11.5 Colligative Properties of Solutions

The properties of solutions are very similar to the properties of their respective pure solvents. This makes sense because the majority of the solution is the solvent. However, some of the properties of solutions differ from pure solvents in measurable and predictable ways. The differences are proportional to the fraction that the solute particles occupy in the solution. These properties are called colligative propertiesA property of solutions related to the fraction that the solute particles occupy in the solution, not their identity.; the word colligative comes from the Greek word meaning “related to the number,” implying that these properties are related to the number of solute particles, not their identities.

Before we introduce the first colligative property, we need to introduce a new concentration unit. The mole fractionThe ratio of the number of moles of a component to the total number of moles in a system. of the ith component in a solution, χ_i, is the number of moles of that component divided by the total number of moles in the sample:

$${\chi }_i=\frac{\text{moles of }i\text{th component}}{\text{total moles}}$$

(χ is the lowercase Greek letter chi.) The mole fraction is always a number between 0 and 1 (inclusive) and has no units; it is just a number.

A useful thing to note is that the sum of the mole fractions of all substances in a mixture equals 1. Thus the mole fraction of C₆H₆ in Example 15 could be calculated by evaluating the definition of mole fraction a second time, or—because there are only two substances in this particular mixture—we can subtract the mole fraction of the C₁₀H₈ from 1 to get the mole fraction of C₆H₆.

Now that this new concentration unit has been introduced, the first colligative property can be considered. As was mentioned in Chapter 10 "Solids and Liquids", all pure liquids have a characteristic vapor pressure in equilibrium with the liquid phase, the partial pressure of which is dependent on temperature. Solutions, however, have a lower vapor pressure than the pure solvent has, and the amount of lowering is dependent on the fraction of solute particles, as long as the solute itself does not have a significant vapor pressure (the term nonvolatile is used to describe such solutes). This colligative property is called vapor pressure depressionThe decrease of a solution’s vapor pressure because of the presence of a solute. (or lowering). The actual vapor pressure of the solution can be calculated as follows:

$$P_{\text{soln}}={\chi }_{\text{solv}}{P}_{\text{solv}}^{*}$$

where P_soln is the vapor pressure of the solution, χ_solv is the mole fraction of the solvent particles, and $P_{\text{solv}}^{*}$ is the vapor pressure of the pure solvent at that temperature (which is data that must be provided). This equation is known as Raoult’s lawThe mathematical formula for calculating the vapor pressure of a solution. (the approximate pronunciation is rah-OOLT). Vapor pressure depression is rationalized by presuming that solute particles take positions at the surface in place of solvent particles, so not as many solvent particles can evaporate.

Two colligative properties are related to solution concentration as expressed in molality. As a review, recall the definition of molality:

$$\text{molality}=\frac{\text{moles solute}}{\text{kilograms solvent}}$$

Because the vapor pressure of a solution with a nonvolatile solute is depressed compared to that of the pure solvent, it requires a higher temperature for the solution’s vapor pressure to reach 1.00 atm (760 torr). Recall that this is the definition of the normal boiling point: the temperature at which the vapor pressure of the liquid equals 1.00 atm. As such, the normal boiling point of the solution is higher than that of the pure solvent. This property is called boiling point elevationThe increase of a solution’s boiling point because of the presence of solute..

The change in boiling point (ΔT_b) is easily calculated:

ΔT_b = mK_b

where m is the molality of the solution and K_b is called the boiling point elevation constantThe constant that relates the molality concentration of a solution and its boiling point change., which is a characteristic of the solvent. Several boiling point elevation constants (as well as boiling point temperatures) are listed in Table 11.3 "Boiling Point Data for Various Liquids".

Remember that what is initially calculated is the change in boiling point temperature, not the new boiling point temperature. Once the change in boiling point temperature is calculated, it must be added to the boiling point of the pure solvent—because boiling points are always elevated—to get the boiling point of the solution.

The boiling point of a solution is higher than the boiling point of the pure solvent, but the opposite occurs with the freezing point. The freezing point of a solution is lower than the freezing point of the pure solvent. Think of this by assuming that solute particles interfere with solvent particles coming together to make a solid, so it takes a lower temperature to get the solvent particles to solidify. This is called freezing point depressionThe decrease of a solution’s freezing point because of the presence of solute..

