Kirchhoff's Law of Thermochemistry Calculator
Convert a reaction enthalpy from one temperature to another using Kirchhoff's law. Enter the known enthalpy at a reference temperature, the target temperature, and the summed heat capacities of reactants and products.
Adjust a Reaction Enthalpy from One Temperature to Another
Kirchhoff's Law of Thermochemistry Formula
Where:
- \(\Delta H_1\) — reaction enthalpy at the reference temperature \(T_1\) (kJ/mol)
- \(\Delta H_2\) — reaction enthalpy at the target temperature \(T_2\) (kJ/mol)
- \(\nu_i\) — stoichiometric coefficient of species \(i\)
- \(C_{p,i}\) — molar heat capacity at constant pressure of species \(i\)
- \(\Delta C_p\) — difference between the total heat capacities of products and reactants
The formula expresses how a reaction enthalpy changes with temperature: heating (or cooling) the reactants and products by the same amount costs different amounts of heat because their heat capacities differ, and this difference accumulates as the shift in \(\Delta H\). It assumes \(\Delta C_p\) is constant over the interval \(T_1\)–\(T_2\).
Kirchhoff's Law – Calculation Example
Example: for the ammonia synthesis reaction N₂ + 3H₂ → 2NH₃, the standard reaction enthalpy at 298.15 K is −92.2 kJ/mol. Estimate the enthalpy at 400 K, using ΣCp(reactants) = 115.5 J/(mol·K) (29.1 + 3 × 28.8) and ΣCp(products) = 71.2 J/(mol·K) (2 × 35.6).
\[ \Delta C_p = 71.2 - 115.5 = -44.3\ \text{J/(mol·K)} = -0.0443\ \text{kJ/(mol·K)} \]
\[ \Delta H_2 = -92.2 + (-0.0443)(400 - 298.15) = -92.2 - 4.51 = -96.71\ \text{kJ/mol} \]
Result: ΔH₂ ≈ −96.71 kJ/mol. The reaction is about 4.5 kJ/mol more exothermic at 400 K than at 298.15 K because ΔCp is negative.
How to Use the Calculator
- Enter the known Reaction Enthalpy at Reference Temperature (ΔH₁), choosing kJ/mol or kcal/mol. Use a negative value for an exothermic reaction, a positive value for an endothermic one.
- Enter the Reference Temperature (T₁) — typically 298.15 K — and select its unit (K, °C, or °F).
- Enter the Target Temperature (T₂) with its unit. This is the temperature at which you want the adjusted enthalpy.
- For every species in the reaction, multiply its molar heat capacity \(C_p\) by its stoichiometric coefficient, then sum the values on each side of the equation. Enter the totals as Sum of Reactant Heat Capacities (ΣCp,reactants) and Sum of Product Heat Capacities (ΣCp,products), selecting J/(mol·K), kJ/(mol·K), or cal/(mol·K).
- Read the result: the Reaction Enthalpy at Target Temperature (ΔH₂) in your chosen enthalpy unit, along with ΔCp, unit conversions, and a full step-by-step breakdown.
What Is Kirchhoff's Law of Thermochemistry?
Kirchhoff's law describes how the enthalpy change of a chemical reaction depends on temperature. Because enthalpy is a state function, the temperature derivative of the reaction enthalpy at constant pressure equals the difference in heat capacities between products and reactants:
\[ \left( \frac{\partial \Delta H}{\partial T} \right)_p = \Delta C_p \]
Integrating this expression from \(T_1\) to \(T_2\) gives \(\Delta H(T_2) = \Delta H(T_1) + \int_{T_1}^{T_2} \Delta C_p\, dT\). When \(\Delta C_p\) is treated as constant over the temperature interval — a good approximation for modest ranges — the integral simplifies to \(\Delta C_p (T_2 - T_1)\), which is the form this calculator uses.
The physical picture is straightforward. Tabulated reaction enthalpies almost always refer to 298.15 K, but real reactors, engines, and calorimeters run at other temperatures. To compare, you imagine heating the reactants from \(T_1\) to \(T_2\), letting the reaction occur, and cooling the products back — the difference between the two heating requirements is exactly \(\Delta C_p (T_2 - T_1)\).
Interpreting the sign is the most useful part. If \(\Delta C_p < 0\) (products have lower heat capacity than reactants), the reaction becomes more exothermic as temperature rises. If \(\Delta C_p > 0\), it becomes more endothermic. If \(\Delta C_p = 0\), the enthalpy is independent of temperature. In practice \(\Delta C_p\) is often small, so \(\Delta H\) typically shifts by only a few percent even over hundreds of kelvin.
Assumptions, Limitations, and Practical Tips
- Constant heat capacities. The linear form is accurate only over intervals where \(C_p\) values change little. For wide temperature ranges (more than a few hundred kelvin), use temperature-dependent fits such as \(C_p = a + bT + cT^2\) or \(C_p = a + bT + c/T^2\) from standard tables and integrate them instead.
