Enthalpy (H)

In everyday chemistry — whether dissolving salt in water, burning fuel in an engine, or baking bread — reactions typically occur in open containers exposed to the atmosphere, meaning they take place at constant (atmospheric) pressure rather than in sealed, rigid vessels. Under these conditions, any gas produced by the reaction pushes back against the surroundings and does work, so not all of the energy change within the system appears as heat. To account for this pressure–volume work automatically, chemists define a new state function called enthalpy, symbol H. Enthalpy is essentially a 'corrected' internal energy that bundles the system's stored energy together with the energy tied up in maintaining its volume against external pressure. Enthalpy is defined as H = U + PV, where U is the internal energy of the system, P is the pressure, and V is the volume. Because H, U, P, and V are all state functions — their values depend only on the current state of the system, not on how it got there — the change in enthalpy ΔH = ΔU + Δ(PV) is also a state function. At constant pressure Δ(PV) = PΔV, and since the First Law tells us that the heat q_p absorbed at constant pressure equals ΔU + PΔV, it follows directly that ΔH = q_p. This is the key experimental meaning of enthalpy: it equals the heat exchanged with the surroundings in a constant-pressure process. A negative ΔH means heat is released (exothermic); a positive ΔH means heat is absorbed (endothermic). For reactions involving ideal gases, the Δ(PV) term can be simplified using the ideal-gas law: Δ(PV) = Δn_g RT, where Δn_g is the change in the number of moles of gas, R is the universal gas constant (8.314 J mol⁻¹ K⁻¹), and T is the absolute temperature. This gives the working relationship ΔH = ΔU + Δn_g RT, which allows chemists to convert between ΔH measured in a constant-pressure calorimeter (e.g., a coffee-cup calorimeter) and ΔU measured in a constant-volume bomb calorimeter, or vice versa. Understanding enthalpy is therefore the bridge between laboratory heat measurements and the deeper energetics described by the First Law of Thermodynamics.

Key takeaways
  • Start with the definition of enthalpy
  • Apply the constant-pressure condition
  • Apply the ideal-gas approximation
  • Interpret the sign and magnitude
  • Standard enthalpy of reaction

Start with the definition of enthalpy

Enthalpy is defined as H = U + PV. Taking the change in a process: ΔH = ΔU + Δ(PV). This is always true regardless of conditions.

Apply the constant-pressure condition

At constant external pressure P, the term Δ(PV) simplifies to PΔV. By the First Law, ΔU = q + w = q_p − PΔV (since the only work done is expansion work w = −PΔV). Substituting: ΔH = ΔU + PΔV = (q_p − PΔV) + PΔV = q_p. Therefore, ΔH equals the heat exchanged at constant pressure.

Apply the ideal-gas approximation

Solids and liquids have negligible volume changes compared to gases. For the gaseous components of a reaction, the ideal gas law gives PV = nRT, so Δ(PV) = Δn_g RT, where Δn_g = (moles of gaseous products) − (moles of gaseous reactants). Substituting back: ΔH = ΔU + Δn_g RT.

Interpret the sign and magnitude

If Δn_g > 0 (more gas moles are produced), the system does extra work pushing back the atmosphere, so ΔH > ΔU. If Δn_g < 0 (fewer gas moles are produced), the surroundings compress the system, doing work on it, so ΔH < ΔU. If Δn_g = 0, ΔH = ΔU exactly. The difference Δn_g RT is often small compared to ΔU (typically a few kJ mol⁻¹), but it is chemically significant and must not be ignored in precise work.

Standard enthalpy of reaction

Standard enthalpies of reaction (ΔH°) are tabulated at 298.15 K and 1 bar. They can be calculated from standard enthalpies of formation ΔH°_f using Hess's Law: ΔH°_rxn = Σ ΔH°_f (products) − Σ ΔH°_f (reactants). Hess's Law works because enthalpy is a state function — the total enthalpy change is independent of the pathway taken.

The Key Equation

ΔH=ΔU+ΔngRT\Delta H = \Delta U + \Delta n_g RT
  • \Delta H — Change in enthalpy (J or kJ): Heat exchanged at constant pressure; negative for exothermic, positive for endothermic reactions.
  • \Delta U — Change in internal energy (J or kJ): Total change in all kinetic and potential energy of molecules; measured directly in a bomb calorimeter at constant volume.
  • \Delta n_g — Change in moles of gas (mol): Moles of gaseous products minus moles of gaseous reactants in the balanced equation; solids and liquids are excluded.
  • R — Ideal gas constant (J mol⁻¹ K⁻¹): Universal constant equal to 8.314 J mol⁻¹ K⁻¹, linking moles, temperature, and energy via the ideal gas law PV = nRT.
  • T — Absolute temperature (K): Temperature measured from absolute zero on the Kelvin scale; standard conditions use T = 298.15 K.

Worked Example

Problem. The combustion of methane at 298 K has ΔU = −885.4 kJ mol⁻¹. Calculate ΔH for the reaction \ce{CH4(g) + 2O2(g) -> CO2(g) + 2H2O(l)} and explain whether ΔH is more or less negative than ΔU.

ΔH=ΔU+ΔngRT=−885.4+(−2)(8.314×10−3)(298)≈−890.4 kJ mol−1\Delta H = \Delta U + \Delta n_g RT = -885.4 + (-2)(8.314 \times 10^{-3})(298) \approx -890.4 \text{ kJ mol}^{-1}

Answer. ΔH = −890.4 kJ mol⁻¹; it is more negative than ΔU because Δn_g = −2 and the surroundings do work on the system.

Figure
Diagram
Enthalpy Level Diagram: Combustion of MethaneReactantsCH₄(g) + 2O₂(g)H = H₁ProductsCO₂(g) + 2H₂O(l)H = H₂ΔH = H₂ − H₁= −890 kJ mol⁻¹(exothermic)|ΔH|Higher enthalpy (reactants)Lower enthalpy (products)Heat is released to surroundings: system loses enthalpy
Origins · 1 minGermain Henri Hess and how this was discovered
GH
Germain Henri Hess
1802–1850 · Swiss-Russian

Hess formulated Hess's Law of Constant Heat Summation in 1840, which states that the total enthalpy change of a reaction is the same regardless of the number of steps taken. This was an early empirical demonstration that heat of reaction is a state function — a foundational idea that underpins modern enthalpy calculations, Kirchhoff's Law, and Born–Haber cycles. Hess conducted his thermochemical experiments long before the concept of 'enthalpy' was formally named — he simply measured heat released in reactions with calorimeters and noticed the additive pattern. The formal definition of enthalpy came decades later, yet his law fitted perfectly into the new framework.

  • Internal Energy and the First Law — Enthalpy is derived from internal energy U using H = U + PV; understanding ΔU and how work and heat contribute to it is essential before extending to ΔH.
  • Hess's Law — Because enthalpy is a state function, reaction enthalpies can be added and subtracted — Hess's Law is the direct practical application of ΔH being path-independent.
  • Calorimetry — Coffee-cup calorimeters measure q_p = ΔH at constant pressure, while bomb calorimeters measure ΔU at constant volume; the equation ΔH = ΔU + Δn_g RT converts between the two.
  • Standard Enthalpy of Formation — Tabulated ΔH°_f values allow calculation of ΔH°_rxn via Hess's Law and are the most common way enthalpy is applied to predict whether reactions release or absorb heat.
  • Bond Enthalpy — Bond enthalpies provide a molecular-level explanation of why ΔH is negative or positive by comparing the energy required to break bonds in reactants with the energy released forming bonds in products.