Van 't Hoff Equation Calculator – How K Changes with Temperature
The van 't Hoff equation connects an equilibrium constant to temperature through the standard reaction enthalpy. This calculator uses it in two ways: to move K from one temperature to another (or find ΔH° from two K values), and to fit a van 't Hoff plot of ln K against 1/T from your own measurements, with uncertainties on ΔH°, ΔS° and ΔCp°. Many textbooks spell it van't Hoff, without the space; it is the same equation.
Where the equation comes from
Two standard results meet here: ΔG° = −RT ln K° and ΔG° = ΔH° − TΔS°. Setting them equal gives ln K° = −ΔH°/(RT) + ΔS°/R. If ΔH° and ΔS° are constant, ln K° is a straight line in 1/T. Comparing two temperatures gives the integrated form ln(K₂/K₁) = −(ΔH°/R)·(1/T₂ − 1/T₁). Jacobus van 't Hoff set out this result in his 1884 book Études de dynamique chimique, and in 1901 he received the first Nobel Prize in Chemistry.
Reading a van 't Hoff plot
Plot ln K against 1/T and the slope is −ΔH°/R, while the intercept is ΔS°/R. Because 1/T grows as T falls, the x-axis runs "backwards": hot is on the left, cold on the right. An endothermic reaction gives a line that falls to the right, so K rises as you heat it.
The solubility preset shows how to handle a bad reading. With all six points the fit gives ΔH° ≈ 107 ± 7 kJ/mol, and the 45 °C point sits far from the line, with a studentized residual of about 16. Excluding it tightens the result to 104.39 ± 0.91 kJ/mol. The tool flags such points but never drops them for you. Check the lab notebook before you exclude anything.
Le Chatelier's principle with numbers
The sign of ΔH° alone decides the direction. Ammonia synthesis is exothermic (ΔH° ≈ −92 kJ/mol), so its K falls by about 6.5 powers of ten between 25 °C and 227 °C. That is why the Haber process trades some yield for a faster rate at high temperature. Dissolving silver chloride is endothermic, so its Ksp rises about eightfold between 25 °C and 50 °C. For a solubility at the new temperature, take the new Ksp to the Solubility Product Calculator.
When "ΔH° is constant" breaks down
ΔH° changes with temperature whenever the heat capacities of products and reactants differ, by ΔH°(T) = ΔH°(T₁) + ΔCp°·(T − T₁). Over a few tens of kelvin this hardly matters. Over hundreds it does. With the illustrative ammonia values, K at 500 K is 0.174 with constant ΔH° but 0.094 once ΔCp° is included, a factor of 1.86. The water example makes the same point from data: a straight line through Kw from 0 °C to 60 °C has R² = 0.9989, yet its residuals form a clear arch, and the curvature test rejects the line. A high R² does not prove you have the right model.
A phase change between the two temperatures, such as a species melting or boiling, also breaks the assumption, because ΔH° jumps there. Split the range at the transition.
Kc, Kp and K°
The equation is written for the dimensionless K°, based on 1 bar and 1 mol/L. Solution constants such as Ka, Kw and Ksp, and Kp in bar, can go in as they are. Kc of a gas reaction cannot: since K° = Kc·(RT)^Δn and that factor depends on T, fitting Kc returns ΔU° rather than ΔH°. The two differ by Δn·R·T, about 2.5 kJ/mol per mole of gas near room temperature. Kp in atm only differs from K° by a constant factor, so ΔH° comes out right and only ΔS° shifts. To convert between the forms at one temperature, use the Kp to Kc Converter.
Same maths, different names
"Logarithm of something against 1/T" turns up across chemistry. The Arrhenius equation does it for rate constants and activation energy, and the Clausius–Clapeyron equation does it for vapour pressure and the enthalpy of vaporisation. Van 't Hoff's name is also on two other ideas: the osmotic-pressure law π = iMRT and the van 't Hoff factor i. Both are covered by the Osmotic Pressure Calculator, not by this tool.