# Clausius-Clapeyron Equation

Skills to Develop

• Apply the Clausius-Clapeyron equation to estimate the vapor pressure at any temperature.
• Estimate the heat of phase transition from the vapor pressures measured at two temperatures.

The vaporization curves of most liquids have similar shape. The vapor pressure steadily increase as the temperature increases. A good approach is to find a mathematical model for the pressure increase as a function of temperature. Experiments showed that the pressure $$P$$, enthalpy of vaporization, $$\Delta{H_{vap}}$$, and temperature $$T$$ are related,

$P = A e^{(- \Delta H_{vap} / RT)} \tag{1}$

where

• $$R$$ (8.3145 J mol-1 K-1) is the gas constant and
• $$A$$ is an unknown constant.

This is known as the Clausius-Clapeyron equation. If $$P_1$$ and $$P_2$$ are the pressures at two temperatures $$T_1$$ and $$T_2$$, the equation has the form:

$\ln \dfrac{P_1}{P_2}= \dfrac{\Delta H_{vap}}{R} \left( \dfrac{1}{T_2}- \dfrac{1}{T_1} \right) \tag{2}$

The Clausius-Clapeyron equation allows us to estimate the vapor pressure at another temperature, if the vapor pressure is known at some temperature, and if the enthalpy of vaporization is known.

Example 1: Vapor Pressure of Water

The vapor pressure of water is 1.0 atm at 373 K, and the enthalpy of vaporization is 40.7 kJ mol-1. Estimate the vapor pressure at temperature 363 and 383 K respectively.

SOLUTION
Using the Clausius-Clapeyron equation (Equation 1), we have:

$P_{363} = 1.0 \;exp \left[- \left(\dfrac{40,700}{8.3145}\right) \left(\dfrac{1}{363\;K} -\dfrac{1}{373\; K}\right) \right]$

$= 0.697\; atm$

$P_{383} = 1.0 \;exp \left[- \left(40,700/8.3145 \right)\left(\dfrac{1}{383\;K} - \dfrac{1}{373\;K} \right) \right]$

$= 1.409\; atm$

Note that the increase in vapor pressure from 363 K to 373 K is 0.303 atm, but the increase from 373 to 383 K is 0.409 atm. The increase in vapor pressure is not a linear process.

Discussion
We can use the Clausius-Clapeyron equation to construct the entire vaporization curve. There is a deviation from experimental value, that is because the enthalpy of vaporization various slightly with temperature.

The Clausius-Clapeyron equation applies to any phase transition. The following example shows its application in estimating the heat of sublimation.

Example 2: Heat of Sublimation of Ice

The vapor pressures of ice at 268 and 273 are 2.965 and 4.560 torr respectively. Estimate the heat of sublimation of ice.

SOLUTION
The enthalpy of sublimation is $$\Delta{H}_{sub}$$. Use a piece of paper and derive the Clausius-Clapeyron equation so that you can get the form:

$\Delta H_{sub} = \dfrac{ R \ln \left(\dfrac{P_{268}}{P_{268}}\right)}{\dfrac{1}{268 \;K} - \dfrac{1}{273\;K}}$

$= \dfrac{8.3145 \ln \left(\dfrac{2.965}{4.560} \right)}{ \dfrac{1}{268\;K} - \dfrac{1}{273\;K} }$

$= 52370\; J\; mol^{-1}$

Note that the heat of sublimation is the sum of heat of melting and the heat of vaporization.

Discussion
Show that the vapor pressure of ice at 274 K is higher than that of water at the same temperature. Note the curve of vaporization is also called the curve of evaporization

Example 3: Heat of Vaporization of Ethanol

Calculate $$\Delta{H_{vap}}$$ for ethanol, given vapor pressure at 40 oC = 150 torr. The normal boiling point for ethanol is 78 oC.

SOLUTION

Recognize that we have TWO sets of $$(p,T)$$ data:

• Set 1: (150 torr at 40+273K)
• Set 2: (760 torr at 78+273K)

$\ln p = \dfrac{-\Delta{H_{vap}}}{RT} + c$

Substituting into the above equation twice produces:

$\ln 150 = \dfrac{-\Delta{H_{vap}}}{(8.314)\times (313)} + c$

and

$\ln 760 = \dfrac{-\Delta{H_{vap}}}{(8.314)\times (351)} + c$

Subtract these two equations, to produce:

$\ln 150 -\ln 760 = \dfrac{-\Delta{H_{vap}}}{8.314} \left[ \dfrac{1}{313} - \dfrac{1}{351}\right]$

$-1.623 = \dfrac{-\Delta{H_{vap}}}{8.314} \left[ 0.0032 - 0.0028 \right]$

Solving for $$\Delta{H_{vap}}$$ :

$\Delta{H_{vap}} = 3.90 \times 10^4 \text{ joule/mole} = 39.0 \text{ kJ/mole}$

### Contributors

• Chung (Peter) Chieh (Chemistry, University of Waterloo)

• Albert Censullo (California Polytechnic State University)