In chapter 10, we examined the properties of pure liquids and solids. It is necessary to understand these states of matter in order to understand what is happening, on a molecular level, when solutions are formed.

Recall that solutions are also known as homogeneous mixtures. Although the most commonly-encountered solutions are those prepared by dissolving a solid solute in a liquid solvent, there are also solutions in the gaseous state (air is a good example) and the solid state (a nickel coin is a solution of 75% copper and 25% nickel.)

Solution Composition
There are many ways to express the concentration of a solution. Probably the most common method of indicating the concentration of a solution of a solid in a liquid is molarity, which is the number of moles of solute divided by the volume of the solution, and was already discussed in chapter 4.

Notice that the molarity of a solution depends on the solution volume, which changes with temperature (most liquids expand at higher temperature). Thus, the molarity of a solution changes with temperature.

Some other methods of expressing concentration are:

• mass percentage : the mass percentage of a solute in a solution is equal to 100 times the mass of the solute divided by the mass of the solution (not solvent!.) Mass percentage has no units.
• mole fraction : the mole fraction of a solute in a solution is equal to the number of moles of solute divided by the total number of moles in the solution (moles of solute plus moles of solvent.) If the solution consists of more than one solute, then the denominator is the sum of the moles of all components. Mole fraction has no units.
• mole percentage: mole percentage is simply mole fraction multiplied by 100.
• molality (abbreviated m): the molality of a solute in a solution is equal to the number of moles of solute divided by the number of kilograms of solvent. Notice that since the molality of a solution is determined by the masses of solute and solvent, it is temperature independent. Do not use m as an abbreviation for moles.
• normality : the normality of a solution is defined as the number of equivalents of solute per liter of solution. This only makes sense, or course, if the meaning of an equivalent is understood. The definition of a chemical equivalent depends on the substance or type of chemical reaction under consideration. Three common situations are described below and in your textbook:
  1. For an acid, one equivalent is that mass which will supply one mole of hydrogen ions. For hydrochloric acid, which is monoprotic, one equivalent is the same as one mole. For sulfuric acid, which is diprotic, one mole contains two equivalents. In other words, the number of equivalents is a whole-number multiple of the number of moles: #eq = (n)(#moles), where n represents the number of ionizable hydrogens in the acid. On the other hand, the equivalent mass of an acid is always either the same as the molar mass or a whole-number fraction (1/2, 1/3, 1/4, etc.) of the molar mass. Thus, the equivalent mass of sulfuric acid is one-half the molar mass, and thus one mole of sulfuric acid contains two equivalents when sulfuric acid participates in an acid/base reaction.
  2. For a base, one equivalent is that mass which will react with one mole of hydrogen ions. For sodium hydroxide NaOH, the equivalent mass is the same as the molar mass. For magnesium hydroxide, Mg(OH)₂, the equivalent mass is one-half the molar mass, and one mole contains two equivalents when magnesium hydroxide participates in an acid/base reaction.
  3. For a substance involved in a redox reaction, one equivalent is that mass which gains (if it is reduced) or loses (if it is oxidized) one mole of electrons per mole of the substance. For example, for the following unbalanced (and incomplete) redox reaction,

the equivalent masses of each of the substances are shown in the following table (determine the oxidation numbers of the transition elements so you'll understand the table):

Normality and equivalents are not used as commonly as they were a few decades ago, but they are useful for a simple reason: by definition, one equivalent of any reactant will react completely with one equivalent of each of the other reactants to yield one equivalent of each of the products. In other words, equivalent ratios for chemical reactions are always 1:1. The same can't be said for mole ratios.

As a result of this concept, two points should be noted:

• Stoichiometric calculations can be performed using equivalents even if the equation isn't balanced. For example, consider the following question: "How many grams of ferric sulfate are produced if 50 g of FeSO₄ react with 30 g of KMnO₄, according to the above reaction?" To answer this using standard stoichiometric calculations, the equation must be balanced. If we use equivalents, we can reason in the following manner to get a ballpark estimate of the answer (refer to the table above for the equivalent masses):
  1. 50 g of FeSO₄ divided by the equivalent mass gives about 1/3 of an equivalent of FeSO₄.
  2. 30 grams of KMnO₄ divided by the equivalent mass gives about one equivalent of KMnO₄.
  3. Thus, this is a limiting reactant problem, and the limiting reactant is FeSO₄. Note that this can be determined directly, without having to worry about mole ratios.
  4. The number of equivalents of Fe₂(SO₄)₃ produced is equal to the number of equivalents of limiting reactant consumed, which is about one-third of an equivalent.
  5. The mass of ferric sulfate formed is equal to the number of equivalents multiplied by the equivalent mass. Thus, about 1/3 of 200 grams, or about 67 grams, of ferric sulfate are produced.

• The equation N₁V₁ = N₂V₂ can be used to do stoichiometric calculations for reactions in solution, where 1 and 2 represent either two reactants, two products, or a reactant and a product in a reaction. (Just as molarity multiplied by volume gives moles, normality multiplied by volume (in liters) gives equivalents.)

