Ideal Solutions: Raoult's Law

When two substances whose molecules are very similar form a liquidA state of matter in which the atomic-scale particles remain close together but are able to change their positions so that the matter takes the shape of its container solutionA mixture of one or more substances dissolved in a solvent to give a homogeneous mixture., the vapor pressureThe pressure (or partial pressure) exerted by the gaseous form of a substance in equilibrium with the liquid form. of the mixtureA combination of two or more substances in which the substances retain their chemical identity. is very simply related to the vaporThe gaseous state of a substance that typically exists as a liquid or solid; a gas at a temperature near or below the boiling point of the corresponding liquid. pressures of the pure substances. Suppose, for example, we mix 1 mol benzene with 1 mol toluene

as shown in Fig. 1. The mole fractionIn a mixture, the chemical amount (moles) of one substance divided by the total chemical amount (moles) of all substances present. of benzene, x_b, and the moleThat chemical amount of a substance containing the same number of units as 12 g of carbon-12. fraction of toluene, x_t, are both equal to 0.5. At 79.6°C the measured vapor pressure of this mixture is 516 mmHg, slightly less than 517 mmHg, the average of the vapor pressures of pure benzene (744 mmHg) and of pure toluene (290 mmHg) at the same temperatureA physical property that indicates whether one object can transfer thermal energy to another object..

It is easy to explain this behavior if we assume that because benzene and toluene molecules are so nearly alike, they behave the same way in solution as they do in the pure liquids. Since there are only half as many benzene molecules in the mixture as in pure benzene, the rate at which benzene molecules escape from the surface of the solution will be half the rate at which they would escape from the pure liquid. In consequence the partial vapor pressure of benzene above the mixture will be one-half the vapor pressure of pure benzene. By a similar argument the partial vapor pressure of the toluene above the solution is also one-half that of pure toluene. Accordingly, we can write

p_b = ½ P_b^*      and      p_t = ½ P_t^*

where p_b and p_t are the partial pressures of benzene and toluene vapors, respectively, and P_b^* and P_t^* are the vapor pressures of the pure liquids. The total vapor pressure of the solution is

Figure 1 Vapor-liquid equilibria for (a) pure toluene; (b) a mixture of equal amounts of toluene and benzene: and (c) pure benzene. In the solution (b) only half the molecules are benzene molecules, and so the concentration of benzene molecules in the vapor phase is only half as great as above pure benzene. Note also that although the initial amounts of benzene and toluene in the solution were equal, more benzene than toluene escapes to the gas phase because of benzene’s higher vapor pressure.

The vapor pressure of the mixture is equal to the mean of the vapor pressures of the two pure liquids.

We can generalize the above argument to apply to a liquid solution of any composition involving any two substances A and B whose molecules are very similar. The partial vapor pressure of A above the liquid mixture, p_A, will then be the vapor pressure of pure A, P_A^*, multiplied by the fraction of the molecules in the liquid which are of type A, that is, the mole fraction of A, x_A. In equation form

p_A = x_AP_A^*      (1a)

Similarly for component B

p_B = x_BP_B^*      (1b)

Adding these two partial pressures, we obtain the total vapor pressure

P = p_A + p_B = x_AP_A^* + x_BP_B^*      (2)

Liquid solutions which conform to Eqs. (1) and (2) are said to obey Raoult’s law and to be ideal mixtures or ideal solutions.

In addition to its use in predicting the vapor pressure of a solution, Raoult’s law may be applied to the solubilityThe extent to which a solute dissolves in a solvent; often expressed as the mass of a substance that will dissolve in 100 mL of solvent. of a gas in a liquid. Dividing both sides of Eq. (1a) by P_A^* gives

Since the vapor pressure of any substance has a specific value at a given temperature, Eq. (3) tells us that the mole fraction x_A of a gaseous soluteThe substance added to a solvent to make a solution. is proportional to the partial pressure p_A of that gas above the solution.

For an ideal solution the proportionality constant k_A is the reciprocal of the vapor pressure of the pure solute at the temperature in question. Since vapor pressure increases as temperature increases, k_A, which is 1/P_A^*, must decrease. Thus we expect the solubility of a gas in a liquid to increase as the partial pressure of gas above the solution increases, but to decrease as temperature increases. Equation (3) is known as Henry’s law. It also applies to gaseous solutes which do not form ideal solutions, but in such cases the Henry’s-law constant k_A does not equal the reciprocal of the vapor pressure.

In actual fact very few liquid mixtures obey Raoult’s law exactly. Even for molecules as similar as benzene and toluene, we noted a deviation of 517 mmHg – 516 mmHg, or 1 mmHg at 79.6°C. Much larger deviations occur if the molecules are not very similar. These deviations are of two kinds. As can be seen from Fig. 2, a plot of the vapor pressure against the mole fraction of one component yields a straight line for an ideal solution. For nonideal mixtures the actual vapor pressure can be larger than the ideal value (positive deviation from Raoult’s law) or smaller (negative deviation). Negative deviations correspond to cases where attractions between unlike molecules are greater than those between like molecules.

Figure 2 Deviations from Raoult's lawThe statement that the partial pressure of a gaseous substance in equilibrium with a solution in which that substane is a component is equal to the mole fraction of the substance in the solution times the vapor pressure of the pure substance.. (a) When Raoult's law is obeyed, a plot of vapor pressure against mole fraction yields a straight line. This is nearly true for the benzene and toluene mixture at 79.6°C. (b) A mixture of acetone and chloroform shows negative deviations from Raoult's law at 35°C, indicating that the two different molecules prefer each other's company to their own. (c) The opposite behavior is shown at 55°C by a benzene-methanol mixture where the polar and nonpolar molecules prefer the company of their own kind.

In the case illustrated, acetone (CH₃COCH₃) and chloroform (CHCl₃) can form a weak hydrogen bondAn attractive force, either intramolecular or intermolecular, between an electronegative atom and a hydrogen atom attached to another electronegative atom.:

Because of this extra intermolecular attraction, molecules have more difficulty escaping the solution and the vapor pressure is lower. The opposite is true of a mixture of benzene and methanol. When C₆H₆ molecules are randomly distributed among CH₃OH molecules, the latter cannot hydrogen bond effectively. Molecules can escape more readily from the solution, and the vapor pressure is higher than Raoult’s law would predict.