van 't Hoff Equation Solver

Free van 't Hoff Equation Solver and Calculator

This calculator solves the van 't Hoff equation, the formula chemists use to work out how an equilibrium constant changes as temperature changes. Give it an equilibrium constant at one temperature, a standard enthalpy change, and a target temperature, and it instantly returns the equilibrium constant at that new temperature — along with a plain-language read on whether the reaction is endothermic or exothermic and which direction the equilibrium shifted.

It also solves the equation in every other direction. If you already have two measured equilibrium constants at two different temperatures, the calculator can back out the standard enthalpy change ΔH° behind them, without ever needing to run a calorimetry experiment. If you know ΔH° and one equilibrium constant and want to know at what temperature a second, target equilibrium constant would hold, it solves for that temperature directly. A second Advanced Tools mode goes further: paste in a full table of temperature and equilibrium-constant readings and it performs a proper van 't Hoff plot, fitting a straight line through ln K versus 1/T to extract both ΔH° and ΔS° at once, exactly the way a real lab experiment would analyze the same data. Every calculation on this page comes with full step-by-step working, free, with no sign-up required.

What Is the van 't Hoff Equation?

The van 't Hoff equation describes a simple but important chemical fact: the equilibrium constant K of a reaction is not a fixed number — it depends on temperature. A reaction that barely proceeds at room temperature might run almost to completion at a higher temperature, or the reverse, depending on whether the reaction absorbs or releases heat.

The relationship is named after Jacobus Henricus van 't Hoff, the Dutch chemist who won the first Nobel Prize in Chemistry in 1901, partly for this exact work connecting thermodynamics to chemical equilibrium. The equation ties together three quantities that otherwise seem unrelated: the equilibrium constant K, the absolute temperature T, and the standard enthalpy change ΔH° of the reaction — the heat absorbed or released under standard conditions.

The Two Forms of the van 't Hoff Equation

The van 't Hoff equation shows up in two closely related forms, and this calculator handles both. The differential form, dlnK/dT = ΔH°/(RT²), describes the instantaneous rate at which ln K changes with temperature. Integrating it (and assuming ΔH° stays roughly constant over the temperature range in question) gives the two-point, integrated form used in the Standard Solver tab of this calculator:

ln(K₂/K₁) = −(ΔH°/R)(1/T₂ − 1/T₁)

Here K₁ is the equilibrium constant at temperature T₁, K₂ is the equilibrium constant at a second temperature T₂, and R is the universal gas constant, 8.314 J/(mol·K). Rearranged slightly, the same equation can also be written as a straight line: ln K = −(ΔH°/R)(1/T) + ΔS°/R. Plotting ln K against 1/T for several temperatures gives a line whose slope is −ΔH°/R and whose y-intercept is ΔS°/R — this is exactly what the Advanced Tools van 't Hoff plot on this page fits automatically.

How to Use This Calculator

On the Standard Solver tab, start by choosing what you're solving for from the dropdown. Solving for K₂ — the most common use — needs K₁, both temperatures (each can be entered in kelvin, °C, or °F), and ΔH° in your choice of J/mol, kJ/mol, cal/mol, or kcal/mol. Solving for ΔH° instead needs both equilibrium constants and both temperatures, which is exactly the situation when a lab has measured K at two different temperatures and wants the enthalpy change behind that shift. Solving for T₂ or T₁ needs one known equilibrium constant, ΔH°, and the other temperature, and answers questions like 'at what temperature would this reaction's equilibrium constant double?'

The Advanced Tools tab is for when you have more than two data points — a proper temperature study. Enter each temperature and its corresponding equilibrium constant as a row (add as many as you need, up to ten), and the calculator fits a best-fit straight line through ln K versus 1/T using linear regression, then reports ΔH°, ΔS°, and the R² goodness-of-fit value so you can judge how well the data actually follows the van 't Hoff relationship.

Worked Example: Finding K₂ at a New Temperature

A reaction has an equilibrium constant K₁ = 0.113 at T₁ = 298 K. Its standard enthalpy change is ΔH° = +50 kJ/mol (endothermic). What is the equilibrium constant K₂ at T₂ = 350 K?

First convert ΔH° to base units: 50 kJ/mol = 50,000 J/mol. Then apply K₂ = K₁ × e^[−(ΔH°/R)(1/T₂ − 1/T₁)] = 0.113 × e^[−(50,000/8.314)(1/350 − 1/298)] = 0.113 × e^[−6,012.03 × (−0.0004977)] = 0.113 × e^2.9925 ≈ 0.113 × 19.95 ≈ 2.25.

K₂ came out much larger than K₁, which makes sense: for an endothermic reaction, raising the temperature always pushes the equilibrium constant higher, shifting the reaction further toward products.

Worked Example: Finding ΔH° from Two Measured Equilibrium Constants

A lab measures K = 0.0025 at 280 K and K = 0.041 at 320 K for the same reaction. What is the reaction's standard enthalpy change?

Apply ΔH° = −R × ln(K₂/K₁) / (1/T₂ − 1/T₁) = −8.314 × ln(0.041/0.0025) / (1/320 − 1/280) = −8.314 × ln(16.4) / (−0.0004464) = −8.314 × 2.797 / (−0.0004464) ≈ +52,090 J/mol, or about +52.1 kJ/mol.