The equation to calculate the change in the freezing point for a solution is similar to the equation for the boiling point elevation:

ΔT_f = mK_f

where m is the molality of the solution and K_f is called the freezing point depression constantThe constant that relates the molality concentration of a solution and its freezing point change., which is also a characteristic of the solvent. Several freezing point depression constants (as well as freezing point temperatures) are listed in Table 11.4 "Freezing Point Data for Various Liquids".

Remember that this equation calculates the change in the freezing point, not the new freezing point. What is calculated needs to be subtracted from the normal freezing point of the solvent because freezing points always go down.

Freezing point depression is one colligative property we use in everyday life. Many antifreezes used in automobile radiators use solutions that have a lower freezing point than normal so that automobile engines can operate at subfreezing temperatures. We also take advantage of freezing point depression when we sprinkle various compounds on ice to thaw it in the winter for safety (Figure 11.2 "Salt and Safety"). The compounds make solutions that have a lower freezing point, so rather than forming slippery ice, any ice is liquefied and runs off, leaving a safer pavement behind.

Before we introduce the final colligative property, we need to present a new concept. A semipermeable membraneA thin membrane that will pass certain small molecules but not others. is a thin membrane that will pass certain small molecules but not others. A thin sheet of cellophane, for example, acts as a semipermeable membrane.

Consider the system in Figure 11.3 "Osmosis"a. A semipermeable membrane separates two solutions having the different concentrations marked. Curiously, this situation is not stable; there is a tendency for water molecules to move from the dilute side (on the left) to the concentrated side (on the right) until the concentrations are equalized, as in Figure 11.3 "Osmosis"b. This tendency is called osmosisThe tendency of solvent molecules to pass through a semipermeable membrane due to concentration differences.. In osmosis, the solute remains in its original side of the system; only solvent molecules move through the semipermeable membrane. In the end, the two sides of the system will have different volumes. Because a column of liquid exerts a pressure, there is a pressure difference Π on the two sides of the system that is proportional to the height of the taller column. This pressure difference is called the osmotic pressureThe tendency of a solution to pass solvent through a semipermeable membrane due to concentration differences., which is a colligative property.

The osmotic pressure of a solution is easy to calculate:

Π = MRT

where Π is the osmotic pressure of a solution, M is the molarity of the solution, R is the ideal gas law constant, and T is the absolute temperature. This equation is reminiscent of the ideal gas law we considered in Chapter 6 "Gases".

Osmotic pressure is important in biological systems because cell walls are semipermeable membranes. In particular, when a person is receiving intravenous (IV) fluids, the osmotic pressure of the fluid needs to be approximately the same as blood serum; otherwise bad things can happen. Figure 11.4 "Osmotic Pressure and Red Blood Cells" shows three red blood cells: Figure 11.4 "Osmotic Pressure and Red Blood Cells"a shows a healthy red blood cell. Figure 11.4 "Osmotic Pressure and Red Blood Cells"b shows a red blood cell that has been exposed to a lower concentration than normal blood serum (a so-called hypotonic solution); the cell has plumped up as solvent moves into the cell to dilute the solutes inside. Figure 11.4 "Osmotic Pressure and Red Blood Cells"c shows a red blood cell exposed to a higher concentration than normal blood serum (hypertonic); water leaves the red blood cell, so it collapses onto itself. Only when the solutions inside and outside the cell are the same (isotonic) will the red blood cell be able to do its job.

Osmotic pressure is also the reason you should not drink seawater if you’re stranded in a lifeboat on an ocean; seawater has a higher osmotic pressure than most of the fluids in your body. You can drink the water, but ingesting it will pull water out of your cells as osmosis works to dilute the seawater. Ironically, your cells will die of thirst, and you will also die. (It is OK to drink the water if you are stranded on a body of freshwater, at least from an osmotic pressure perspective.) Osmotic pressure is also thought to be important—in addition to capillary action—in getting water to the tops of tall trees.