Chapter 11 Solutions

11.1 Some Definitions

The major component of a solution is called the solventThe major component of a solution.. The minor component of a solution is called the soluteThe minor component of a solution.. By major and minor we mean whichever component has the greater presence by mass or by moles. Sometimes this becomes confusing, especially with substances with very different molar masses. However, here we will confine the discussion to solutions for which the major component and the minor component are obvious.

Solutions exist for every possible phase of the solute and the solvent. Salt water, for example, is a solution of solid NaCl in liquid water; soda water is a solution of gaseous CO₂ in liquid water, while air is a solution of a gaseous solute (O₂) in a gaseous solvent (N₂). In all cases, however, the overall phase of the solution is the same phase as the solvent.

One important concept of solutions is in defining how much solute is dissolved in a given amount of solvent. This concept is called concentrationHow much solute is dissolved in a given amount of solvent.. Various words are used to describe the relative amounts of solute. DiluteA solution with very little solute. describes a solution that has very little solute, while concentratedA solution with a lot of solute. describes a solution that has a lot of solute. One problem is that these terms are qualitative; they describe more or less but not exactly how much.

In most cases, only a certain maximum amount of solute can be dissolved in a given amount of solvent. This maximum amount is called the solubilityThe maximum amount of a solute that can be dissolved in a given amount of a solvent. of the solute. It is usually expressed in terms of the amount of solute that can dissolve in 100 g of the solvent at a given temperature. Table 11.2 "Solubilities of Some Ionic Compounds" lists the solubilities of some simple ionic compounds. These solubilities vary widely: NaCl can dissolve up to 31.6 g per 100 g of H₂O, while AgCl can dissolve only 0.00019 g per 100 g of H₂O.

When the maximum amount of solute has been dissolved in a given amount of solvent, we say that the solution is saturatedA solution with the maximum amount of solute dissolved in it. with solute. When less than the maximum amount of solute is dissolved in a given amount of solute, the solution is unsaturatedA solution with less than the maximum amount of solute dissolved in it.. These terms are also qualitative terms because each solute has its own solubility. A solution of 0.00019 g of AgCl per 100 g of H₂O may be saturated, but with so little solute dissolved, it is also rather dilute. A solution of 36.1 g of NaCl in 100 g of H₂O is also saturated but rather concentrated. Ideally, we need more precise ways of specifying the amount of solute in a solution. We will introduce such ways in Section 11.2 "Quantitative Units of Concentration".

In some circumstances, it is possible to dissolve more than the maximum amount of a solute in a solution. Usually, this happens by heating the solvent, dissolving more solute than would normally dissolve at regular temperatures, and letting the solution cool down slowly and carefully. Such solutions are called supersaturatedA unstable solution with more than the normal maximum amount of solute in it. solutions and are not stable; given an opportunity (such as dropping a crystal of solute in the solution), the excess solute will precipitate from the solution.

It should be obvious that some solutes dissolve in certain solvents but not others. NaCl, for example, dissolves in water but not in vegetable oil. Beeswax dissolves in liquid hexane but not water. What is it that makes a solute soluble in some solvents but not others?

The answer is intermolecular interactions. The intermolecular interactions include London dispersion forces, dipole-dipole interactions, and hydrogen bonding (as described in Chapter 10 "Solids and Liquids"). From experimental studies, it has been determined that if molecules of a solute experience the same intermolecular forces that the solvent does, the solute will likely dissolve in that solvent. So, NaCl—a very polar substance because it is composed of ions—dissolves in water, which is very polar, but not in oil, which is generally nonpolar. Nonpolar wax dissolves in nonpolar hexane but not in polar water. This concept leads to the general rule that “like dissolves like” for predicting whether a solute is soluble in a given solvent. However, this is a general rule, not an absolute statement, so it must be applied with care.

11.2 Quantitative Units of Concentration

Rather than qualitative terms (Section 11.1 "Some Definitions"), we need quantitative ways to express the amount of solute in a solution; that is, we need specific units of concentration. In this section, we will introduce several common and useful units of concentration.

MolarityThe number of moles of solute divided by the number of liters of solution. (M) is defined as the number of moles of solute divided by the number of liters of solution:

$$\text{molarity = }\frac{\text{moles of solute}}{\text{liters of solution}}$$

which can be simplified as

$$\text{M}=\frac{\text{mol}}{\text{L}}\text{, or mol/L}$$

As with any mathematical equation, if you know any two quantities, you can calculate the third, unknown, quantity.

