# Energy cycles

> IB Chemistry Higher Level · R1: What drives chemical reactions?
> Source: https://www.owlsprep.com/study/ib-chemistry-hl-u4-energy-cycles/

Energy cycles apply Hess's Law to calculate unknown enthalpy changes when direct experimental measurement is impossible. This sub-topic covers general enthalpy cycles and the HL-specific application to ionic solids via Born-Haber cycles, a core exam topic.

**Prerequisites:** [Standard enthalpy definitions (formation, combustion, atomization)](https://www.owlsprep.com/study/ib-chemistry-hl-u3-enthalpy-fundamentals/)

## Learning objectives

- Apply Hess's Law to construct valid enthalpy energy cycles
- Calculate unknown enthalpy changes from given experimental data
- Construct Born-Haber cycles to find lattice enthalpy for ionic compounds
- Avoid common calculation errors in energy cycle problems

## Hess's Law and General Enthalpy Cycles

**Hess's Law** — The total enthalpy change for a reaction is the same regardless of the path taken between reactants and products, since enthalpy is a state function.

To construct an energy cycle, connect reactants and products to a common set of intermediate species (usually elements for enthalpy of formation, or combustion products for enthalpy of combustion). The unknown enthalpy change is found by adding and reversing known enthalpy changes to match the alternative route.

**Worked example:** Calculate the standard enthalpy of formation of ethanol ($C_2H_5OH(l)$) given: $
\Delta H_c^\circ (C(graphite)) = -393.5\ kJ\ mol^{-1}, 
\Delta H_c^\circ (H_2(g)) = -285.8\ kJ\ mol^{-1}, 
\Delta H_c^\circ (C_2H_5OH(l)) = -1367\ kJ\ mol^{-1}$

1. Write the target equation for formation of 1 mole of ethanol:

   $$2C(graphite) + 3H_2(g) + \frac{1}{2}O_2(g) \rightarrow C_2H_5OH(l) \quad \Delta H_f^\circ = ?$$
2. Apply the rule for enthalpy of reaction from combustion data: sum of reactant combustion enthalpies minus sum of product combustion enthalpies:

   $$\Delta H_f^\circ = 2\Delta H_c^\circ(C) + 3\Delta H_c^\circ(H_2) - \Delta H_c^\circ(C_2H_5OH)$$
3. Substitute values and calculate the result:

   $$(2 \times -393.5) + (3 \times -285.8) - (-1367) = -277.4\ kJ\ mol^{-1}$$

> **Exam tip:** Always check the direction of your route: reverse the sign of any enthalpy change you traverse opposite to its standard definition.

*Calculator:* allowed

## Born-Haber Cycles for Ionic Compounds

**Lattice Enthalpy** — The enthalpy change when one mole of an ionic solid is formed from its gaseous ions under standard conditions. Dissociation of the lattice has the opposite sign.

*Notation:* \Delta H_{latt}

*Example:* Dissociation of NaCl: $\Delta H_{latt} = +787\ kJ\ mol^{-1}$

A Born-Haber cycle is a specialized energy cycle that applies Hess's Law to calculate lattice enthalpy, which cannot be measured directly. The cycle relates lattice enthalpy to measurable terms: enthalpy of formation, atomization enthalpy, ionization energy, and electron affinity.

**Worked example:** Calculate the lattice enthalpy of dissociation for KCl(s) given: 
$\Delta H_f^\circ(KCl(s)) = -437\ kJ\ mol^{-1}$, 
$\Delta H_{at}^\circ(K(s)) = +90\ kJ\ mol^{-1}$, 
$\Delta H_{at}^\circ(\frac{1}{2}Cl_2(g)) = +122\ kJ\ mol^{-1}$, 
$IE_1(K(g)) = +419\ kJ\ mol^{-1}$, 
$EA_1(Cl(g)) = -349\ kJ\ mol^{-1}$

1. Apply Hess's Law to the full Born-Haber cycle, solving for lattice enthalpy of formation first:

   $$\Delta H_f^\circ = \Delta H_{at}(K) + \Delta H_{at}(Cl_2) + IE_1(K) + EA_1(Cl) + \Delta H_{latt(formation)}$$
2. Rearrange to find $\Delta H_{latt(formation)}$:

   $$\Delta H_{latt(formation)} = -437 - (90 + 122 + 419 - 349) = -719\ kJ\ mol^{-1}$$
3. Dissociation is the reverse process, so reverse the sign:

   $$\Delta H_{latt(dissociation)} = +719\ kJ\ mol^{-1}$$

> **info**
>
> Exam questions will often ask for dissociation lattice enthalpy, which is always positive. Always confirm which value is requested.

