Hess's Law

Introduction

Some reaction enthalpies are difficult to measure directly, but related reactions are known. Hess's law lets us combine their enthalpy changes to find the target value. The law works because enthalpy is a state function: when initial and final chemical states match, the sum is the same regardless of the imagined route.

Core explanation

Suppose A → B has enthalpy ΔH₁ and B → C has enthalpy ΔH₂. Adding the equations cancels B and gives A → C with ΔH = ΔH₁ + ΔH₂. The intermediate B appears once as a product and once as a reactant in the same physical state, so it disappears from the overall equation. This is an accounting identity for enthalpy differences, not a claim that the direct reaction literally follows those two experimental steps.

The operations are precise. Reversing a known reaction changes the sign of its ΔH. Multiplying every coefficient by a factor multiplies ΔH by the same factor. Adding reactions adds their enthalpies. Before doing arithmetic, manipulate equations until their left and right sides sum to the desired target; then verify that all unwanted species cancel. If the summed chemical equation is wrong, the enthalpy sum cannot be trusted.

A classic carbon example uses C(s, graphite) + O₂(g) → CO₂(g) and CO(g) + ½O₂(g) → CO₂(g). Reverse the second and add it to the first. CO₂ cancels, leaving C(s, graphite) + ½O₂(g) → CO(g). Its enthalpy is the first known enthalpy minus the second. This determines a formation enthalpy for CO that may be harder to measure by direct partial oxidation because CO₂ can also form.

Physical-state labels must match during cancellation. H₂O(l) on one side will not cancel H₂O(g) on the other without adding a vaporization or condensation step. Likewise, graphite and diamond are different starting states. Hess's law is not a license to erase chemically identical formulas with different phases or reference conditions.

The law follows from energy conservation. If two routes between the same states had different total enthalpy changes, completing one route and reversing the other would form a cycle with nonzero net state-function change, contradicting the fact that a cycle returns to its initial state. Heat transferred along particular paths can differ, but their constant-pressure reaction enthalpy changes between matched states must agree.

Hess calculations underpin formation-enthalpy formulas, Born–Haber cycles and phase-change relations. Their reliability depends on data from compatible temperatures, pressures and states. A result should include the target balanced reaction and units per that reaction as written.

Step-by-step reasoning

1. Write the target reaction prominently with physical states. 2. Inspect known equations for target reactants and products. 3. Reverse or scale known equations to place species correctly. 4. Add equations and cancel identical intermediates. 5. Apply the same reversals and scaling to ΔH, then sum.

Visual explanation

Draw a triangle with A, B and C at corners. Put ΔH₁ on A→B, ΔH₂ on B→C and ΔH target on A→C. A highlighted route through B shows ΔH target = ΔH₁ + ΔH₂. Beneath it write the chemical equations stacked vertically with B struck out on both sides.

Real-world analogy

Elevation change between two locations is the same whether one walks directly or via a third town. Adding the two leg changes gives the direct change. Enthalpy behaves similarly as a state function, provided the exact starting and ending states match.

Real-world example

The enthalpy of forming CO from graphite can be inferred from complete carbon combustion and CO combustion. The indirect route avoids relying on a reaction that may produce both CO and CO₂ in an uncontrolled mixture.

Why?

Why may an experimentally impossible step appear in a Hess cycle? The cycle is a mathematical path between states. Enthalpy differences add even if a particular step is only hypothetical, provided its value is known or reliably inferred.

Common misconception

“If two species have the same formula, they cancel even when phases differ.” H₂O(l) and H₂O(g) are different states with a vaporization enthalpy between them; they cannot be silently canceled.

Worked example

Let C(graphite) + O₂ → CO₂ have ΔH = −393.5 kJ, and CO + ½O₂ → CO₂ have ΔH = −283.0 kJ. Reverse the second: CO₂ → CO + ½O₂, ΔH = +283.0 kJ. Add to the first and cancel CO₂ and half the oxygen: C + ½O₂ → CO. The target ΔH = −393.5 + 283.0 = −110.5 kJ per reaction as written.

Quick check

1. What happens to ΔH when a known equation is reversed? Answer: Its sign changes while the magnitude remains the same for the same states and conditions.

Exam focus

Manipulate chemical equations before manipulating numbers. Show cancellation with phases visible, then carry the same scaling to enthalpies. State the target equation with the final value.

Advanced insight

Hess's law reflects the mathematical exactness of enthalpy differences. It extends to entropy and Gibbs energy changes between matched states, although the experimentally accessible data and standard-state conventions differ. A thermochemical network can be solved as a system of linear equations in reaction extents.

Summary

Hess's law adds reaction enthalpies along any route connecting the same initial and final states. Reverse, scale and add equations consistently, and cancel only identical species in identical states. The final enthalpy belongs to the resulting balanced reaction.

Practice questions

1. A→B is +20 kJ and B→C is −35 kJ. Find A→C. Answer: +20 + (−35) = −15 kJ for the combined reaction. 2. What is ΔH for 2B→2A if A→B is +10 kJ? Answer: −20 kJ: reverse sign and double the equation. 3. Why can H₂O(l) not cancel H₂O(g) in a Hess sum? Answer: They are different physical states separated by a nonzero phase-change enthalpy.