10.2: pH, pOH, and Relative Strengths of Acids and Bases

Learning Objectives

By the end of this section, you will be able to:

• Explain the characterization of aqueous solutions as acidic, basic, or neutral
• Express hydronium and hydroxide ion concentrations on the pH and pOH scales
• Perform calculations relating pH and pOH
• Assess the relative strengths of acids and bases according to their ionization constants
• Rationalize trends in acid–base strength in relation to molecular structure
• Carry out equilibrium calculations for weak acid–base systems

As discussed earlier, hydronium and hydroxide ions are present both in pure water and in all aqueous solutions, and their concentrations are inversely proportional as determined by the ion product of water (K_w). The concentrations of these ions in a solution are often critical determinants of the solution’s properties and the chemical behaviors of its other solutes, and specific vocabulary has been developed to describe these concentrations in relative terms. A solution is neutral if it contains equal concentrations of hydronium and hydroxide ions; acidic if it contains a greater concentration of hydronium ions than hydroxide ions; and basic if it contains a lesser concentration of hydronium ions than hydroxide ions.

A common means of expressing quantities that may span many orders of magnitude is to use a logarithmic scale. One such scale that is very popular for acid and base concentrations is the p-function, defined as shown where “X” is the quantity of interest and “log” is the base-10 logarithm. The pH of a solution is therefore defined as shown here, where [H₃O⁺] is the molar concentration of hydronium ion in the solution:

pH = −log[H₃O⁺]

$$\text{pH}=\text{−log }[{\text{H}}_3{\text{O}}^{}{\text{⁺}}^{}]$$

Rearranging this equation to isolate the hydronium ion molarity yields the equivalent expression:

$$[{\text{H}}_3{\text{O}}^{}{\text{⁺}}^{}]={10}^{\text{−pH}}$$

Likewise, the hydroxide ion molarity may be expressed as a p-function, or pOH:

$$\text{pOH}=\text{−log}[{\text{OH}}^{\text{−}}]$$

or

$$\left[{\text{OH}}^{\text{−}}\right]={10}^{\text{−pOH}}$$

Finally, the relation between these two ion concentration expressed as p-functions is easily derived from the K_w expression:

$$K_{\text{w}}=[{\text{H}}_3{\text{O}}^{}{\text{⁺}}^{}]\left[{\text{OH}}^{\text{−}}\right]$$

$$$$

$$=\text{pH}+\text{pOH}$$

At 25 °C, the value of K_w is 1.0 $\times$ 10⁻¹⁴, and so:

$$14.00=\text{pH}+\text{pOH}$$

As was shown in Example 14.1, the hydronium ion molarity in pure water (or any neutral solution) is 1.0 $\times$ 10⁻⁷ M at 25 °C. The pH and pOH of a neutral solution at this temperature are therefore:

$$\text{pH}=\text{−log}[{\text{H}}_3{\text{O}}^{}{\text{⁺}}^{}]=\text{−log}(1.0\times 10^{\mathrm{-7}})=7.00$$

$$\text{pOH}=\text{−log}[{\text{OH}}^{\text{−}}]=\text{−log}(1.0\times 10^{\mathrm{-7}})=7.00$$

And so, at this temperature, acidic solutions are those with hydronium ion molarities greater than 1.0 $\times$ 10⁻⁷ M and hydroxide ion molarities less than 1.0 $\times$ 10⁻⁷ M (corresponding to pH values less than 7.00 and pOH values greater than 7.00). Basic solutions are those with hydronium ion molarities less than 1.0 $\times$ 10⁻⁷ M and hydroxide ion molarities greater than 1.0 $\times$ 10⁻⁷ M (corresponding to pH values greater than 7.00 and pOH values less than 7.00).

Figure \(\PageIndex{1}\) shows the relationships between [H₃O⁺], [OH⁻], pH, and pOH for solutions classified as acidic, basic, and neutral.

Figure \(\PageIndex{1}\) The pH and pOH scales represent concentrations of H₃O⁺ and OH⁻, respectively. The pH and pOH values of some common substances at 25 °C are shown in this chart.

