1. Write the ion product constant for the autoionization of water, and use it to relate [H⁺] and [OH^–] in aqueous solutions.
2. Describe the pH scale, and calculate pH from a knowledge of [H⁺] or [OH^–].
3. Carry out numerical calculations involving the relationships among [H⁺], [OH^–], pH and pOH.

Autoionization and the Ion-Product of Water. Pure water is itself a very weak electrolyte and ionizes according to the equation:

H₂O H⁺(aq) + OH^–(aq)

According to the Brønsted theory, the reaction is viewed as a proton transfer from one water molecule to another.

H2O(l)	+	H2O(l)		H3O+(aq)	+	OH–(aq)
acid1		base2		acid2		base1

Since water can act as both an acid and a base, it is an amphoteric substance.

This reaction is reversible and H₂O, H₃O⁺ and OH^– are in equilibrium. In pure water at 25°C, [H⁺] = [OH^–] = 1.0 x 10^–7 M. These are low concentrations and tell us that very few H₂O molecules are ionized, and that the equilibrium lies to the left.

At equilibrium, the product of the hydrogen ion concentration and hydroxide ion concentration equals a constant called the ion-product constant for water, K_w.

K^w = [H⁺][OH^–] = (1.0 x 10^–7)(1.0 x 10^–7) = 1.0 x 10^–14

Like other equilibrium constants we treat K_w as unitless. The value of K_w applies to all aqueous solutions at 25°C. When an acid is added to water, the [H⁺] increases. Therefore, the [OH^–] must decrease in order for K_w to remain constant. In acidic solutions [H⁺] > [OH^–]. Similarly, when a base is added to water, and [OH^–] increases, then [H⁺] must decrease. In basic solutions [OH^–] > [H⁺]. Acidic, basic, and neutral solutions are characterized by the following conditions:

neutral	[H+] = [OH–]
acidic	[H+] > [OH–]
basic	[H+] < [OH–]

The ion product provides a useful relationship for aqueous solutions. If the value of [H⁺] is known, then the concentration of OH^– can be calculated. Similarly, the H⁺ ion concentration can be calculated, if the value of [OH^–] is known. Example 15.3 illustrates this type of calculation. In Table 15.1 below each row corresponds to a solution with the given H⁺ and OH^– concentrations. The table covers the entire practical range of concentrations found in aqueous solutions. Note that the product of the two concentrations in all aqueous solutions is 1.0 x 10^–14.

EXAMPLE Using the Ion-Product Constant, K_w

The H⁺ ion concentration in a certain solution is 5.0 x 10^–5 M. What is the OH^– ion concentration?

x 10^ M

          Correct!

Click a Hint button for help.

The pH Scale. The concentration of H⁺(aq) in a solution can be expressed in terms of the pH scale. The pH of a solution is defined as the negative logarithm of the hydrogen ion concentration.

pH = –log [H⁺]

Recall that the logarithm of a number is the power to which 10 must be raised in order to equal the number. For example, the logarithm of 100 is 2.0 because raising 10 to the 2^nd power gives 100.

100 = 10²
log 100 = 2

The log of a fraction or number less than 1 is a negative number.

= 0.01 = 10^–2
log 0.01 = –2

First, let's find the pH of a neutral solution. In pure water at 25°C; [H⁺] = 1 x 10^–7 M. Using the definition of pH given above, take the log of the H⁺ ion concentration first:

pH = –log (1.0 x 10^–7)
pH = –(–7.0) = 7.0

The pH of a neutral solution is 7.0.

Likewise, for an acidic solution where, for example, the H⁺ ion concentration is 1 x 10^–5 M, the pH is 5.0.

pH = –log (1 x 10^–5) = –(– 5.0) = 5.0

All acidic solutions have a pH < 7.0.

When [H⁺] is not an exact power of 10, the pH is not a round number. Take the following basic solution, for example, if [H⁺] = 2.5 x 10^–9 M, the pH is

pH = –log (2.5 x 10^–9) = –(–8.60)
pH = 8.60

Note that all basic solutions have a pH > 7.0. The pH values corresponding to selected sets of H⁺(aq) and OH^–(aq) concentrations are given in Table 15.1. In terms of pH, solutions that are acidic, basic, and neutral are defined as follows:

neutral	pH = 7.0O
acidic	pH < 7.0
basic	pH > 7.0

The pOH Scale. A scale just like the pH scale has been devised for the hydroxide ion concentration, where

pOH = –log [OH^–]

Just as the H⁺ ion and OH^– ion concentrations are related by the ion-product constant of water, K_w, the pH and pOH are also related.

K_w = [H⁺][OH^–] = 1.0 x 10^–14
pH + pOH = 14

The sum of the pH and pOH values of any solution is always 14 at 25°C. You can see this in Table 15.1. Sum the pH and pOH for each set of H⁺ and OH^– concentrations, and see what you get.

It is also important to notice that a change in pH of one unit corresponds to a 10-fold change in [H⁺]. As H⁺ drops from 10^–8 to 10^–9 M, for instance, the pH changes from 8 to 9. A change of 2.0 pH units corresponds to a 100-fold change in H⁺ ion concentration. Never say "A pH of 2 is twice as acidic as a pH of 4." It is really 100 times more acidic!

Table 15.1 Relationship of pH and pOH in Aqueous Solutions

[H+]	[OH–]	pH	pOH	Nature of Solution
100	10–14	0	14	acidic
10–1	10–13	1	13	acidic
10–2	10–12	2	12	acidic
10–3	10–11	3	11	acidic
10–6	10–8	6	8	acidic
10–7	10–7	7	7	neutral
10–8	10–6	8	6	basic
10–11	10–3	11	3	basic
10–12	10–2	12	2	basic
10–13	10–1	13	1	basic
10–14	100	14	0	basic

The OH^– ion concentration in a certain ammonia solution is 7.2 x 10^–4 M. What is the pOH and pH?

Changing pH Values to [H⁺]. Given the pH, how do we calculate the [H⁺]? Rearrange the equation for pH:

pH = –log [H⁺]

log [H⁺] = –pH

taking the antilog of both sides:

antilog (log [H⁺]) = antilog (–pH)

gives:

[H⁺] = 10^–pH

Any electronic calculator with a 10^x key will easily make the calculation of H⁺ ion concentrations from pH values. Just enter –pH for x and push 10^x.

EXAMPLE H⁺ Ion Concentration from pH

What is the H⁺ ion concentration in a solution with a pOH of 3.9?

x 10^ M

          Correct!

Click a Hint button for help.

The pH of many cola-type soft drinks is about 3.0. How many times greater is the H⁺ concentration in these drinks than in neutral water?

The [H⁺] is about four ten a hundred a thousand ten thousand a hundred thousand a million times greater.

          Correct!

Click a Hint button for help.

Complete the following questions to check your understanding of the material. Select the check button to see if you answered correctly.

3. Calculate the concentration of OH^– ions in an HNO₃ solution where [H⁺] = 0.0010 M.

4. The OH^– ion concentration in an ammonia solution is 7.5 x 10^–3 M. What is the H⁺ ion concentration?

5. Calculate the concentration of H⁺ ions in an acid solution with a pH of 2.29.

6. What is the concentration of OH^– ions in a NaOH solution which has a pOH of 4.90?

7. The pH of solution A is 2.0 and the pH of solution B is 4.0. How many times greater is the H⁺(aq) concentration in solution A than in solution B?

8. The pH of a certain solution is 3.0. How many moles of H₃O⁺(aq) ions are there in 0.10 L of this solution?