Hess cycles

A Hess cycle is a diagram that finds an enthalpy change you can't measure directly, by going round it a different way. It uses Hess's law: the total enthalpy change for a reaction is the same whichever route you take from the reactants to the products. You add the enthalpy changes along the arrows you follow, and subtract the ones you go against.

What is Hess's law?

OpenStax Chemistry states it this way: if a process can be written as the sum of several steps, the enthalpy change of the total process equals the sum of the enthalpy changes of the various steps. It follows from conservation of energy. If two routes from the same start to the same finish had different enthalpy changes, you could go out one way and back the other and create energy from nothing.

AQA's A-level specification asks you to use Hess's law to calculate enthalpy changes for reactions from enthalpies of combustion or from enthalpies of formation. Both use the same kind of cycle.

How to draw a Hess cycle

1. Write the reaction you want across the top, reactants on the left and products on the right, with an arrow labelled ΔrH (or ΔfH, ΔcH, whatever you are finding).
2. Below, write the substances that both sides can be turned into: the elements for a formation cycle, or the combustion products (CO₂ and H₂O) for a combustion cycle.
3. Draw arrows between the top and the bottom, and label each with its enthalpy change, multiplied by the number of moles.
4. Choose two routes between the same start and end, and set their totals equal.

Hess cycles with enthalpies of combustion

Use this when you have combustion data. Everything at the top can be burned to the same products, so the arrows point down to CO₂ and H₂O.

Example 1: the enthalpy of formation of ethanol

Find ΔfH of ethanol, 2C(s) + 3H₂(g) + ½O₂(g) → C₂H₅OH(l), using these standard enthalpies of combustion from OpenStax Chemistry: C(s) −393.5, H₂(g) −285.8 and C₂H₅OH(l) −1366.8 kJ mol⁻¹. (Oxygen doesn't burn, so it has no combustion enthalpy.)

1. Burning the elements: 2 × (−393.5) + 3 × (−285.8) = −787.0 − 857.4 = −1644.4 kJ.
2. Route 1, straight down from the elements: −1644.4 kJ.
3. Route 2, along the top then down: ΔfH + (−1366.8).
4. The routes are equal: ΔfH − 1366.8 = −1644.4, so ΔfH = −277.6 kJ mol⁻¹.

You can't measure this directly, because carbon, hydrogen and oxygen won't react cleanly to make ethanol. The cycle gets round that.

Hess cycles with enthalpies of formation

Use this when you have formation data. Every compound at the top can be made from its elements, so the arrows point up from the elements.

Elements in their standard states, such as O₂(g) or C(s), have ΔfH = 0.

Example 2: burning ethanol

Check the combustion enthalpy of ethanol from formation data: ΔfH of C₂H₅OH(l) is −277.6, CO₂(g) is −393.5 and H₂O(l) is −285.8 kJ mol⁻¹. The reaction is C₂H₅OH(l) + 3O₂(g) → 2CO₂(g) + 3H₂O(l).

• Products: 2 × (−393.5) + 3 × (−285.8) = −1644.4 kJ.
• Reactants: −277.6 + 3 × 0 = −277.6 kJ.
• ΔrH = −1644.4 − (−277.6) = −1366.8 kJ mol⁻¹, matching the measured value.

Note the order of subtraction: products minus reactants for formation data, but reactants minus products for combustion data. Drawing the cycle stops you mixing them up.

Hess's law with given equations

Sometimes you are given equations instead of a table. Rearrange them so they add up to the reaction you want: reversing an equation changes the sign of its ΔH, and multiplying it multiplies ΔH.

Example 3: carbon monoxide

Burning carbon in a limited supply of oxygen always makes some CO₂ too, so the enthalpy change for C(s) + ½O₂(g) → CO(g) can't be measured directly. Use:

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

Keep the first and reverse the second, CO₂(g) → CO(g) + ½O₂(g), ΔH = +283.0. Adding them, the CO₂ cancels, leaving C(s) + ½O₂(g) → CO(g) with ΔH = −393.5 + 283.0 = −110.5 kJ mol⁻¹.

The enthalpy calculator checks Hess's law answers from formation data, combustion data or a set of equations, and the enthalpy of combustion guide explains where combustion values come from. The Born-Haber cycle is a bigger Hess cycle for ionic compounds.

Common mistakes

• Forgetting to multiply by the moles. Three moles of water means 3 × ΔfH(H₂O).
• Subtracting the wrong way round. Formation data: products minus reactants. Combustion data: reactants minus products.
• Giving elements a value. ΔfH of an element in its standard state is zero, and oxygen has no combustion enthalpy.
• Not flipping the sign when you go against an arrow or reverse an equation.
• Using the wrong state. ΔfH of H₂O(l) and H₂O(g) are different; use the one in the equation.

To practise energetics in a short timed paper, try today's A-level Chemistry: Year 1 paper.

Read next

Chemistry · EnergeticsEnthalpy calculator: ΔH by Hess's law or bond enthalpiesFind the enthalpy change of a reaction from enthalpies of formation or combustion, combined equations or bond enthalpies, with working and worked examples.