*Calculator:* allowed

## Drawing and Interpreting Exam Energy Cycles

IB examiners regularly require you to draw fully labelled energy cycles, not just complete calculations. Every arrow must be correctly directed from starting species to product species, and each step must be clearly labelled with the correct enthalpy term name.

**Exam command terms**

Common command terms for this topic have specific exam expectations:

- **Construct** — Draw a fully labelled diagram with all steps and enthalpy changes *(Construct a Born-Haber cycle for MgO)*

- **Calculate** — Use Hess's Law to find the unknown enthalpy change, show all working *(Calculate the enthalpy of formation using the cycle)*

**Check your understanding**

Check your understanding of enthalpy sign conventions:

1. What is the sign of atomization enthalpy for solid sodium?

   - Negative, because bonds are broken
   - Positive, because bonds are broken
   - Negative, because bonds are formed
   - Positive, because bonds are formed

   *Answer:* Positive, because bonds are broken

   *Why:* Atomization breaks metallic bonds in solid sodium to form gaseous atoms, which is endothermic, so ΔH is positive.

*Calculator:* allowed

## Common pitfalls

- **Wrong:** Forgetting to reverse the sign of an enthalpy change when traversing a step opposite to its definition
  - Why it fails: Enthalpy change sign depends on direction of heat flow; ignoring this flips the sign of the final result
  - Correct: Always label arrows on your cycle, and add a negative sign to any enthalpy term you move through backwards
- **Wrong:** Not scaling enthalpy values by the stoichiometric coefficients in the target equation
  - Why it fails: Standard enthalpy values are reported per mole, so they must be scaled for the number of moles in the reaction
  - Correct: Check the stoichiometry of every step before calculating, and multiply each enthalpy value by its mole count
- **Wrong:** Confusing formation and dissociation lattice enthalpy in Born-Haber calculations
  - Why it fails: Formation of solid from ions is exothermic, dissociation is endothermic, so they have opposite signs
  - Correct: Always read the question carefully to confirm whether a positive (dissociation) or negative (formation) answer is required
- **Wrong:** Drawing arrows pointing in the wrong direction for steps in the cycle
  - Why it fails: Examiners penalize incorrectly directed arrows even if the final numerical answer is correct
  - Correct: Always draw the arrow starting at the reactant of the step and pointing to the product of the step
- **Wrong:** Missing bonds when calculating enthalpy change from average bond enthalpies
  - Why it fails: Students often miss bonds that do not change in the reaction or double count bonds in cyclic structures
  - Correct: Draw full structural formulas for all reactants and products, count every bond before calculating

## Cheatsheet

| Calculation Type | Formula | Key Rule |
| --- | --- | --- |
| Enthalpy from formation | $\Delta H_{rxn} = \sum \Delta H_f(products) - \sum \Delta H_f(reactants)$ | Reverse sign for reactants |
| Enthalpy from combustion | $\Delta H_{rxn} = \sum \Delta H_c(reactants) - \sum \Delta H_c(products)$ | Reverse sign for products |
| Bond enthalpy reaction | $\Delta H_{rxn} = \sum bonds\ broken - \sum bonds\ formed$ | Breaking = positive, forming = negative |
| Born-Haber dissociation | $\Delta H_{latt(dissoc)} = -\Delta H_{latt(formation)}$ | Dissociation is always positive |

## What's next

Energy cycles are the foundation of all thermochemical calculations in IB Chemistry, and underpin all advanced topics related to reaction spontaneity. Mastering Hess's Law and energy cycle construction will make it much easier to tackle enthalpy problems across all papers, from multiple choice to extended response. Next, you will build on this knowledge to explore trends in lattice enthalpy and their relationship to ionic bonding, before moving on to entropy and Gibbs free energy, which explain what truly drives chemical reactions.

- [AHL: Extended enthalpy and entropy calculations](https://www.owlsprep.com/study/ib-chemistry-hl-u4-ahl-extended-enthalpy-and-entropy/)
- [AHL: Gibbs free energy and reaction spontaneity](https://www.owlsprep.com/study/ib-chemistry-hl-u4-ahl-gibbs-free-energy-and/)
- [R2: How much / how fast / how far?](https://www.owlsprep.com/study/ib-chemistry-hl-u5-overview/)

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