Imagine a hypothetical ammonia-synthesis reaction with an internal-energy change of −87.0 kJ per reaction as written at 298.15 K. Counting the gas molecules reveals a further correction of about −4.96 kJ, so its corresponding ideal-gas enthalpy change is about −91.96 kJ. The two energies are not interchangeable labels.
The distinction begins with H=U+pV and becomes especially important when a reaction changes the number of gas molecules. We will derive the ideal-gas correction, calculate it for a balanced equation, distinguish constant-volume and constant-pressure calorimetry, and show why phase labels and reaction coefficients must remain attached to every reported result.
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Define energy and heat · Chapters 1–2 · Start this route
Derive the gas correction · Chapters 3–5 · Start this route
Calculate and compare reactions · Chapters 6–8 · Start this route
Check the measurement and limits · Chapters 9–10 · Start this route
Practise and transfer · Chapters 11–12 · Start this route
Full chapter index · Worked practice · Thermochemistry subject root
Chapter index
Define energy and heat · Chapters 1–2
Derive the gas correction · Chapters 3–5
Calculate and compare reactions · Chapters 6–8
Check the measurement and limits · Chapters 9–10
Practise and transfer · Chapters 11–12
Internal energy U is a state property of the chemical system; it includes the microscopic forms of energy represented by the thermodynamic model. Enthalpy H is another state property, defined as H=U+pV. They are related but not the same numerical quantity when the pV contribution changes.
In many laboratory discussions, a reaction energy is quoted as though ‘heat released’ were a property independent of how the process is measured. Heat is actually energy transferred along a process path. Under suitably specified conditions, a measured heat is associated with a change in U or H.
Before calculating, name the chosen reacting system, initial and final states, temperature, pressure or volume constraint, and whether forms of work other than pressure–volume work occur. Only then decide which state-function change a reported heat can represent.
By definition, H=U+pV. For a reaction between specified initial and final states, ΔH=ΔU+Δ(pV). This is a difference of state-function products, not always the single expression pΔV when pressure also varies.
At constant pressure, Δ(pV)=pΔV, so ΔH=ΔU+pΔV. In a closed system where only pressure–volume work occurs, the heat exchanged at constant external pressure corresponds to ΔH with the standard chemistry sign convention. At constant volume with no other work, the heat corresponds to ΔU.
These are statements about appropriate thermodynamic conditions. It is unsafe to take any warmed cup, any rigid container or any open reactor and simply relabel its thermometer reading as a reaction enthalpy without considering heat balance and work.
Use the balanced gaseous reaction N₂(g)+3H₂(g)→2NH₃(g). This equation has four moles of gaseous reactants and two moles of gaseous products for one mole of reaction as written.
The signed change in total gas amount is Δngas=2−(1+3)=−2 mol of gas per mole of reaction as written. This is a stoichiometric change, not the number of gas molecules literally counted in one laboratory flask.
The balance is crucial. An analyst who counts N₂ and H₂ as two gaseous species and NH₃ as one would compute 1−2=−1, which is wrong. We count molar coefficients of gases, not distinct chemical names.
For gases treated as ideal, pV=nRT. Compare reactants and products at the same temperature T, using the same definition of reaction extent. The difference in the gases’ pV terms is Δ(pV)≈ΔngasRT if condensed-phase pV contributions are negligible.
Combining this with H=U+pV gives the familiar relation ΔH≈ΔU+ΔngasRT. With all species ideal gases the gas contribution follows exactly from the stated equation of state; the approximation sign flags omitted condensed-phase or nonideal terms when present.
The correction is not determined by the change in total stoichiometric coefficients when solids or liquids are involved. A liquid mole does not contribute an ideal-gas RT term merely because it appears in the balanced equation.
Take the universal gas constant R=8.31446 J/(mol·K) and temperature T=298.15 K. Their product is RT≈2478.96 J/mol, or 2.47896 kJ/mol.
For Δngas=−2 mol per reaction as written, the correction is (−2)(8.31446)(298.15) J≈−4957.9 J=−4.9579 kJ. The gas amount and the inverse-mole units in R cancel to give energy for the stated amount of reaction.
Keep the Kelvin temperature. Substituting ’25’ because the experiment is at 25 °C instead of 298.15 K would produce a grossly wrong correction. A temperature difference of 25 °C may equal 25 K, but absolute temperature in RT must be expressed on the Kelvin scale.
Suppose a properly corrected hypothetical internal-energy change for the reaction as written is ΔU=−87.0 kJ at 298.15 K. This is invented teaching input, not a quoted experimentally measured enthalpy for ammonia manufacture.
Then ΔH≈−87.0+(−4.9579)=−91.96 kJ per mole of reaction as written. Both changes are negative. In this example, the enthalpy is more negative because the gaseous product amount is lower than the gaseous reactant amount.
