How To Calculate Vapor Pressure

When you are in the presence of water that is starting to boil, your main safety concern is most likely not being scalded owing to the high temperature of the water and the escaping steam. But you may have noticed something else about steam, or for that matter, any type of matter in the form of gas: It doesn't like to be contained, and will "fight," often quite forcefully, to escape. Accounts of accidents involving exploding steam boilers hearken to this threat.

When water or another liquid boils, in physical terms, it is undergoing a phase transition or change of state from liquid to gas. Put another way, the vapor pressure of the liquid has begun to exceed that of the gas above it, usually the Earth's atmosphere. ("Vapor" is a loose term meaning gas, e.g., "water vapor" is H₂O in the gaseous state.)

Solid can also enter the gaseous state directly, "bypassing" the liquid state altogether in a process known as sublimation. In this case, the underlying reason for the phase transition is the same: Solids have their own vapor pressure, and under certain conditions the value of this pressure can exceed atmospheric pressure. But more often, solids transition into liquids.

On Earth, under natural conditions, matter exists in one of three states: solid, liquid or gas. For any one substance, these phases represent sequential increases in the average kinetic energy of the molecules of the substance, reflected in increasing temperature. Some substances, however, exist as gases at room temperature, whereas others are liquids, and still others are solids; this is the result of some molecules being more easily separated within a substance by a given input of thermal energy (heat).

Every element and molecule exists as a solid at 0 K, or absolute zero (about –273 °C). The structure of matter at very low temperatures is a solid crystalline lattice. As temperature increases, the molecules, effectively locked in place, are able to vibrate with enough energy to break free of the lattice, and when this happens substance-wide, the substance is in the liquid state.

In the liquid state, matter assumes the shape of its container, but within the limits of gravity. When kinetic energy increases still more, molecules begin to escape the air-liquid interface and enter the gaseous state, where the only thing limiting the shape of the gas is the container limiting the movement of the high-energy molecules.

When you observe a pot of water at room temperature, it may not be evident, but some water molecules are flitting about above the water's surface, with an equal (and very small) number returning to the water phase at the same time. The system is therefore in equilibrium, and the vapor pressure created by the minimal escaping of H₂O molecules is the equilibrium vapor pressure of water.

As you will see, different substances in the liquid state have different characteristic levels of vapor pressure P_vapor at room temperature, with this value depending on the nature of the intermolecular forces between molecules in the liquid. For example, substances that have weaker intermolecular forces, such as hydrogen bonds, will have higher levels of equilibrium P_vapor because it is easier for molecules to break free of the liquid.

If equilibrium conditions are perturbed by the addition of heat, however, the vapor pressure of the liquid rises toward atmospheric pressure (101.3 kilopascal, 1 atm or 762 torr). If the value of vapor pressure were not temperature-dependent, it would be difficult to get any liquids (or solids) to boil, or evaporate, especially those with high inherent vapor-pressure values.

Once enough heat is added to a liquid to drive its vapor pressure to the level of atmospheric pressure, the liquid begins to boil. How much heat needs to be added depends on the characteristics of the substance. But what if the substance is not pure water, but instead a solution in which a solid substance is dissolved in a liquid such as water?

The addition of solute typically has effects on many of the parameters of a liquid, including its boiling and melting (i.e., freezing) points. The parameters affected by solute concentration are known as colligative ("connection-related") properties. Vapor pressure is lowered by the addition of solute, and the extent to which this occurs depends on the amount of solute added and ultimately the molar ratio of solute to solvent.

• What does lowering vapor pressure do to the boiling point of a solution? When you think about the math, it means that the liquid will then have a larger gap between its own vapor pressure and atmospheric pressure, and you will need more heat added to get it to boil. Its boiling point is therefore increased by some amount. 

The equation of interest in these situations, which you'll see demonstrated below, is a form of what is known as Raoult's law: P_total=∑P_iX_i. Here P_total is the vapor pressure of the solution as a whole, and the right-hand side represents the sum of the products of the individual vapor pressures and mole fractions of the solute and solvent.

Since water is a ubiquitous liquid and solvent, it is worth investigating the factors that determine its vapor pressure equation in more detail.

Water has a P_vapor of 0.031 atm, or less than 1/30th of atmospheric pressure. This helps explain its relatively high boiling point for such a simple molecule; this low value in turn is explained by the hydrogen bonds between oxygen atoms and hydrogen atoms on adjacent molecules (these are intermolecular forces, not true chemical bonds).

When heated from room temperature (about 25 °C) to about 60 °C, the vapor pressure of water rises only slightly. It then begins to rise more sharply before reaching a value of 1 atm at 100 °C (by definition).

Now, it's time for you to see Raoult's law in action. Know as you approach these problems that you can always look up values for P_vapor for particular substances.

A solution contains a mixture of 1 mole (mol) H₂O, 2 mol ethanol (C₂H₅OH), and 1 mol acetaldehyde (CH₃CHO) at 293 K. What is the total vapor pressure of this solution? Note: The partial pressures of these substances at room temperature are 18 torr, 67.5 torr and 740 torr respectively.

First, set up your equation. From above, you have

P_total = P_watX_wat + P_ethX_eth + P_aceX_ace

The mole fractions of the respective substances are the number of moles of each divided by the total moles of substance in the solution, which is 1 + 2 + 1 = 4. Thus you have X_wat = 1/4 – 0.25, X_eth = 2/4 = 0.5, and X_ace = 1/4 = 0.25. (Note that the sum of the mole fractions must always be exactly 1.) Now, you are ready to plug in the given values for the individual vapor pressures and solve for the total vapor pressure of the mixture of solutions:

P_total = (0.25)(18 torr) + (0.5)(67.5 torr) + (0.25)(740 torr) = 223.25 torr.