Each of the two components will be present in each of the two phases, but they are very unlikely to be present in the same proportions. We will therefore need a formalism for indicating their concentration in each phase. While we could still express the concentration in molarity (moles/liter), or any other unit of concentration for that matter, a more convenient way for binary component phase equilibria is with mole fractions. The mole fraction is defined as the number of moles of the molecule we wish to quantify divided by the total number of moles of molecules. For example, let's say our solution consisted of 0.25 moles of benzene and 0.50 moles of toluene. The mole fraction of each of the components in this solution is:

x_benzene = (0.25 moles )/(0.25 moles + 0.50 moles) = 0.33

x_toluene = (0.50 moles)/(0.25 moles + 0.50 moles) = 0.67

Because you are dividing moles by moles, concentrations expressed in mole fractions are unitless. To distinguish the molecules in the liquid phase from that of the vapor phase, we express the liquid phase mole fractions as x_i and the vapor phase mole fractions as y_i where i = the component of interest, in this example either benzene or toluene. In any single phase, the mole fractions of all the components in that phase must sum to 1.

At 20 ^oC, the pressure of benzene vapor in equilibrium with pure benzene liquid is 99.4 mbar. At this temperature, the vapor pressure of pure toluene is 29.7 mbar. Let's consider what happens when we mix the two pure liquids so that the mole fraction of benzene in solution (the liquid phase) is x_benzene = 0.5 when the system reaches equilibrium (i.e., after all mixing has taken place and the vapor pressure above the solution has reached a constant value). Note that if the mole fraction of benzene is 0.5, the mole fraction of toluene has to equal x_toluene = 1 - x_benzene = 0.5. Now we ask, "What are the partial pressures of the two individual components and what is the total vapor pressure above the solution"?

There is a reason that we have chosen benzene and toluene for this example. The two molecules are very much alike and have similar types of intermolecular interactions holding them together in the condensed phase. A benzene molecule dissolved in toluene finds itself in a very similar chemical environment to that of a benzene molecule in pure liquid benzene. Solutions in which interactions between the different types of molecules are similar to those of the individual pure liquids are called ideal solutions and their partial pressure above the solution can be approximated by Raoult's law:

P₁ = x₁P₁^o

P₂ = x₂P₂^o

P = P₁ + P₂

Here, we have generalized our binary solution composition to the two components 1 and 2; which you call 1 and which two is totally arbitrary. P_i is the partial pressure of i = 1 or 2 over the solution and P_i^o is the vapor pressure of pure liquid i. Dalton's Law tells us that P, the total pressure above the solution, is simply the sum of the partial pressures of the individual components. A demonstration of Raoult's law suitable for high school students can be found by clicking here.

If we assume the molecules in the vapor behave as ideal gases, we can calculate the mole fraction of each component in the gas phase simply by measuring its partial pressure. For a binary solution:

y₁ = P₁/(P₁ + P₂)

y₂ = P₂/(P₁ + P₂)

Remember that x₁≠ y₁ and x₂ ≠ y₂ and the composition of the vapor will not be the same as the composition of the condensed phase. Rather, the vapor will be enriched in the more volatile component. This enrichment forms the basis for fractional distillation, useful in producing petrochemicals and distilled spirits, among other processes. A simple distillation column, as might be used in a college-level organic chemistry course can be viewed by clicking here.

Many solutions are not ideal; they do not follow Raoult's law very well. Please do not get the concept of an ideal solution (a condensed phase-vapor phase system that follows Raoult's law) confused with that of an ideal gas (a gas that follows the ideal gas law). Both are ideal because their interactions are modeled in a very basic and simple way, but the ideal solution and the ideal gas are two entirely separate concepts. Solutions deviate from ideal solution behavior when molecular interactions between the different components are measurably different than those found in the pure liquids.

If the molecular interactions are more favorable (lower energy) in solution than in the pure liquids, a negative deviation from Raoult's law will occur. This means that the partial pressure above the solution will be less than x_iP_i^o predicted by Raoult's law. If the molecular interactions are less favorable (higher energy) in solution than in the pure liquids, a positive deviation from Raoult's law will occur and the partial pressure above the solution will be greater than that predicted by Raoult's law. The partial pressure of both components of the solution must deviate in the same direction; either both partial pressures are less than predicted or both are greater than predicted by Raoult's law. And even for solutions that are not ideal, there will be at least a very small range in which Raoult's law is an acceptable approximation. This will occur for the solvent (the component present in the greater amount) in a very dilute solution, in other words for each component as x_i --> 1.

One way of displaying vapor pressure data for binary solutions is to plot the mole fraction of one component along the x-axis and the partial pressure of each component along the y-axis. Here is an example that I took from Professor Joel Bowman's webpage on negative deviations from Raoult's law:

The two components, acetone (nail polish remover) and chloroform (an obsolete anesthetic) are completely miscible over the entire range of possible compositions. The mole fraction of chloroform is plotted from 0 to 1 along the x-axis and the vapor pressure is plotted along the y-axis. However, the x-axis can also be used to measure the mole fraction of acetone since x_acetone = 1 - x_chloroform; the mole fraction of the acetone decreases from 1 to 0 as you go left to right. The dashed lines are the ideal solution behavior predicted by Raoult's law whereas the solid lines are actual vapor pressures. Note that the chloroform still follows Raoult's law fairly well for x_chloroform greater than about 0.95 mole fraction and that acetone follows Raoult's law on the opposite side of the graph at values higher than x_acetone = 0.99 (x_chloroform less than 0.01).

For positive deviations from Raoult's law, see the propanone (another name for acetone)-carbon disulfide graph on Dr. Walter's Chemistry 260 lecture note web page.

Regardless of how non-ideal a solution's behavior is over much of the concentration range, there will always be a solution dilute enough in component 2 that the solvent behavior can be approximated by Raoult's law. This makes sense because by Raoult's law, P₁ = x₁P₁^o, as x₁ gets closer and closer to 1 (pure component 1), P₁ has to approach P₁^o (the vapor pressure of pure component 1). But what about the dilute component, x₂? When x₁ approaches 1, x₂ approaches 0 and we can model the behavior of P₂ for the very dilute component by Henry's law:

P₂ = x₂P₂^*

We only give the equation for component 2 because Henry's law is only applied to the dilute component and never to the solvent, which will follow Raoult's law. P₂^* is the Henry's law constant and in the equation above, it will have units of pressure (Torr, mbar, kPa, etc.). As you can see, Henry's law looks a lot like Raoult's law and you use it much the same way but the Henry's law constant is not an actual vapor pressure for component 2 at any x₂. Rather it is the slope of linear fit of component 2's partial pressure data at low concentrations obtained by extrapolation of the data to small x₂ values.

One application of Henry's law that is particularly useful in in determining how much gas will dissolve in a liquid as a function of its partial pressure over the solution. An example calculation. In dissolved-gas applications, concentrations can be given in any number of units besides mole fractions. This will change the units on P₂^*, which might also be given a different symbol (k is common). Don't let this confuse you; it is simple units conversion. To help with the units, here is a Henry's law unit converter. There are some interesting uses of Henry's law that might make good projects for science students. These include gas dissolution in the blood stream in scuba and the related physiological phenomenon of decompression, the use of breath tests to determine intoxication, environmental chemistry, and carbonation (soda pop!).

Although it is called a "constant", P₂^* is much more accurately referred to as the Henry's law coefficient because it will vary with temperature. In general, solids and liquids increase their solubility with an increase in temperature and gases tend to decrease their solubility.