The positive ΔH° confirms this is an endothermic reaction — exactly what the rising K value with rising temperature already suggested before doing any arithmetic at all.

Worked Example: A Full van 't Hoff Plot from Multiple Data Points

Suppose a reaction's equilibrium constant is measured at five temperatures: K = 0.00037 at 280 K, K = 0.00134 at 300 K, K = 0.00414 at 320 K, K = 0.01119 at 340 K, and K = 0.0271 at 360 K.

Converting each point to (1/T, ln K) and fitting a straight line by least squares gives a slope of about −5,412 and a y-intercept of about 11.43. Since slope = −ΔH°/R, ΔH° = −(−5,412) × 8.314 ≈ 45,000 J/mol, or 45 kJ/mol. Since intercept = ΔS°/R, ΔS° = 11.43 × 8.314 ≈ 95 J/(mol·K).

This is exactly how ΔH° and ΔS° are found experimentally in a real lab: rather than trusting a single pair of readings (which is sensitive to measurement error), a full range of temperatures is measured and a best-fit line is drawn through all of them, giving a far more reliable pair of values along with an R² statistic that shows how well the data actually behaves linearly.

Endothermic vs Exothermic: What the Sign of ΔH° Tells You

The sign of ΔH° completely determines which way K moves as temperature changes, and this calculator reports that interpretation automatically alongside the numeric result.

For an endothermic reaction (ΔH° > 0), heat behaves like a reactant that's being consumed. Adding more heat by raising the temperature pushes the equilibrium further toward products, so K increases as temperature rises. For an exothermic reaction (ΔH° < 0), heat behaves like a product being released. Raising the temperature pushes the equilibrium back toward reactants, so K decreases as temperature rises. This is simply Le Chatelier's principle applied to temperature as the disturbance, and the van 't Hoff equation is the quantitative version of that same qualitative rule.

The Key Assumption: ΔH° and ΔS° Are Treated as Constant

The two-point van 't Hoff equation used in the Standard Solver assumes that ΔH° (and, in the linear plot form, ΔS° too) doesn't change meaningfully over the temperature range being considered. In reality, both ΔH° and ΔS° do drift somewhat with temperature, because the heat capacities of reactants and products aren't perfectly equal and aren't perfectly constant either.

For most everyday temperature ranges — a lab bench study spanning 20-80 K, for instance — this assumption holds well enough that the van 't Hoff equation gives accurate, useful results. Across very large temperature swings, or for reactions where heat capacity changes sharply, the relationship becomes noticeably curved rather than a straight line, and the R² value from the Advanced Tools van 't Hoff plot is exactly the number that reveals whether that curvature is significant in your own data.

Where the van 't Hoff Equation Is Used in Real Life

Predicting how far an industrial reaction (like ammonia synthesis in the Haber process) will proceed at different operating temperatures, so engineers can pick a temperature that balances yield against reaction speed. Determining the binding enthalpy of a drug to its target protein from binding-affinity measurements taken at several temperatures — a standard technique in pharmacology and structural biology. Studying protein folding and denaturation, where the 'reaction' is folded protein in equilibrium with unfolded protein and the van 't Hoff enthalpy describes the thermal stability of the fold. Estimating how solubility, vapor pressure, or reaction yield will shift with seasonal or process temperature changes in chemical engineering and environmental chemistry.

Common Mistakes to Avoid

A handful of small errors account for most incorrect answers when working with the van 't Hoff equation by hand.

• Using °C or °F directly in the equation — both temperatures must be absolute temperature in kelvin, since 1/T only makes physical sense on an absolute scale.
• Mixing energy units mid-calculation — keep ΔH° in J/mol (not kJ/mol) when substituting directly, or convert consistently throughout the whole calculation.
• Flipping which temperature is T₁ and which is T₂ — this flips the sign of the whole right-hand side and gives the reciprocal-looking wrong answer.
• Forgetting that K must always be positive — K comes from concentrations or pressures, which can't be negative or zero, so ln K is undefined for K ≤ 0.
• Assuming a single pair of (T, K) readings gives a trustworthy ΔH° — a two-point calculation is exact only if both measurements are error-free; whenever more than two temperatures are available, a full van 't Hoff plot (Advanced Tools) gives a far more reliable value.

Limitations to Keep in Mind

This calculator assumes ΔH° (and, for the linear plot, ΔS°) is constant across the temperature range entered, which is an approximation rather than an exact law. It also assumes ideal behavior for the species involved in the equilibrium. For very wide temperature ranges, reactions with a large change in heat capacity between reactants and products, or high-precision thermodynamic work, a temperature-dependent ΔCp correction to the van 't Hoff equation is needed, which is beyond a simple two-point or linear-fit calculation. Always compare results against published thermodynamic data where precision matters.

Quick Reference: Every Formula on This Page

ln(K₂/K₁) = −(ΔH°/R)(1/T₂ − 1/T₁) — the integrated, two-point van 't Hoff equation used in the Standard Solver. ln K = −(ΔH°/R)(1/T) + ΔS°/R — the linear plot form used in the Advanced Tools van 't Hoff plot, where slope = −ΔH°/R and intercept = ΔS°/R. dlnK/dT = ΔH°/(RT²) — the original differential form the equation is derived from. R = 8.314 J/(mol·K) — the gas constant used throughout.