Reaction Enthalpy per Mole of Extent

Introduction

A calorimeter reports heat for the particular amount that reacted in one experiment. A thermochemical equation usually reports energy per mole of reaction as written. The bridge is reaction extent: determine how many moles of the balanced equation occurred, then divide the run's enthalpy change by that extent. A coefficient of two means two moles of that species are consumed per mole of reaction extent.

Core explanation

For a balanced reaction aA + bB → cC, if n A,reacted moles A are consumed, the extent is ξ = n A,reacted/a. The same ξ should equal n B,reacted/b and n C,formed/c when the stated equation is the only process. If an experimentally inferred reaction enthalpy for the run is ΔH run, then the reaction enthalpy associated with one mole extent is Δ rH = ΔH run/ξ, with units such as kJ mol⁻¹ of reaction as written. Do not divide by raw moles of A unless its coefficient is one.

Suppose 2A + B → 2C releases 12.0 kJ when 0.200 mol A reacts. The reaction extent is ξ = 0.200/2 = 0.100 mol, so Δ rH = −12.0/0.100 = −120 kJ mol⁻¹ extent. Dividing by 0.200 mol A would give −60.0 kJ per mole A consumed, a valid different basis . The two numbers are not contradictory; their labels differ. If the equation is halved to A + 0.5B → C, its reaction enthalpy is halved to −60.0 kJ mol⁻¹ of the new equation.

At constant external pressure, with only appropriate pressure–volume work and well-defined initial and final states, reaction heat q p corresponds to the enthalpy change for the run. A coffee-cup calorimeter often approximates this, but the measurement includes heat exchanged with solution and apparatus. The energy balance from calorimetry first gives q rxn = −(q solution + q calorimeter) under an insulated approximation. Then calculate ξ from stoichiometry and report q rxn/ξ as an estimate of Δ rH. At constant volume, measured heat relates more directly to internal-energy change, so it should not automatically be labelled ΔH.

If the reaction does not go to completion, initial limiting-reactant moles divided by a coefficient are not the actual extent. Use the measured amount consumed or product formed. If a sample is impure, correct its active mass before converting to moles. If a side reaction contributes heat, a single measured q may not represent the desired reaction's enthalpy without additional data. These checks matter as much as arithmetic.

The enthalpy of reaction depends on the equation's chemical species and their states. Forming liquid water and forming water vapour are different thermochemical equations because vaporisation has its own enthalpy. Temperature also matters when heat capacities change. Keep the equation beside the numerical value so the molar basis is unambiguous.

Step-by-step reasoning

1. Write the balanced equation with physical states and choose its exact coefficient scaling. 2. Obtain the run's signed reaction heat from calorimetry or supplied data. 3. Find actual reacted moles or formed product moles and divide by the relevant coefficient to get ξ. 4. Calculate Δ rH = ΔH run/ξ, retaining the heat sign. 5. Label the result “per mole of reaction as written” and check whether experimental conditions justify an enthalpy interpretation.

Visual explanation

Draw a row of reaction packets. Each packet consumes two A tokens and one B token and releases a fixed energy amount. If a lab run consumes 0.200 mol A, it completed 0.100 mol packets. Put the measured −12.0 kJ beside those packets, then scale to one mole of packets to show −120 kJ mol⁻¹ extent.

Real-world analogy

A bakery recipe uses two cups of flour per batch. If a kitchen uses 20 cups, it made ten recipe-batches, not twenty. Energy used per batch is total energy divided by ten. Stoichiometric coefficients play the role of ingredient requirements per reaction batch.

Real-world example

An acid–base neutralisation experiment warms a solution. The observed temperature change yields the run's heat, while titration data tell how many moles actually reacted. Dividing by the correct reaction extent makes results from different sample sizes comparable, provided both refer to the same balanced reaction and state conditions.

Why?

Why is dividing by ξ essential? Extensive heat scales with how much reaction occurs. A molar reaction enthalpy removes sample size by referring to one mole of the entire balanced equation . Dividing by an arbitrarily chosen reactant amount can silently change the stated basis.

Common misconception

“Reaction enthalpy is the same number no matter how the equation is written.” Reversing the equation changes its sign, and multiplying coefficients changes its per-equation magnitude. The physical heat for one particular run remains the same, but its reported molar basis changes.

Worked example

An insulated constant-pressure experiment follows A + 2B → C. A 50.0 g solution with c = 4.00 J g⁻¹ K⁻¹ warms 4.00 K, and the cup alone has C cal = 10.0 J K⁻¹. Solution heat gain is 800 J; cup heat gain is 40.0 J, so q rxn = −840 J. Analysis shows 0.0100 mol A reacted, giving ξ = 0.0100 mol because A's coefficient is one. Estimated Δ rH = −0.840 kJ/0.0100 mol = −84.0 kJ mol⁻¹ for A + 2B → C. If B had been the measured reactant at 0.0200 mol consumed, dividing it by coefficient two would give the same ξ.

Quick check

1. For 2A + B → C, 0.50 mol A reacts and 5.0 kJ is released. What is Δ rH per mole extent? Answer: ξ = 0.50/2 = 0.25 mol, so Δ rH = −5.0/0.25 = −20 kJ mol⁻¹ extent.

Exam focus

Put the balanced equation next to Δ rH. Convert measured q to a signed reaction quantity, divide reacted species moles by its coefficient, and only then divide heat by ξ. Keep constant-pressure and constant-volume measurements conceptually distinct.

Advanced insight

For multiple simultaneous reactions, total enthalpy change is the sum of each reaction's extent times its own Δ rH. One observed heat value cannot generally determine all unknown extents; composition measurements or additional experiments are needed to separate contributions.

Summary

Measured heat belongs to a particular run. Reaction enthalpy per mole extent equals that signed heat divided by actual moles of the balanced reaction as written. Coefficient scaling, physical states and calorimeter conditions determine how the number should be interpreted.

Practice questions

1. For 3A → B, how much extent occurs when 0.60 mol A is consumed? Answer: ξ = 0.60/3 = 0.20 mol reaction extent. 2. If that run absorbs +8.0 kJ, what is Δ rH for 3A → B? Answer: +8.0/0.20 = +40 kJ mol⁻¹ extent. 3. What is Δ rH when the same equation is doubled to 6A → 2B? Answer: The enthalpy number doubles to +80 kJ mol⁻¹ of the doubled reaction equation.