Reverse any equation whose reactant/product appears on the "wrong" side of the target; flip its ΔH sign.
4. Multiply any equation so its coefficients match the target; scale its ΔH.
5. Add the equations, canceling identical species on opposite sides.
6. Add the ΔH values to get ΔH_rxn.

Worked Example

Find ΔH for C(s) + ½O₂(g) → CO(g), given:

(1) C(s) + O₂(g) → CO₂(g), ΔH = −393.5 kJ (2) CO(g) + ½O₂(g) → CO₂(g), ΔH = −283.0 kJ

Step 1 — Target has C(s) and ½O₂ as reactants, CO(g) as the sole product. CO₂ must cancel.

Step 2 — Keep equation (1) as written (C on the reactant side, correct):

C(s) + O₂(g) → CO₂(g), ΔH = −393.5 kJ

Step 3 — Equation (2) has CO(g) as a reactant, but we need CO(g) as a product. Reverse it and flip the sign:

CO₂(g) → CO(g) + ½O₂(g), ΔH = +283.0 kJ

Step 4 — Add the two equations:

C(s) + O₂(g) + CO₂(g) → CO₂(g) + CO(g) + ½O₂(g)

CO₂ cancels (one on each side); O₂ on the left minus ½O₂ on the right leaves ½O₂ on the left:

C(s) + ½O₂(g) → CO(g)

Step 5 — Add the enthalpies:

ΔH = (−393.5 kJ) + (+283.0 kJ) = −110.5 kJ

This is exactly the standard enthalpy of formation of carbon monoxide, −110.5 kJ/mol — a value that is hard to measure directly because burning carbon tends to over-oxidize to CO₂, so Hess's law is how it is determined.

Common Confusions

• Forgetting to flip the sign when reversing. This is the most frequent error. Reversed reaction = reversed sign, always.
• Forgetting to scale ΔH when changing coefficients. If you double an equation to balance it, you must double its ΔH.
• Canceling incorrectly. Only cancel species that are identical and on opposite sides. If one side has 2O₂ and the other has ½O₂, they don't fully cancel.
• Treating ΔH as temperature. ΔH is a heat quantity with a sign; don't confuse "adding equations" with arithmetic on temperatures.
• Assuming the direct reaction can't be measured. Hess's law is often introduced with measurable steps but is most useful for unmeasurable targets (like CO formation).