Apply Hess's law with energy cycles
Construct enthalpy cycles and use path independence to find reaction changes that are difficult to measure directly.
Section 5 · Chemical energetics
AS Level; Cambridge International Chemistry 9701, 2025–2027 syllabus. Report signs, states, units and significant figures consistently. Standard conditions use 298 K and 101 kPa in this syllabus. Written practice is self-checked
Before you begin
- Rearrange a formula, balance a chemical equation and use temperature differences.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
What does Hess's law say about two routes between the same initial and final states?
What you will learn
- State and apply Hess's law
- Construct simple energy cycles
- Reverse and scale enthalpy equations consistently
Enthalpy change is independent of the route
Hess's law states that the total enthalpy change is the same whichever route is taken between the same initial and final states, provided conditions and states are consistent.
Represent the initial and final substances at the bottom of a cycle and connect them with alternative routes. Since the overall change is path-independent, add known step enthalpies algebraically to obtain an unknown route.
Equation direction and coefficients control ΔH
Reversing a thermochemical equation reverses the sign of ΔH. Multiplying every coefficient by a factor multiplies ΔH by that factor. Add equations so intermediate substances cancel and the target equation remains.
Use physical states and standard conditions consistently. The target enthalpy is for the balanced reaction as written; check that the final algebra reproduces that equation exactly.
Pause and explain
What happens to ΔH when a thermochemical equation is reversed?
Worked example
C(s) + O₂(g) → CO₂(g), ΔH = −393.5 kJ mol⁻¹. CO(g) + ½O₂(g) → CO₂(g), ΔH = −283.0 kJ mol⁻¹. Find ΔH for C(s) + ½O₂(g) → CO(g).
Show the worked solution
- Reverse the CO combustion equation: CO₂ → CO + ½O₂, ΔH = +283.0.
- Add it to carbon combustion; CO₂ and half O₂ cancel appropriately.
- Add enthalpy changes: −393.5 + 283.0.
Answer ΔH = −110.5 kJ mol⁻¹.
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Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
A(s)+O₂(g)→AO₂(g), ΔH=−500 kJ mol⁻¹. B(s)+O₂(g)→BO₂(g), ΔH=−400 kJ mol⁻¹. Find ΔH for A(s)+BO₂(g)→AO₂(g)+B(s).
Give me a hint
Use A combustion forward and B combustion in reverse.
Compare my reasoning
- A(s)+O₂→AO₂ has ΔH=−500.
- Reverse B combustion: BO₂→B+O₂ has ΔH=+400.
- Add and cancel O₂.
ΔH = −100 kJ mol⁻¹.
Look for these in your work
- Equation directions correct
- Reversed sign applied
- Intermediate oxygen cancelled
A known reaction has ΔH=−85 kJ mol⁻¹. Find ΔH if its equation is reversed and then multiplied by two.
Give me a hint
Reverse first, then scale the enthalpy with the coefficients.
Compare my reasoning
- Reversal changes −85 to +85.
- Doubling the equation doubles ΔH.
- Apply the factor to the positive value.
ΔH = +170 kJ mol⁻¹.
Look for these in your work
- Sign reversal correct
- Coefficient scaling correct
- Final sign and magnitude correct
Two routes from the same reactants to the same products give totals −120 kJ mol⁻¹ and +35 kJ mol⁻¹. What does this tell you about the proposed cycle?
Give me a hint
Hess's law requires path independence for the same endpoints and conditions.
Compare my reasoning
- The total enthalpy changes should be equal.
- The endpoint states or equation directions may not match.
- Check signs, coefficients and intermediate cancellation.
The cycle is inconsistent as written: paths between identical endpoints under the same conditions must have equal total ΔH. Recheck equations and signs.
Look for these in your work
- Path independence applied
- Inconsistency identified
- Useful checks named
Common mistakes
- Keeping the same sign after reversing an equation
- Changing an equation coefficient without scaling ΔH
- Failing to cancel intermediates
- Combining equations that do not reproduce the target reaction
What can you explain now?
What does Hess's law say about two routes between the same initial and final states?
Compare with the explanation
Their total enthalpy changes are equal
Enthalpy change depends on endpoints, not route.
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Make it stick
How do equation reversal, scaling and cancellation let you use Hess's law?
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.