Example 14.4

Calculation of pH from [H₃O⁺]

What is the pH of stomach acid, a solution of HCl with a hydronium ion concentration of 1.2 $\times$ 10⁻³ M?

Solution

$$\text{pH}=\text{−log}[{\text{H}}_3{\text{O}}^{}{\text{+}}^{}]$$

$$=\text{−log(}1.2\times {10}^{\mathrm{-3}})$$

$$=\text{−}(\mathrm{-2.92})=2.92$$

(The use of logarithms is explained in Appendix B. When taking the log of a value, keep as many decimal places in the result as there are significant figures in the value.)

Check Your Learning

Water exposed to air contains carbonic acid, H₂CO₃, due to the reaction between carbon dioxide and water:

$${\text{CO}}_2(aq)+{\text{H}}_2\text{O}\text{(}l)\rightleftharpoons {\text{H}}_2{\text{CO}}_3(aq)$$

Air-saturated water has a hydronium ion concentration caused by the dissolved CO₂ of 2.0 $\times$ 10⁻⁶ M, about 20-times larger than that of pure water. Calculate the pH of the solution at 25 °C.

Answer:

5.70

Example 14.5

Calculation of Hydronium Ion Concentration from pH

Calculate the hydronium ion concentration of blood, the pH of which is 7.3.

Solution

\[ \begin{gathered}\mathrm{pH}=-\log \left[\mathrm{H}_3 \mathrm{O}^+\right]=7.3 \\ \log \left[\mathrm{H}_3 \mathrm{O}^+\right]=-7.3 \\ {\left[\mathrm{H}_3 \mathrm{O}^+\right]=10^{-7.3} \text { or } \left[\mathrm{H}_3 \mathrm{O}^+\right]=\text { antilog of }-7.3} \\ {\left[\mathrm{H}_3 \mathrm{O}^+\right]=5 \times 10^{-8} M}\end{gathered} \]

(On a calculator take the antilog, or the “inverse” log, of −7.3, or calculate 10^−7.3.)

Check Your Learning

Calculate the hydronium ion concentration of a solution with a pH of −1.07.

Answer:

12 M

How Sciences Interconnect

Environmental Science

Normal rainwater has a pH between 5 and 6 due to the presence of dissolved CO₂ which forms carbonic acid:

$${\text{H}}_2\text{O}(l)+{\text{CO}}_2(g)⟶{\text{H}}_2{\text{CO}}_3(aq)$$

$${\text{H}}_2{\text{CO}}_3(aq)\rightleftharpoons {\text{H}}^{\text{+}}(aq)+{\text{HCO}}_3{}^{\text{−}}(aq)$$

Acid rain is rainwater that has a pH of less than 5, due to a variety of nonmetal oxides, including CO₂, SO₂, SO₃, NO, and NO₂ being dissolved in the water and reacting with it to form not only carbonic acid, but sulfuric acid and nitric acid. The formation and subsequent ionization of sulfuric acid are shown here:

$${\text{H}}_2\text{O}(l)+{\text{SO}}_3(g)⟶{\text{H}}_2{\text{SO}}_4(aq)$$

$${\text{H}}_2{\text{SO}}_4(aq)⟶{\text{H}}^{}{\text{+}}^{}(aq)+{\text{HSO}}_4{}^{\text{−}}(aq)$$

Carbon dioxide is naturally present in the atmosphere because most organisms produce it as a waste product of metabolism. Carbon dioxide is also formed when fires release carbon stored in vegetation or fossil fuels. Sulfur trioxide in the atmosphere is naturally produced by volcanic activity, but it also originates from burning fossil fuels, which have traces of sulfur, and from the process of “roasting” ores of metal sulfides in metal-refining processes. Oxides of nitrogen are formed in internal combustion engines where the high temperatures make it possible for the nitrogen and oxygen in air to chemically combine.

