The presence of solute particles can make the physical properties (such as boiling point and freezing point) of the solution different from those of the pure solvent. Such effects are more easily studied systematically in cases where the solution is relatively dilute and the solute is nonvolatile (negligible presence in the gas phase; does not exert vapor pressure of its own). The more numerous these solute particles are in solution, the more pronounced the changes on the physical properties. Physical properties that depend on the number of dissolved particles in the solution but not on their chemical identity or nature are known as colligative properties.
When solute B is added to pure solvent A, the vapor pressure of A above the solvent decreases. If the vapor pressure of A above pure solvent A is designated by P°A and the vapor pressure of A above the solution containing B is PA, the change in vapor pressure, defined as ∆P = P°A – PA, is
where XB is the mole fraction of the solute B in solvent A. For a two-component system (that is, no other kind of solute present), XB = 1 – XA, and so by substituting this into the equation, together with the definition ∆P = P°A – PA, one obtains Raoult’s law:
The more solute particles there are in solution, the lower the mole fraction of the solvent would be, and hence the lower the vapor pressure. One limiting case is the trivial scenario where only the solute is present; without any solvent, XA is zero, and so the vapor pressure of A is zero by the equation, which certainly makes sense because there simply isn’t any solvent around to exert a vapor pressure. In the other extreme, when no solute is present, the system is composed entirely of solvent A; its mole fraction is therefore one and its vapor pressure would be the same as that of pure A. In between these two cases, Raoult’s law states that the vapor pressure is linearly proportional to the mole fraction of the solvent.
It should be pointed out that even though we have introduced Raoult’s law in the study of colligative properties, it is not limited to the solvent-nonvolatile solute systems on which we have focused. In a solution with several volatile components (a mixture of benzene and toluene, for example), Raoult’s law states that the vapor pressure of each component is proportional to its mole fraction in the solution below.
The sum of all the mole fractions has to equal one. The total vapor pressure over the solution, then, is the sum of the partial vapor pressures of each component:
This last result is simply an application of Dalton’s law of partial pressures.
Raoult’s law is actually only an idealized description of the behavior of solutions, and holds only when the attraction between molecules of the different components of the mixture is equal to the attraction between the molecules of any one component in its pure state. When this condition does not hold, the relationship between mole fraction and vapor pressure will deviate from Raoult’s law. Solutions that obey Raoult’s law are called ideal solutions, much in the same way that gases obeying PV = nRT are called ideal gases.
Pure water (H2O) freezes at 0°C at 1 atm; however, for every mole of solute particles dissolved in 1 L of water, the freezing point is lowered by 1.86°C. This is because the solute particles interfere with the process of crystal formation that occurs during freezing; the solute particles lower the temperature at which the molecules can align themselves into a crystalline structure.
The formula for calculating this freezing-point depression is:
where ∆Tf is the freezing-point depression (the number of degrees or Kelvin the freezing point is lowered by), Kf is a proportionality constant characteristic of a particular solvent, and m is the molality of the solution (mol solute/kg solvent). Each solvent has its own characteristic Kf . The larger the value, the more sensitive its freezing point is to the presence of solutes.
Note that the molality in question is the total molality of all particles present. A 1 m aqueous solution of NaCl, for example, would correspond to 2 m in solute particles since it dissociates to give 1 m of Na+ ions and 1 m of Cl− ions. It would lead to a freezing-point depression that is twice the magnitude of that of a 1 m aqueous solution of sugar.
Freezing-point depression is the principle behind spreading salt on ice: The freezing point of water is lowered by the presence of the salt, and so the ice melts. Antifreeze (mostly ethylene glycol) also operates by the same principle.
A liquid boils when its vapor pressure equals the atmospheric pressure. Since, as we have seen above, the vapor pressure of a solution is lower than that of the pure solvent, more energy (and consequently a higher temperature) will be required before its vapor pressure equals atmospheric pressure. In other words, the boiling point of a solution is higher than that of the pure solvent. The extent to which the boiling point of a solution is raised relative to that of the pure solvent is given by the following formula:
where ∆Tb is the boiling-point elevation, Kb is a proportionality constant characteristic of a particular solvent, and m is the molality of the solution. Note how similar this equation is in form to that for freezing-point depression: The only difference is that a solvent will have different values for Kf and Kb, and that it is important to keep in mind that in one case the temperature is raised (∆T > 0), while in the other case the temperature is lowered (∆T < 0).
Consider a container separated into two compartments by a semipermeable membrane (which, by definition, selectively permits the passage of certain molecules). One compartment contains pure water, while the other contains water with dissolved solute. The membrane allows water but not the solute molecules to pass through. Because it is more favorable for the two compartments to equalize their concentration, water will diffuse from the compartment containing pure water to the compartment containing the water-solute mixture. This net flow will cause the water level in the compartment containing the solution to rise above the level in the compartment containing pure water.
Because the solute cannot pass through the membrane, the concentrations of solute in the two compartments can never be equal. The pressure exerted by the water level in the solute-containing compartment will eventually oppose the influx of water, and thus the water level will rise only to the point at which it exerts a sufficient pressure to counterbalance the tendency of water to flow across the membrane. This pressure is defined as the osmotic pressure (Π) of the solution, and is given by the formula:
where M is the molarity of the solution, R is the ideal gas constant, and T is the temperature on the Kelvin scale. This equation clearly shows that molarity and osmotic pressure are directly proportional; that is, as the concentration of the solution increases, the osmotic pressure also increases.
The setup behind the concept of osmotic pressure, as described above, may seem at first glance artificial and contrived, but actually is very important in cellular biology, because the cell membrane is a semipermeable membrane that allows only certain types of molecules to diffuse through. The solute concentration in the cytoplasm of a cell relative to that of its environment determines the net direction of the flow of water, and may lead to either shrinking or swelling (maybe even bursting, or lysing) of the cell.