For example, suppose you have 0.500 L of solution that has 0.24 mol of NaOH dissolved in it. The concentration of the solution can be calculated as follows:

$$\text{molarity}=\frac{\text{0}\text{.24 mol NaOH}}{\text{0}\text{.500 L}}=0.48\text{ M NaOH}$$

The concentration of the solution is 0.48 M, which is spoken as “zero point forty-eight molarity” or “zero point forty-eight molar.” If the quantity of the solute is given in mass units, you must convert mass units to mole units before using the definition of molarity to calculate concentration. For example, what is the molar concentration of a solution of 22.4 g of HCl dissolved in 1.56 L? First, convert the mass of solute to moles using the molar mass of HCl (36.5 g/mol):

$$\text{22}\text{.4}\cancel{\text{ g HCl}}\times \frac{\text{1 mol HCl}}{\text{36}\text{.5}\cancel{\text{ g HCl}}}=0.614\text{ mol HCl}$$

Now we can use the definition of molarity to determine a concentration:

$$\text{M}=\frac{\text{0}\text{.614 mol HCl}}{\text{1}\text{.56 L}}=0.394\text{ M}$$

The definition of molarity can be used to determine the amount of solute or the volume of solution, if the other information is given. Example 4 illustrates this situation.

If you need to determine volume, remember the rule that the unknown quantity must be by itself and in the numerator to determine the correct answer. Thus rearrangement of the definition of molarity is required.

A similar unit of concentration is molalityThe number of moles of solute per kilogram of solvent. (m), which is defined as the number of moles of solute per kilogram of solvent, not per liter of solution:

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

Mathematical manipulation of molality is the same as with molarity.

Another way to specify an amount is percentage composition by massRatio of mass of solute to the total mass of a sample times 100. (or mass percentage, % m/m). It is defined as follows:

$$\%\text{ m/m}=\frac{\text{mass of solute}}{\text{mass of entire sample}}\times 100\%$$

It is not uncommon to see this unit used on commercial products (Figure 11.1 "Concentration in Commercial Applications").

Related concentration units are parts per thousand (ppth)Ratio of mass of solute to total mass of sample times 1,000., parts per million (ppm)Ratio of mass of solute to total mass of sample times 1,000,000., and parts per billion (ppb)Ratio of mass of solute to total mass of sample times 1,000,000,000.. Parts per thousand is defined as follows:

$$\text{ppth}=\frac{\text{mass of solute}}{\text{mass of sample}}\times \mathrm{1,000}$$

There are similar definitions for parts per million and parts per billion:

$$\text{ppm}=\frac{\text{mass of solute}}{\text{mass of sample}}\times \mathrm{1,000,000}$$ and $$\text{ppb}=\frac{\text{mass of solute}}{\text{mass of sample}}\times \mathrm{1,000,000,000}$$

Each unit is used for progressively lower and lower concentrations. The two masses must be expressed in the same unit of mass, so conversions may be necessary.

As with molarity and molality, algebraic rearrangements may be necessary to answer certain questions.

For ionic solutions, we need to differentiate between the concentration of the salt versus the concentration of each individual ion. Because the ions in ionic compounds go their own way when a compound is dissolved in a solution, the resulting concentration of the ion may be different from the concentration of the complete salt. For example, if 1 M NaCl were prepared, the solution could also be described as a solution of 1 M Na⁺(aq) and 1 M Cl⁻(aq) because there is one Na⁺ ion and one Cl⁻ ion per formula unit of the salt. However, if the solution were 1 M CaCl₂, there are two Cl⁻(aq) ions for every formula unit dissolved, so the concentration of Cl⁻(aq) would be 2 M, not 1 M.

In addition, the total ion concentration is the sum of the individual ion concentrations. Thus for the 1 M NaCl, the total ion concentration is 2 M; for the 1 M CaCl₂, the total ion concentration is 3 M.

11.3 Dilutions and Concentrations

Often, a worker will need to change the concentration of a solution by changing the amount of solvent. DilutionThe addition of solvent, which decreases the concentration of the solute in the solution. is the addition of solvent, which decreases the concentration of the solute in the solution. ConcentrationThe removal of solvent, which increases the concentration of the solute in the solution. is the removal of solvent, which increases the concentration of the solute in the solution. (Do not confuse the two uses of the word concentration here!)

In both dilution and concentration, the amount of solute stays the same. This gives us a way to calculate what the new solution volume must be for the desired concentration of solute. From the definition of molarity,

$$\text{molarity}=\frac{\text{moles of solute}}{\text{liters of solution}}$$

we can solve for the number of moles of solute:

moles of solute = (molarity)(liters of solution)

A simpler way of writing this is to use M to represent molarity and V to represent volume. So the equation becomes

moles of solute = MV

Because this quantity does not change before and after the change in concentration, the product MV must be the same before and after the concentration change. Using numbers to represent the initial and final conditions, we have

M₁V₁ = M₂V₂

as the dilution equationThe mathematical formula for calculating new concentrations or volumes when a solution is diluted or concentrated.. The volumes must be expressed in the same units. Note that this equation gives only the initial and final conditions, not the amount of the change. The amount of change is determined by subtraction.