Acid rain is a particular problem in industrial areas where the products of combustion and smelting are released into the air without being stripped of sulfur and nitrogen oxides. In North America and Europe until the 1980s, it was responsible for the destruction of forests and freshwater lakes, when the acidity of the rain actually killed trees, damaged soil, and made lakes uninhabitable for all but the most acid-tolerant species. Acid rain also corrodes statuary and building facades that are made of marble and limestone ( Figure \(\PageIndex{2}\) ). Regulations limiting the amount of sulfur and nitrogen oxides that can be released into the atmosphere by industry and automobiles have reduced the severity of acid damage to both natural and manmade environments in North America and Europe. It is now a growing problem in industrial areas of China and India.

For further information on acid rain, visit this website hosted by the US Environmental Protection Agency.

Figure \(\PageIndex{2}\) (a) Acid rain makes trees more susceptible to drought and insect infestation, and depletes nutrients in the soil. (b) It also is corrodes statues that are carved from marble or limestone. (credit a: modification of work by Chris M Morris; credit b: modification of work by “Eden, Janine and Jim”/Flickr)

Measuring pH

The acidity of a solution is typically assessed experimentally by measurement of its pH. The pOH of a solution is not usually measured, as it is easily calculated from an experimentally determined pH value. The pH of a solution can be directly measured using a pH meter ( Figure \(\PageIndex{3}\) ).

Figure \(\PageIndex{3}\) (a) A research-grade pH meter used in a laboratory can have a resolution of 0.001 pH units, an accuracy of ± 0.002 pH units, and may cost in excess of $1000. (b) A portable pH meter has lower resolution (0.01 pH units), lower accuracy (± 0.2 pH units), and a far lower price tag. (credit b: modification of work by Jacopo Werther)

The pH of a solution may also be visually estimated using colored indicators ( Figure \(\PageIndex{4}\) ). The acid-base equilibria that enable use of these indicator dyes for pH measurements are described in a later section of this chapter.

Figure \(\PageIndex{4}\) (a) A solution containing a dye mixture, called universal indicator, takes on different colors depending upon its pH. (b) Convenient test strips, called pH paper, contain embedded indicator dyes that yield pH-dependent color changes on contact with aqueous solutions.(credit: modification of work by Sahar Atwa)

Acid and Base Ionization Constants

The relative strength of an acid or base is the extent to which it ionizes when dissolved in water. If the ionization reaction is essentially complete, the acid or base is termed strong; if relatively little ionization occurs, the acid or base is weak. As will be evident throughout the remainder of this chapter, there are many more weak acids and bases than strong ones. The most common strong acids and bases are listed in Figure \(\PageIndex{5}\).

Figure \(\PageIndex{5}\) Some of the common strong acids and bases are listed here.

The relative strengths of acids may be quantified by measuring their equilibrium constants in aqueous solutions. In solutions of the same concentration, stronger acids ionize to a greater extent, and so yield higher concentrations of hydronium ions than do weaker acids. The equilibrium constant for an acid is called the acid-ionization constant, K_a. For the reaction of an acid HA:

$$\text{HA}(aq)+{\text{H}}_2\text{O}(l)\rightleftharpoons {\text{H}}_3{\text{O}}^{}{\text{+}}^{}(aq)+{\text{A}}^{\text{−}}(aq),$$

the acid ionization constant is written

$$K_{\text{a}}=\frac{[{\text{H}}_3{\text{O}}^{}{\text{+}}^{}][{\text{A}}^{\text{−}}]}{\text{[HA]}}$$

where the concentrations are those at equilibrium. Although water is a reactant in the reaction, it is the solvent as well, so we do not include [H₂O] in the equation. The larger the K_a of an acid, the larger the concentration of ${\text{H}}_3{\text{O}}^{}{\text{+}}^{}$ and A⁻ relative to the concentration of the nonionized acid, HA, in an equilibrium mixture, and the stronger the acid. An acid is classified as “strong” when it undergoes complete ionization, in which case the concentration of HA is zero and the acid ionization constant is immeasurably large (K_a ≈ ∞). Acids that are partially ionized are called “weak,” and their acid ionization constants may be experimentally measured. A table of ionization constants for weak acids is provided in Appendix H.