If the reacting direction is reversed, both ΔU and ΔH change sign, as does Δngas. A correct reverse-reaction calculation will therefore yield ΔH≈+91.96 kJ per mole of the reversed equation under the same idealised reference conditions.
| Quantity | Value per reaction as written | Meaning |
|---|---|---|
| Reaction | N₂(g) + 3H₂(g) → 2NH₃(g) | Balanced illustrative equation |
| Δn gas | 2 − 4 = −2 | Change in gaseous stoichiometric amount |
| R and T | 8.31446 J/(mol·K), 298.15 K | Consistent SI units |
| ΔnRT | −4.9579 kJ | Ideal-gas pV correction |
| Hypothetical ΔU | −87.0 kJ | Illustrative input, not experimental data |
| Estimated ΔH | −91.96 kJ | ΔU + ΔnRT |
Consider H₂(g)+½O₂(g)→H₂O(g). Counting only gaseous coefficients gives Δngas=1−1.5=−0.5. At 298.15 K its ideal-gas pV correction would be approximately −1.2395 kJ for one mole of reaction.
Change the product to liquid water: H₂(g)+½O₂(g)→H₂O(l). Now the gas count goes from 1.5 moles to zero, so Δngas=−1.5, giving an ideal-gas correction of roughly −3.7184 kJ, while neglecting the liquid pV contribution.
Do not conclude that the two reactions have the same ΔU or ΔH with only a modified gas correction. Forming liquid rather than vapour changes the product state and includes the energetic effect of condensation. The reaction energy itself must be determined for the specified phases.
The illustrative ammonia reaction forms two moles of NH₃ per mole of reaction as written. A reported ΔH≈−91.96 kJ for that written reaction corresponds to about −45.98 kJ per mole of NH₃ formed, provided the amounts and conditions are understood consistently.
If every balanced-equation coefficient were doubled, Δngas would double and both ΔU and ΔH for the doubled reaction would double. The per mole of NH₃ value would not double, because the normalisation also changes.
This is why thermochemical equations should travel with reaction enthalpy figures. A number marked only ‘kJ/mol’ leaves unanswered the vital question: a mole of which reaction or product?
A rigid, closed bomb calorimeter approximates a constant-volume experiment. After proper corrections and where no other work applies, the reaction heat at constant volume is related to ΔU. A constant-pressure calorimeter instead may measure heat associated with ΔH when its assumptions and energy balance are satisfied.
Neither apparatus directly hands you an abstract state function without analysis. Vessel heat capacity, solution or water heat, ignition energy, heat losses, actual reaction extent and temperature corrections influence the inferred result.
OpenStax’s chemistry discussion distinguishes calorimetry at constant volume from the enthalpy changes associated with constant pressure. The essential reporting question is not which instrument sounds more sophisticated, but which physical constraint and heat terms the experiment actually used.
The simple correction assumes the relevant gaseous components behave ideally and initial and final species are compared at the same temperature. At high pressure, strong intermolecular interactions or near phase transitions, real-fluid pV and enthalpy behaviour can require a more complete equation of state.
Condensed phases make small but not automatically negligible pV contributions. Processes at different initial and final temperatures also involve sensible enthalpy and internal-energy changes, so do not insert one RT term and forget heat capacities.
Work other than expansion work, open-system flow, incomplete reaction and mixtures with uncertain composition need their own models. ΔH=ΔU+Δ(pV) always defines the state relation, but the convenient ΔngasRT shortcut belongs to a narrower idealisation.
Practice A: For 2SO₂(g)+O₂(g)→2SO₃(g), count gaseous coefficients: products 2, reactants 3, so Δngas=−1. At 298.15 K the ideal-gas correction is about −2.479 kJ per reaction as written.
Practice B: Reverse that equation. The correction becomes approximately +2.479 kJ. If an exothermic forward reaction has negative ΔH, the reversed reaction has positive ΔH of equal magnitude at the same states.
Practice C: For A(g)+B(g)→C(g)+D(g), Δngas=0. Under the ideal-gas, equal-temperature assumptions, ΔH≈ΔU even though heat may still be released or absorbed. Zero correction does not mean zero reaction energy.
Write and balance the reaction including phase labels. Count only gaseous stoichiometric coefficients to obtain signed Δngas. State T in kelvins and R in consistent energy units. Identify whether the known energy is ΔU or ΔH and apply ΔH≈ΔU+ΔnRT in the correct direction.
Finally label the reporting basis, inspect the plausibility of the correction, and check whether real-gas behaviour, other phases or temperature changes make the approximation inadequate. This procedure keeps energy, matter and measurement constraints together.
Connect to The Gold Standard of Thermochemistry and the sibling Limiting Reagents and the Correct Mole in Calorimetry. The first explains the larger field; the second protects the mole denominator before an enthalpy is reported.
Sources and connected reading
The ammonia reaction energy is an invented teaching value. The relation is a state-property calculation under stated ideal-gas and equal-temperature conditions, not a report of a measured experimental enthalpy.