# Clausius Clapeyron Equation

The Clausius Clapeyron equation is a relation that describes the phase transition between the liquid and vapor phases of matter. Depending on the form of the equation, it predicts the vapor pressure of a liquid or the heat of vaporization from the vapor pressures at two temperatures. Another use is predicting whether or not a phase transition will occur under certain conditions.

The premise is that the vaporization curves of most liquids have similar shapes. Over a certain range, the change in vapor pressure as a function of temperature is a nearly linear relationship. In other words, vapor pressure increases as temperature increases.

The equation takes its name for Rudolf Clausius and Benoît Paul Émile Clapeyron. It is also called the Clapeyron equation or the Clausius-Clapeyron relation. Whichever name you choose, the equation finds use in physical chemistry, thermodynamics, and meteorology.

### Clausius Clapeyron Equation Formula

There are several equivalent forms of the Clausius Clapeyron equation, including some forms which are differential equations. But, the most common forms simply relate heat of vaporization, temperature, and vapor pressure or else the vapor pressure and corresponding temperatures for one chemical substance.

ln P = RT/ ΔHvap + C

ln P = – ΔHvap/RT + C

ln(P1/P2) = ΔHvap/R (1/T2 – 1/T1)

ln(P1/P2) = – ΔHvap/R(1/T1 – 1/T2)

ln(P2/P1) = – ΔHvap/R(1/T2 – 1/T1)

Here, P is vapor pressure, R is the ideal gas constant (0.008314 kJ/K·mol), ΔHvap is the heat of vaporization of a substance, T is absolute temperature (in Kelvin), and C is a constant.

The molar enthalpy of vaporization of a liquid is always a positive number, so the Clausius Clapeyron equation predicts that vapor pressure always increases as temperature increases. Note that while you can use the equation over the entire vaporization curve, the prediction deviates from experimental values because enthalpy of vaporization varies slightly according to temperature.

### Clausius Clapeyron Equation Example Problem

For example, we can use the Clausius Clapeyron equation for predicting the vapor pressure of a solution. Calculate the vapor pressure of 1-propanol at 52.8 °C if the vapor pressure is 10.0 torr at 14.7 °C and its heat of vaporization is 47.2 kJ/mol.

The first step is converting units so they work in the equation. Convert Celsius temperatures to Kelvin:

TK = °C + 273.15
T1 = 14.7 °C + 273.15
T1 = 287.85 K
T2 = 52.8 °C + 273.15
T2 = 325.95 K

Next, apply the Clausius Clapeyron equation and solve for P2:

ln(P1/P2) = ΔHvap/R (1/T2 – 1/T1)
ln[10 torr/P2] = (47.2 kJ/mol/0.008314 kJ/K·mol)[1/325.95 K – 1/287.85 K]
ln[10 torr/P2] = 5677(-4.06 x 10-4)
ln[10 torr/P2] = -2.305

Take the antilog of both sides of the equation:

10 torr/P2 = 0.997
P2/10 torr = 10.02
P2 = 100.2 torr

### Applying the Equation to Solids

Most calculations using the Clausius Clapeyron equation involve the phase transition between the liquid and vapor phases. But, the equation also applies to sublimation.

For example, estimate the heat of enthalpy for the sublimation of ice if the vapor pressures of ice at 268 K and 273 K are 2.965 and 4.560 torr, respectively.

ΔHsub = R ln(P2 / P1) / (1/T1 – 1/T2)
ΔHsub = R ln(P273 / P268) / (1/268 – 1/273)
ΔHsub = (0.008314 kJ/K·mol) ln(4.560 / 2.965) / (1/268 – 1/273)
ΔHsub = 52.370 kJ/mol

### References

• Callen, H.B. (1985). Thermodynamics and an Introduction to Thermostatistics. Wiley. ISBN 978-0-471-86256-7.
• Clapeyron, M. C. (1834). “Mémoire sur la puissance motrice de la chaleur“. Journal de l’École polytechnique (in French). 23: 153–190.
• Clausius, R. (1850). “Ueber die bewegende Kraft der Wärme und die Gesetze, welche sich daraus für die Wärmelehre selbst ableiten lassen” [On the motive power of heat and the laws which can be deduced therefrom regarding the theory of heat]. Annalen der Physik (in German). 155 (4): 500–524. doi:10.1002/andp.18501550403
• Iribarne, J.V.; Godson, W.L. (2013). “4. Water-Air systems § 4.8 Clausius–Clapeyron Equation”. Atmospheric Thermodynamics. Springer. ISBN 978-94-010-2642-0.
• Wark, Kenneth (1988) [1966]. “Generalized Thermodynamic Relationships”. Thermodynamics (5th ed.). New York, NY: McGraw-Hill, Inc. ISBN 978-0-07-068286-3.
• Yau, M.K.; Rogers, R.R. (1989). Short Course in Cloud Physics (3rd ed.). Butterworth–Heinemann. ISBN 978-0-7506-3215-7.