Vapour pressure and boiling

Evaporation

Molecules in a liquid have many different speeds, just as in a gas. A molecule near the surface that moves fast enough, and in the right direction, can break free from its neighbours and enter the air. This is evaporation. It happens at every temperature. The molecules that leave are the fastest ones, so the average energy of those left behind falls. This is why a wet cloth feels cold, and why sweat cools the body.

The reverse process is condensation. A molecule of vapour that hits the liquid surface can be caught and stay.

Vapour pressure

Pour some ether into a closed flask, and leave it. At first, evaporation wins. Vapour molecules build up in the space above. More of them hit the surface and condense. At last the two rates become equal. The system is in dynamic equilibrium: molecules still move both ways, but the amount of vapour does not change. The pressure of the vapour at this point is the vapour pressure of the liquid at that temperature.

Three facts about vapour pressure:

• It depends on the liquid. Weak forces between molecules mean easy escape and a high vapour pressure. A liquid with a high vapour pressure is called volatile.
• It rises steeply with temperature, because more molecules have enough energy.
• It does not depend on the amount of liquid or the size of the vessel, as long as some liquid is left. A bigger vessel just means that more liquid evaporates before equilibrium.

At $20\ ^\circ\text{C}$ the vapour pressures are about $58.7\ \text{kPa}$ for diethyl ether, $5.95\ \text{kPa}$ for ethanol and $2.34\ \text{kPa}$ for water. Ether has only dipole-dipole and dispersion forces. Ethanol and water form hydrogen bonds. The order of vapour pressure is the reverse of the order of boiling points.

Boiling

When the temperature is high enough, bubbles of vapour can form inside the liquid. A bubble can grow only if the pressure of the vapour inside it is at least equal to the pressure outside. So a liquid boils when its vapour pressure equals the pressure above it. The temperature is the boiling point.

The normal boiling point is the boiling point at $1\ \text{atm} = 101.325\ \text{kPa}$. For water it is $100.0\ ^\circ\text{C}$. The IUPAC standard boiling point uses $1\ \text{bar}$, and for water it is $99.6\ ^\circ\text{C}$.

Evaporation and boiling are different. Evaporation happens only at the surface and at any temperature. Boiling happens throughout the liquid at one temperature, set by the outside pressure.

Because the boiling point depends on the pressure, water boils at a lower temperature on a mountain, where the air pressure is low. At a height of about $3000\ \text{m}$ the pressure is about $70\ \text{kPa}$, and water boils at $89.8\ ^\circ\text{C}$. Food cooks slowly there. A pressure cooker does the opposite. It keeps the pressure at about $200\ \text{kPa}$, and water boils at $120\ ^\circ\text{C}$, so food cooks faster.

The Clausius-Clapeyron equation

How does the vapour pressure change with temperature? The answer depends on the enthalpy of vaporisation $\Delta H_{\text{vap}}$, the heat needed to turn one mole of liquid into vapour at constant temperature and pressure. For water it is $40.7\ \text{kJ/mol}$ at $100\ ^\circ\text{C}$. A liquid with a larger $\Delta H_{\text{vap}}$ has a vapour pressure that rises more steeply.

The Clausius-Clapeyron equation says that

A graph of $\ln P$ against $1/T$ is a straight line with slope $-\Delta H_{\text{vap}}/R$. If we know the pressure at two temperatures, the constant $C$ cancels:

Use $T$ in kelvin, $R = 8.314\ \text{J K}^{-1}\ \text{mol}^{-1}$ and $\Delta H_{\text{vap}}$ in joules per mole. Take $T_1$ as the normal boiling point, where $P_1 = 101.325\ \text{kPa}$.

Example: water on a mountain. At $P_2 = 70\ \text{kPa}$,

so $T_2 = 362.9\ \text{K}$, which is $89.8\ ^\circ\text{C}$.

The equation assumes that $\Delta H_{\text{vap}}$ does not change with temperature. This is good over a short range. At $25\ ^\circ\text{C}$, far from $100\ ^\circ\text{C}$, it gives $3.74\ \text{kPa}$ for water, while the measured value is $3.17\ \text{kPa}$. The enthalpy of vaporisation of water is $44.0\ \text{kJ/mol}$ at $25\ ^\circ\text{C}$, and the difference matters.

Many nonpolar liquids have $\dfrac{\Delta H_{\text{vap}}}{T_b}$ close to $85$ to $88\ \text{J K}^{-1}\ \text{mol}^{-1}$. This is Trouton's rule. Diethyl ether, with $26.5\ \text{kJ/mol}$ and $307.8\ \text{K}$, gives $86\ \text{J K}^{-1}\ \text{mol}^{-1}$. Water gives $109\ \text{J K}^{-1}\ \text{mol}^{-1}$. The extra comes from hydrogen bonds, which make the liquid more ordered than the vapour.

Set the pressure above the liquid to $101\ \text{kPa}$, and then to $50\ \text{kPa}$. The dashed vertical line moves to the left. A lower pressure means a lower boiling point.

Practise with new numbers

Try it in Python

The program uses the Clausius-Clapeyron equation for water. It finds the vapour pressure at four temperatures and the boiling point at three pressures.

Common mistakes

• Temperature in °C. In the Clausius-Clapeyron equation $T$ is in kelvin.
• $\Delta H$ in kJ/mol with $R$ in J. Use joules with $R = 8.314\ \text{J K}^{-1}\ \text{mol}^{-1}$, or use $R = 0.008314\ \text{kJ K}^{-1}\ \text{mol}^{-1}$.
• Thinking that vapour pressure depends on the amount of liquid. It depends on the liquid and the temperature only.
• Mixing up evaporation and boiling. Evaporation is at the surface at any temperature. Boiling is throughout the liquid at the boiling point.
• Saying that a liquid boils at one temperature only. The normal boiling point is the value at $1\ \text{atm}$. At another pressure the boiling point changes.

Remember these

• Vapour pressure: the pressure of the vapour in equilibrium with its liquid. It depends on the liquid and on the temperature.
• A liquid boils when its vapour pressure equals the outside pressure.
• $\ln\dfrac{P_2}{P_1} = -\dfrac{\Delta H_{\text{vap}}}{R}\left(\dfrac{1}{T_2} - \dfrac{1}{T_1}\right)$ with $T$ in K and $\Delta H$ in J/mol.
• Weak intermolecular forces: high vapour pressure, low boiling point and a small $\Delta H_{\text{vap}}$.
• The normal boiling point is the boiling point at $101.325\ \text{kPa}$.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. A closed flask holds a little liquid ether. A second flask, ten times larger, holds the same amount of ether at the same temperature. Which flask has the higher vapour pressure?

2. Why is the vapour pressure of diethyl ether at $20\ ^\circ\text{C}$ much higher than that of water?

3. On the top of a mountain, water boils at $85\ ^\circ\text{C}$. Is the air pressure there higher or lower than at sea level?

4. Use the Clausius-Clapeyron equation to find the vapour pressure of water at $80\ ^\circ\text{C}$.

5. Why must $\Delta H_{\text{vap}}$ be in joules per mole in the equation, when it is usually printed in kJ/mol?

References

• OpenStax, Chemistry 2e, Chapter 10, sections 10.2 (vapour pressure and boiling) and 10.3 (enthalpy of vaporisation, the Clausius-Clapeyron equation).
• The vapour pressures, boiling points and enthalpies of vaporisation are standard values from the CRC Handbook of Chemistry and Physics.