Kirchhoff's Law ΔH(T) Calculator

Calculate reaction enthalpy at any temperature using Kirchhoff's law and heat capacities.

ΔH° at 298K (kJ/mol) (kJ/mol)

Cp of products (J/mol·K) (J/mol·K)

Sum Cp of all products (stoich×Cp)

Cp of reactants (J/mol·K) (J/mol·K)

Sum Cp of all reactants

Target temperature T₂ (K) (K)

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Why ΔH Changes with Temperature

Tabulated enthalpies apply at 298 K. At other temperatures the value shifts, because products and reactants absorb heat at different rates as they warm. Kirchhoff's law quantifies that difference.

ΔH(T₂) = ΔH(T₁) + ΔC_p(T₂ − T₁)

ΔC_p is the heat capacity of products minus reactants. Its sign determines the direction of the shift:

ΔC_p	Effect of heating	Physical meaning
Positive	ΔH becomes more positive	Products absorb heat faster than reactants
Negative	ΔH becomes more negative	Reactants absorb heat faster
Near zero	ΔH nearly constant	Similar heat capacities on both sides

Reactions with similar numbers and types of molecules on each side have small ΔC_p, so the correction is negligible. Reactions that change the number of gas moles substantially — ammonia synthesis converts four moles to two — have large ΔC_p and need the correction.

When the Correction Matters

Temperature range	Typical correction	Recommendation
298–400 K	Usually under 2%	Often ignorable
298–700 K	5–20%	Apply the correction
Above 1,000 K	Can exceed 30%	Use polynomial C_p(T)

The linear form assumes ΔC_p is constant, which is reasonable over a few hundred kelvin. Over wider ranges heat capacities themselves vary substantially, and the integral form with polynomial C_p(T) expressions is needed — standard practice in industrial process simulation.

Worked Examples

Example 1: N2+3H2→2NH3: ΔH298=-92.4kJ, Cp_prod=2×35.7=71.4, Cp_react=29+3×28.8=115.4

ΔCp=71.4-115.4=-44 J/mol·K

Result: ΔH(700K)=-92.4+(-44)×402/1000=-110.1kJ/mol

More exothermic at higher T

Example 2: CO+½O2→CO2: ΔH298=-283kJ

ΔCp small for this reaction

Result: ΔH changes only slightly with T

Cp correction often small for simple combustion

Example 3: Ammonia synthesis at reactor temperature

ΔH₂₉₈ = −92.4 kJ, ΔC_p = −44 J/(mol·K), T = 700 K

Result: ΔH(700) = −110.1 kJ

The reaction becomes 19% more exothermic at operating temperature. Reactor heat removal must be sized on this figure, not the tabulated one.

Example 4: When it can be ignored

Reaction with ΔC_p = −2 J/(mol·K) at 400 K

Result: Correction is 0.2 kJ

With similar heat capacities on both sides, the correction is negligible and the 298 K value can be used directly.

Common Mistakes

⚠️

Using 298 K values at reaction temperature

Industrial reactions often run hundreds of kelvin above standard conditions. Ignoring the correction can introduce errors above 20%.

⚠️

Assuming ΔC_p is constant over wide ranges

Heat capacities rise with temperature. Above about 700 K the linear approximation degrades and polynomial expressions are required.

⚠️

Mixing J and kJ

C_p is tabulated in J/(mol·K) while ΔH is in kJ/mol. The product ΔC_pΔT must be divided by 1000 before adding.

⚠️

Forgetting stoichiometric coefficients in ΔC_p

Each species' heat capacity is multiplied by its coefficient. Two moles of ammonia contributes twice its C_p.

Frequently Asked Questions

When is Kirchhoff correction important?

For >200K temperature difference, or when ΔCp is large (>50 J/mol·K). For combustion reactions in adiabatic flame temperature calculations, always needed. For small ΔT or small ΔCp, ΔH298 is adequate approximation.

Polynomial Cp?

For greater accuracy: Cp = a + bT + cT² + dT³ (Shomate equation). Then integrate: ΔH(T) = ΔH(298) + ∫(ΔCp dT) from 298 to T. NIST WebBook provides Shomate constants for most compounds.

Why does reaction enthalpy change with temperature?

Because products and reactants have different heat capacities, so warming them from 298 K to the reaction temperature costs different amounts of energy on each side.

When should I apply Kirchhoff's correction?

Above about 400 K, or whenever ΔC_p is large. Corrections of 5–20% are typical by 700 K.

When does the linear form break down?

Above roughly 700–1,000 K, where heat capacities themselves vary substantially with temperature. Polynomial C_p(T) expressions are then needed.

What does a negative ΔC_p mean?

Reactants absorb heat faster than products as temperature rises, so the reaction becomes more exothermic at higher temperature.

Formula Explorer connections

Interpretation: This relationship tracks energy transfer, state-function change or the balance between enthalpy and entropy in a chemical process. Assumption: Keep energy units compatible, use kelvin for absolute temperature, and match standard states and reaction stoichiometry. Thermodynamic favorability does not determine reaction speed.

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