The Clausius-Clapeyron equation describes how the boiling point (or freezing point) of a substance changes when the surrounding pressure changes. More precisely, it relates the pressure at which a liquid becomes a gas (its vapor pressure) to temperature, using the energy required to vaporize that substance. It’s one of the most practical tools in thermodynamics, used in everything from predicting weather to designing oil refineries.
The Core Idea
Every liquid has a vapor pressure: the pressure at which its molecules escape into the gas phase at a given temperature. Heat it up and the vapor pressure rises. The Clausius-Clapeyron equation tells you exactly how steep that rise is. At its most general, it says the slope of the pressure-versus-temperature curve on a phase diagram equals the heat energy needed for the phase change divided by the temperature times the change in volume:
dP/dT = ΔH / (T × ΔV)
Here, ΔH is the enthalpy of vaporization (the energy needed to turn liquid into gas), T is the absolute temperature in Kelvin, and ΔV is the difference in volume between the gas and liquid phases. This version is sometimes called the Clapeyron equation and applies to any phase transition: liquid to gas, solid to liquid, or solid to gas.
The Integrated Form
The version most people use in practice is a simplified, integrated form that applies specifically to liquid-gas and solid-gas transitions. With a couple of reasonable assumptions (more on those below), the equation becomes:
ln(P₂/P₁) = (ΔH_vap / R) × (1/T₁ − 1/T₂)
This lets you calculate a new boiling point or vapor pressure when conditions change. The variables break down like this:
- P₁ and P₂ are the vapor pressures at two different temperatures, measured in Pascals (or any consistent pressure unit).
- T₁ and T₂ are the corresponding absolute temperatures, always in Kelvin.
- ΔH_vap is the enthalpy of vaporization, the energy required to convert liquid to gas (in joules per mole).
- R is the universal gas constant, 8.314 J/(mol·K).
You can also write it in a more compact form: ln(P) = A − B/T, where A is a constant and B equals ΔH_vap/R. Plot the natural log of vapor pressure against 1/T and you get a roughly straight line. The slope of that line gives you the enthalpy of vaporization.
How to Use It: A Quick Example
Say you know water boils at 100°C (373 K) at standard atmospheric pressure (1 atm), and you want to find its boiling point at the top of a mountain where pressure drops to 0.7 atm. You look up water’s enthalpy of vaporization (about 40,700 J/mol), plug into the equation, and solve for T₂:
T₂ = T₁ / (1 + (R × T₁ / ΔH_vap) × ln(P₁/P₂))
That gives you a boiling point around 90°C, which is why cooking takes longer at high altitude. The equation works the same way in reverse: if you pull a vacuum on a container of liquid, you can calculate the new, lower boiling point under reduced pressure.
The Assumptions Behind It
The integrated form relies on two key assumptions that hold well under everyday conditions but break down in extreme situations.
First, it treats the vapor as an ideal gas and assumes the volume of the liquid is negligible compared to the volume of the gas. A liter of liquid water becomes about 1,700 liters of steam at atmospheric pressure, so ignoring the liquid volume introduces very little error. This assumption works well at temperatures far below a substance’s critical point, where the gas is sparse enough to behave ideally.
Second, it assumes the enthalpy of vaporization stays constant across the temperature range you’re looking at. In reality, ΔH_vap changes with temperature, but over modest ranges (say, a few tens of degrees), the change is small enough to ignore.
Where It Breaks Down
Near a substance’s critical point (the temperature and pressure where liquid and gas become indistinguishable), both assumptions fail. The gas is dense enough that it no longer behaves ideally, and the liquid and gas volumes converge, so you can’t ignore one relative to the other. The enthalpy of vaporization also drops toward zero as you approach the critical point. For conditions anywhere near critical, engineers use more complex equations of state rather than the Clausius-Clapeyron equation.
Applications in Weather and Climate
Meteorologists rely on the Clausius-Clapeyron equation to understand how much moisture the atmosphere can hold. Warmer air has a higher saturation vapor pressure, meaning it can carry more water vapor before condensation occurs. The equation predicts that atmospheric water-holding capacity increases by about 7% for every 1°C of warming. This is sometimes called the “Clausius-Clapeyron rate” and is a foundational number in climate science.
In practice, the global water cycle appears to be intensifying at roughly 3 to 4% per degree Celsius of warming, about half the theoretical rate, because large-scale atmospheric circulation patterns constrain how quickly evaporation and precipitation actually respond. Still, the equation sets the physical ceiling, and it explains why warmer climates produce more intense rainstorms: there’s simply more water vapor available to condense at once.
In meteorological applications, the constants are often expressed per kilogram rather than per mole. The latent heat of vaporization for water is about 2.5 × 10⁶ J/kg, and the gas constant for water vapor is 461.5 J/(kg·K). A common reference point is that water’s saturation vapor pressure is 6.11 hPa at 273 K (0°C).
Applications in Chemical Engineering
Vapor pressure is the backbone of thermodynamic equilibrium calculations in industrial chemistry. The Clausius-Clapeyron equation is used in designing distillation columns and separators, where engineers need to know the vapor pressure of each component at different temperatures to predict how mixtures will separate. In petroleum refining, for instance, researchers use the equation to calculate heats of vaporization for heavy oil fractions from measured vapor pressure data, information that feeds directly into energy balance calculations for distillation columns and heat exchangers.
Origins of the Equation
The equation carries the names of two 19th-century physicists. Émile Clapeyron, a French engineer, first expressed Sadi Carnot’s ideas about heat in mathematical form in 1834, developing the general relationship between pressure, temperature, and phase changes. Rudolf Clausius, a German physicist, later refined this work around 1850, applying the emerging second law of thermodynamics to simplify the equation into the form used for vapor-liquid transitions. Clapeyron’s graphical and analytical approach gave the equation its structure; Clausius provided the thermodynamic framework that made it broadly useful.