Concentrating solutions involves removing solvent. Usually this is done by evaporating or boiling, assuming that the heat of boiling does not affect the solute. The dilution equation is used in these circumstances as well.

11.4 Concentrations as Conversion Factors

Concentration can be a conversion factor between the amount of solute and the amount of solution or solvent (depending on the definition of the concentration unit). As such, concentrations can be useful in a variety of stoichiometry problems. In many cases, it is best to use the original definition of the concentration unit; it is that definition that provides the conversion factor.

A simple example of using a concentration unit as a conversion factor is one in which we use the definition of the concentration unit and rearrange; we can do the calculation again as a unit conversion, rather than as a definition. For example, suppose we ask how many moles of solute are present in 0.108 L of a 0.887 M NaCl solution. Because 0.887 M means 0.887 mol/L, we can use this second expression for the concentration as a conversion factor:

$$0.108\cancel{\text{ L NaCl}}\times \frac{0.887\text{ mol NaCl}}{\cancel{\text{ L NaCl}}}=0.0958\text{ mol NaCl}$$

(There is an understood 1 in the denominator of the conversion factor.) If we used the definition approach, we get the same answer, but now we are using conversion factor skills. Like any other conversion factor that relates two different types of units, the reciprocal of the concentration can be also used as a conversion factor.

Of course, once quantities in moles are available, another conversion can give the mass of the substance, using molar mass as a conversion factor.

More complex stoichiometry problems using balanced chemical reactions can also use concentrations as conversion factors. For example, suppose the following equation represents a chemical reaction:

2AgNO₃(aq) + CaCl₂(aq) → 2AgCl(s) + Ca(NO₃)₂(aq)

If we wanted to know what volume of 0.555 M CaCl₂ would react with 1.25 mol of AgNO₃, we first use the balanced chemical equation to determine the number of moles of CaCl₂ that would react and then use concentration to convert to liters of solution:

$$1.25\cancel{{\text{ mol AgNO}}_{\text{3}}}\times \frac{\text{1}\cancel{{\text{ mol CaCl}}_{\text{2}}}}{2\cancel{{\text{ mol AgNO}}_{\text{3}}}}\times \frac{1\text{ L solution}}{0.555\cancel{{\text{ mol CaCl}}_{\text{2}}}}=1.13{\text{ L CaCl}}_{\text{2}}$$

This can be extended by starting with the mass of one reactant, instead of moles of a reactant.

We can extend our skills even further by recognizing that we can relate quantities of one solution to quantities of another solution. Knowing the volume and concentration of a solution containing one reactant, we can determine how much of another solution of another reactant will be needed using the balanced chemical equation.

We have used molarity exclusively as the concentration of interest, but that will not always be the case. The next example shows a different concentration unit being used.

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.

11.6 Colligative Properties of Ionic Solutes

In Section 11.5 "Colligative Properties of Solutions", we considered the colligative properties of solutions with molecular solutes. What about solutions with ionic solutes? Do they exhibit colligative properties?

There is a complicating factor: ionic solutes separate into ions when they dissolve. This increases the total number of particles dissolved in solution and increases the impact on the resulting colligative property. Historically, this greater-than-expected impact on colligative properties was one main piece of evidence for ionic compounds separating into ions (increased electrical conductivity was another piece of evidence).

For example, when NaCl dissolves, it separates into two ions:

NaCl(s) → Na⁺(aq) + Cl⁻(aq)

This means that a 1 M solution of NaCl actually has a net particle concentration of 2 M. The observed colligative property will then be twice as large as expected for a 1 M solution.

It is easy to incorporate this concept into our equations to calculate the respective colligative property. We define the van’t Hoff factorThe number of particles each solute formula unit breaks apart into when it dissolves. (i) as the number of particles each solute formula unit breaks apart into when it dissolves. Previously, we have always tacitly assumed that the van’t Hoff factor is simply 1. But for some ionic compounds, i is not 1, as shown in Table 11.5 "Ideal van’t Hoff Factors for Ionic Compounds".

The ideal van’t Hoff factor is equal to the number of ions that form when an ionic compound dissolves.

It is the “ideal” van’t Hoff factor because this is what we expect from the ionic formula. However, this factor is usually correct only for dilute solutions (solutions less than 0.001 M). At concentrations greater than 0.001 M, there are enough interactions between ions of opposite charge that the net concentration of the ions is less than expected—sometimes significantly. The actual van’t Hoff factor is thus less than the ideal one. Here, we will use ideal van’t Hoff factors.

Revised equations to calculate the effect of ionization are then easily produced:

ΔT_b = imK_b ΔT_f = imK_g Π = iMRT

where all variables have been previously defined. To calculate vapor pressure depression according to Raoult’s law, the mole fraction of solvent particles must be recalculated to take into account the increased number of particles formed on ionization.