To illustrate this idea, three acid ionization equations and K_a values are shown below. The ionization constants increase from first to last of the listed equations, indicating the relative acid strength increases in the order CH₃CO₂H < HNO₂ < ${\text{HSO}}_4{}^{\text{−}}:$

$${\text{CH}}_3{\text{CO}}_2\text{H}(aq)+{\text{H}}_2\text{O}(l)\rightleftharpoons {\text{H}}_3{\text{O}}^{}{\text{+}}^{}(aq)+{\text{CH}}_3{\text{CO}}_2{}^{\text{−}}(aq)K_{\text{a}}=1.8\times {10}^{\mathrm{-5}}$$

$${\text{HNO}}_2(aq)+{\text{H}}_2\text{O}(l)\rightleftharpoons {\text{H}}_3{\text{O}}^{}{\text{+}}^{}(aq)+{\text{NO}}_2{}^{\text{−}}(aq)K_{\text{a}}=4.6\times {10}^{\mathrm{-4}}$$

$${\text{HSO}}_4{}^{\text{−}}(aq)+{\text{H}}_2\text{O}(aq)\rightleftharpoons {\text{H}}_3{\text{O}}^{}{\text{+}}^{}(aq)+{\text{SO}}_4{}^2{\mathrm{-}}^{}(aq)K_{\text{a}}=1.2\times {10}^{\mathrm{-2}}$$

Link to Learning

View the simulation of strong and weak acids and bases at the molecular level.

Relative Strengths of Conjugate Acid-Base Pairs

Brønsted-Lowry acid-base chemistry is the transfer of protons; thus, logic suggests a relation between the relative strengths of conjugate acid-base pairs. The strength of an acid or base is quantified in its ionization constant, K_a or K_b, which represents the extent of the acid or base ionization reaction. For the conjugate acid-base pair HA / A⁻, ionization equilibrium equations and ionization constant expressions are

$$\text{HA}(aq)+{\text{H}}_2\text{O}(l)\rightleftharpoons {\text{H}}_3{\text{O}}^{}{\text{+}}^{}(aq)+{\text{A}}^{\text{−}}(aq)K_{\text{a}}=\frac{[{\text{H}}_3{\text{O}}^{}{\text{+}}^{}]\left[{\text{A}}^{\text{−}}\right]}{\left[\text{HA}\right]}$$

$${\text{A}}^{\text{−}}(aq)+{\text{H}}_2\text{O}(l)\rightleftharpoons {\text{OH}}^{\text{−}}(aq)+\text{HA}(aq)K_{\text{b}}=\frac{\left[\text{HA}\right]\left[{\text{OH}}^{–}\right]}{\left[{\text{A}}^{\text{−}}\right]}$$

Adding these two chemical equations yields the equation for the autoionization for water:

$$\sout{\text{HA(}aq)}+{\text{H}}_2\text{O}(l)+\sout{{\text{A}}^{\text{−}}(aq)}+{\text{H}}_2\text{O}(l)\rightleftharpoons {\text{H}}_3{\text{O}}^{}{\text{+}}^{}(aq)+\sout{{\text{A}}^{\text{−}}(aq)}+{\text{OH}}^{\text{−}}(aq)+\sout{\text{HA(}aq)}$$

$${\text{2H}}_2\text{O}(l)\rightleftharpoons {\text{H}}_3{\text{O}}^{}{\text{+}}^{}(aq)+{\text{OH}}^{\text{−}}(aq)$$

As discussed in another chapter on equilibrium, the equilibrium constant for a summed reaction is equal to the mathematical product of the equilibrium constants for the added reactions, and so

\( K_a \times K_b=\frac{\left[\mathrm{H}_3 0^{+}\right]\left[\mathrm{A}^{-}\right]}{\left[\mathrm{HA}^{-}\right]} \times \frac{[\mathrm{HA}]\left[\mathrm{OH}^{-}\right]}{\left[\mathrm{A}^{-}\right]} \)