The Formation of Solutions
Whether or not two substances mix with one another to form a solution depends on the relative magnitude of three quantities (note: we symbolize solvent by S and solute by U):

1. U-U interactions: which types of intermolecular attractive forces exist between solute molecules? Can you see now why an understanding of the previous chapter is necessary to answer this question?
2. S-S interactions: which types of intermolecular attractive forces exist between solvent molecules?
3. S-U interactions: which types of intermolecular attractive forces exist between solute molecules and solvent molecules?

If S-U is comparable in magnitude to U-U and/or S-S, then the two substances will generally form a solution. If, on the other hand, U-U and/or S-S are significantly stronger than S-U, then a solution will generally not result. Consider the following four situations:

Color coding of U-U, S-S, and S-U columns: red = strong attractions, yellow = intermediate attractions, green = weak attractions.

Colligative Properties
Colligative properties which depend only on the number, not type, of solute particles present in solution. The four colligative properties considered in chapter 11 are:

• Vapor pressure lowering: the vapor pressure of a solvent is reduced by the presence of a solute in the solvent. The solute can be either a nonvolatile solute or another liquid which has an appreciable vapor pressure. Raoult's Law can be used to calculate the vapor pressure of a solution. For a two-component solution,
  
  

  where X represents the Greek letter chi, for the mole fraction of the component, and p⁰ represents the vapor pressure of the pure component. The above equation can be used directly for a mixture of two liquids. See Sample Exercise 11.7, page 526, for an application of this equation.)
  

  When the solute is a nonvolatile solid, the vapor pressure is zero, so the above equation reduces to
  

  
  

  Sample Exercise 11.5, page 521, is an example of the use of this equation.
  

  It is important to remember that solutes which dissociate in water produce more particles than a nonelectrolyte. For example, if separate solutions of sugar and sodium chloride have equal mole fractions, the salt solution will depress the vapor pressure of the water twice as much as the sugar solution, since the sodium chloride produces two moles of particles in solution (one mole of each of the ions) per mole of substance dissolved, while the sugar produces one mole of dissolved particles per mole of sugar.

• Boiling Point Elevation: the presence of a solute raises the boiling point of a solvent. The extent of elevation is directly proportional to the molality of the solution. Since molarity is temperature-dependent, the concentration unit molality is used instead:
  
  

  where delta T_b is the change in boiling point, i (called the van't Hoff factor) represents the number of particles in solution for each mole of solute dissolved, m is the molality, and K_b is called the molal boiling point elevation constant. The only thing which determines K_b is the solvent. The table below illustrates the meaning of the van't Hoff factor:

  Example	Ideal Value for i	Comments
  Any nonelectrolyte	1	no dissociation
  Weak electrolyte	between 1 and 2	partial ionization
  NaCl	2	1:1 strong electrolyte
  CaCl2	3	1:2 strong electrolyte
  Na2SO4	3	2:1 strong electrolyte
  AlCl3	4	1:3 strong electrolyte

  Actually, the ideal value for i in the table is only for 1:1 electrolytes, and even then only at infinite dilution. At higher concentrations, ion pairing occurs (some ions move through the solution as pairs, effectively reducing the number of particles in solution.) Furthermore, for multiply-charged ions, ion pairing is more likely (because of stronger coulombic attraction) even at lower concentrations. Table 11.6 (page 536) compares the actual and expected values of i for several electrolytes at equimolar concentrations.

• Freezing Point Depression: the presence of a solute lowers the freezing point of a solvent. The extent of depression is directly proportional to the molality of the solution:
  
  

  where delta T_f is the change in freezing point, i (called the van't Hoff factor) represents the number of particles in solution for each mole of solute dissolved, m is the molality, and K_f is called the molal freezing point depression constant. The only thing which determines K_f is the solvent.
  

• Osmotic Pressure: The term osmotic pressure only has meaning for a solution; in other words, pure solvents don't have osmotic pressures. If a solution is placed on one side of a semipermeable membrane and the pure solvent is placed on the other side, solvent moves across the membrane (thereby diluting the solution) at a faster rate than solvent moves from the solution to the side containing pure solvent. As a result, the volume of the solution increases and the volume of the pure solvent necessarily decreases. At equilibrium, when there is no longer any change in the volumes, the difference in the two levels, expressed as pressure, is called the osmotic pressure. An equivalent way of defining osmotic pressure is the pressure which has to be applied to the solution side to prevent any increase in volume. Osmotic pressure, like vapor pressure lowering, boiling point elevation, and freezing point depression, is a colligative property. The osmotic pressure of a solution can be determined by the equation,
  
  

  where pi is the osmotic pressure, i is the van't Hoff factor, M is the molarity of the solution, R is the gas constant (in the form 0.08206 liter-atm/mol-K), and T is the absolute temperature. Note that units cancel in such a way that the osmotic pressure is expressed in atmospheres.
  

Last modified December 15, 1998