Reversing and Scaling Thermochemical Equations

Introduction

Most Hess-law errors arise before the final addition: a step is reversed but its sign is not, or coefficients are halved while enthalpy stays unchanged. Manipulating thermochemical equations is algebra on both matter and energy. Every operation applied to a reaction must also be applied to its ΔH.

Core explanation

If A + B → C has enthalpy ΔH, the reverse C → A + B has −ΔH. This follows because the enthalpy difference swaps final and initial states. If the equation is multiplied by two, 2A + 2B → 2C has 2ΔH because twice as much matter changes state. Halving the equation gives ½A + ½B → ½C with ΔH/2. Fractional coefficients are legitimate in thermochemical equations when the target reaction requires them.

The order of operations does not alter the result. Reversing and then doubling gives −2ΔH; doubling and then reversing also gives −2ΔH. What matters is applying operations to the entire equation, including all physical-state labels and the energy change. Multiplying only the species of interest while leaving another reactant unchanged would no longer describe the same stoichiometric event.

When building a target reaction, select a species that appears in only one provided equation. If the target needs it on the product side but it appears as a reactant, reverse that equation. If the target coefficient is half the known coefficient, halve the equation and ΔH. Continue until all target species have the desired sides and amounts, then check that intermediates cancel. This systematic method is safer than guessing a combination of enthalpy numbers first.

For example, if a known reaction is 2CO(g) + O₂(g) → 2CO₂(g), ΔH = −566 kJ, then the one-mole-CO oxidation CO(g) + ½O₂(g) → CO₂(g) has ΔH = −283 kJ. The reverse formation of CO and oxygen from one mole CO₂ has ΔH = +283 kJ. The oxygen coefficient also halves or reverses with the equation; it is not optional.

Sometimes a target uses a phase different from a known equation. Scaling does not solve that mismatch. Add an explicit vaporization, condensation, fusion or other state-change equation with its own ΔH. The algebra is valid only when species being canceled are chemically and physically identical.

Units should be attached to the reaction scale. A table may state kJ mol⁻¹ for a conventional reaction extent. When multiplying a displayed equation by a factor, one may report a total kJ change for the scaled equation or redefine a molar reaction value for the new extent. Avoid vague “kJ/mol” without saying per mole of what.

After summing, count every element and charge in the target. A thermochemical equation is a chemical equation first. An energy result with an unbalanced target is invalid even if the numerical arithmetic is flawless.

Step-by-step reasoning

1. Write the target equation and mark desired species sides and coefficients. 2. Reverse each known step whose needed species is on the wrong side. 3. Multiply or divide each whole equation to match coefficients. 4. Carry sign and scale changes to ΔH immediately. 5. Add, cancel intermediates and verify atom and charge balance.

Visual explanation

Draw an equation card with a reversible arrow. Flipping it changes a red ΔH label from −x to +x. A photocopier doubles every species icon and doubles the energy label; a half-size card halves all coefficients and energy. Stack cards so intermediary icons cancel in matching phases.

Real-world analogy

A recipe for two cakes uses twice every ingredient and twice the total oven-energy budget under the simplified model. Reversing an accounting transaction reverses its sign. Changing only flour while leaving all other ingredients unchanged would not produce the same recipe.

Real-world example

To combine carbon and carbon-monoxide oxidation data, one commonly reverses the CO oxidation equation and scales it to one mole before adding. Writing the modified chemical equation first makes the needed +283 kJ sign evident.

Why?

Why does halving a reaction halve ΔH? Enthalpy is extensive: half as many moles undergo the same specified state change, giving half the total energy difference.

Common misconception

“ΔH is a fixed property of a chemical formula independent of reaction coefficients.” It belongs to a balanced reaction as written. Changing stoichiometric scale changes the quoted total enthalpy for that event.

Worked example

Given 2H₂(g) + O₂(g) → 2H₂O(l), ΔH = −571.6 kJ, find the reverse formation of one mole H₂ and half a mole O₂ from liquid water. Reverse to 2H₂O(l) → 2H₂ + O₂, ΔH = +571.6 kJ. Halve all coefficients: H₂O(l) → H₂(g) + ½O₂(g), ΔH = +285.8 kJ. Both the sign and magnitude change.

Quick check

1. A reaction has ΔH = −80 kJ. What is ΔH when it is reversed and tripled? Answer: +240 kJ for the reversed, tripled equation.

Exam focus

Write each transformed equation next to its transformed ΔH before adding. Fractional coefficients are allowed. Never cancel species with different phases without a separate phase-change step.

Advanced insight

Thermochemical equation manipulation is linear algebra in the space of stoichiometric vectors. Reversal multiplies the vector and enthalpy by −1; scaling multiplies both by the same scalar. Hess's law says enthalpy is a linear function of the net reaction vector under fixed conditions.

Summary

Reverse a reaction and reverse ΔH's sign; scale a reaction and scale ΔH by the same factor. Perform these operations on every species and state label. Only then add equations and enthalpies to obtain a valid target.

Practice questions

1. If 2A→2B has ΔH = +50 kJ, find A→B. Answer: +25 kJ, because every coefficient and ΔH are halved. 2. If A→B is −12 kJ, find 3B→3A. Answer: +36 kJ after reversal and tripling. 3. Can H₂O(g) cancel H₂O(l) after reversing one equation? Answer: No. The phases differ and require an additional phase-change enthalpy step.