\( \frac{\left[\mathrm{H}_3 0^{+}\right]\left[\mathrm{A}^{-}\right]}{\left[\mathrm{HA}^{-}\right]} \times \frac{[\mathrm{HA}]\left[\mathrm{OH}^{-}\right]}{\left[\mathrm{A}^{-}\right]} =\left[\mathrm{H}_3 0^{+}\right]\left[\mathrm{OH}^{-}\right] \)

\( \left[\mathrm{H}_3 0^{+}\right]\left[\mathrm{OH}^{-}\right] = K_w \)

This equation states the relation between ionization constants for any conjugate acid-base pair, namely, their mathematical product is equal to the ion product of water, K_w. By rearranging this equation, a reciprocal relation between the strengths of a conjugate acid-base pair becomes evident:

$$K_{\text{a}}=K_{\text{w}}\text{/}{K}_{\text{b}}\text{or}{K}_{\text{b}}=K_{\text{w}}\text{/}{K}_{\text{a}}$$

The inverse proportional relation between K_a and K_b means the stronger the acid or base, the weaker its conjugate partner. Figure \(\PageIndex{6}\) illustrates this relation for several conjugate acid-base pairs.

Figure \(\PageIndex{6}\) Relative strengths of several conjugate acid-base pairs are shown.

Effect of Molecular Structure on Acid-Base Strength

Binary Acids and Bases

The acid strength of binary compounds of hydrogen with nonmetals (A) increases as the H-A bond strength decreases down a group in the periodic table. For group 17, the order of increasing acidity is HF < HCl < HBr < HI.

Across a row in the periodic table, the acid strength of binary hydrogen compounds increases with increasing electronegativity of the nonmetal atom because the polarity of the H-A bond increases. Thus, the order of increasing acidity (for removal of one proton) across the second row is CH₄ < NH₃ < H₂O < HF; across the third row, it is SiH₄ < PH₃ < H₂S < HCl (see Figure \(\PageIndex{7}\)).

Figure \(\PageIndex{7}\) The figure shows trends in the strengths of binary acids and bases.

Ternary Acids and Bases

Ternary compounds composed of hydrogen, oxygen, and some third element (“E”) may be structured as depicted in the image below. In these compounds, the central E atom is bonded to one or more O atoms, and at least one of the O atoms is also bonded to an H atom as shown in Figure \(\PageIndex{8}\).

Figure \(\PageIndex{8}\) The figure shows trends in the strengths of binary acids and bases.

If the central atom, E, has a low electronegativity, its attraction for electrons is low. Little tendency exists for the central atom to form a strong covalent bond with the oxygen atom, and bond a between the element and oxygen is more readily broken than bond b between oxygen and hydrogen. Hence bond a is ionic, hydroxide ions are released to the solution, and the material behaves as a base—this is the case with Ca(OH)₂ and KOH. Lower electronegativity is characteristic of the more metallic elements; hence, the metallic elements form ionic hydroxides that are by definition basic compounds.

If, on the other hand, the atom E has a relatively high electronegativity, it strongly attracts the electrons it shares with the oxygen atom, making bond a relatively strongly covalent. The oxygen-hydrogen bond, bond b, is thereby weakened because electrons are displaced toward E. Bond b is polar and readily releases hydrogen ions to the solution, so the material behaves as an acid. High electronegativities are characteristic of the more nonmetallic elements. Thus, nonmetallic elements form covalent compounds containing acidic −OH groups that are called oxyacids.

Increasing the oxidation number of the central atom E also increases the acidity of an oxyacid because this increases the attraction of E for the electrons it shares with oxygen and thereby weakens the O-H bond. Sulfuric acid, H₂SO₄, or O₂S(OH)₂ (with a sulfur oxidation number of +6), is more acidic than sulfurous acid, H₂SO₃, or OS(OH)₂ (with a sulfur oxidation number of +4). Likewise nitric acid, HNO₃, or O₂NOH (N oxidation number = +5), is more acidic than nitrous acid, HNO₂, or ONOH (N oxidation number = +3). In each of these pairs, the oxidation number of the central atom is larger for the stronger acid (Figure \(\PageIndex{9}\)).

Figure \(\PageIndex{9}\) As the oxidation number of the central atom E increases, the acidity also increases.