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# The 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 Clausius-Clapeyron Equation

The vaporization curves of most liquids have similar shape. The vapour 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, DHvap, and temperature T are related, P = A exp (- DHvap / R T) where R (= 8.3145 J mol-1 K-1) and A are the gas constant and unknown constant. This is known as the Clausius- Clapeyron equation. If P1 and P2 are the pressures at two temperatures T1 and T2, the equation has the form:
```     P1    DHvap    1     1
ln (---) = ----  (--- - ---)
P2     R      T2    T1
```

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

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, we have:

P363 = 1.0 exp (- (40700/8.3145)(1/363 - 1/373)
= 0.697 atm

P383 = 1.0 exp (- (40700/8.3145)(1/383 - 1/373)
= 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

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 DHsub. Use a piece of paper and derive the Clausius-Clapeyron equation so that you can get the form:

DHsub = R ln (P268 / P268) (1/268 - 1/273)
= 8.3145*ln(2.965/4.560) / (1/268 - 1/273)
= 52370 J mol-1
Note that the heat of sublimation is the sum of heat of melting and the heat of vaproization.

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.

### Confidence Building Questions

• What is the boiling point of water if the atmosphere is 3.0 atm?

Skill -
Apply the Clausius-Clapeyron equation to estimate the temperature for a desirable vapor pressure.

• Skill -