- No phase changes. The calculation assumes every species keeps the same phase between \(T_1\) and \(T_2\). A melting, vaporization, or any other transition inside the interval would add a term \(\nu \Delta H_{\text{transition}}\) at the transition temperature.
- Consistent data. All heat capacities must refer to the same pressure (usually 1 bar) and the same physical state as the species in the balanced equation. Double-check the stoichiometric coefficients — the sums ΣνCp must match the equation as written.
- Signs matter. Keep ΔH₁ negative for exothermic reactions. The calculator reports whether the reaction is exo- or endothermic at the target temperature and how far the enthalpy shifted.
- Watch for T₁ = T₂. If both temperatures are the same after conversion to kelvin, ΔH₂ simply equals ΔH₁ and the calculator will ask for two different values.
If you do not yet have a reference enthalpy, you can build one from standard enthalpies of formation with the Hess's Law calculator or estimate it from average bond strengths with the reaction enthalpy from bond energies calculator. For other temperature-dependent chemistry estimates, the Antoine equation calculator is a useful companion when vapor pressures are involved.
Frequently Asked Questions
How do I apply Kirchhoff's law step by step without a calculator?
First, balance the equation and list every species. Multiply each molar heat capacity by its stoichiometric coefficient and sum separately for reactants and products. Subtract: \( \Delta C_p = \sum C_p(\text{products}) - \sum C_p(\text{reactants}) \). Convert \( \Delta C_p \) to kJ/(mol·K) if it is in J/(mol·K). Finally compute \( \Delta H_2 = \Delta H_1 + \Delta C_p (T_2 - T_1) \), using temperatures in kelvin or any consistent unit, since only the difference matters.
Why is it okay to use °C for the temperatures when enthalpy tables use kelvin?
Because the formula only uses the difference \( T_2 - T_1 \), and a change of 1 °C equals a change of 1 K, Celsius and kelvin give identical results. Fahrenheit also works if both temperatures are in °F, but never mix units — entering one temperature in K and the other in °C is a common error that shifts the result by 273.15 units. The calculator converts everything to kelvin internally.
How accurate is the constant-ΔCp approximation, and over what temperature range can I use it?
For intervals of roughly 100–200 K near room temperature, treating heat capacities as constant typically introduces errors well under 1 kJ/mol. Over wider ranges, Cp values drift — especially for gases — and the error grows. For serious work, take polynomial Cp fits from tables (e.g. NIST or textbook Shomate data), integrate \( \int_{T_1}^{T_2} \Delta C_p\, dT \) analytically or numerically, and add the result to \( \Delta H_1 \) instead.
What happens if a substance melts, boils, or changes phase between T₁ and T₂?
The simple formula no longer applies, because a phase transition absorbs or releases latent heat at a fixed temperature. You must split the interval at the transition temperature, apply Kirchhoff's law to each segment, and insert \( \nu \, \Delta H_{\text{transition}} \) at the crossing point. For example, water going from liquid at 25 °C to vapor at 150 °C needs the vaporization enthalpy (about 40.7 kJ/mol) added explicitly. Also make sure the Cp values you enter match the actual phase.
What are the most common mistakes when using Kirchhoff's law?
The top errors are: forgetting to multiply each Cp by its stoichiometric coefficient (e.g. 3 × 28.8 for 3H₂, not 28.8); mixing J and kJ in the heat capacities while the enthalpy is in kJ/mol; subtracting reactants from products instead of the reverse; losing the sign of an exothermic \( \Delta H_1 \); and using Cp values for the wrong phase or pressure. Always double-check the balanced equation before summing.
Where can I find the molar heat capacity values I need to enter?
Standard textbooks and reference tables (NIST WebBook, CRC Handbook) list \( C_p^\circ \) at 298.15 K for common substances; typical gas values range from about 20 to 40 J/(mol·K). Use the value for the correct physical state — Cp differs substantially between liquid water (75.3 J/(mol·K)) and water vapor (33.6 J/(mol·K)). If only a mean heat capacity over your temperature interval is tabulated, that is actually preferable, since it improves the constant-ΔCp approximation.
Does Kirchhoff's law also apply to Gibbs energy or equilibrium constants?
No — Kirchhoff's law concerns enthalpy only. The temperature dependence of the Gibbs free energy follows the Gibbs–Helmholtz equation, \( \partial(G/T)/\partial T = -H/T^2 \), and equilibrium constants shift with temperature according to the van 't Hoff equation. If a reaction crosses from exothermic to temperature-neutral as T rises, its equilibrium behavior changes too, but you need the entropy or Gibbs energy data to calculate that, not \( \Delta C_p \) alone.