1.6.1 Acids and Bases

The acid-base reaction class has been studied for quite some time. In 1680, Robert Boyle reported traits of acid solutions that included their ability to dissolve many substances, to change the colors of certain natural dyes, and to lose these traits after coming in contact with alkali (base) solutions. In the eighteenth century, it was recognized that acids have a sour taste, react with limestone to liberate a gaseous substance (now known to be CO₂), and interact with alkalis to form neutral substances. In 1815, Humphry Davy contributed greatly to the development of the modern acid-base concept by demonstrating that hydrogen is the essential constituent of acids. Around that same time, Joseph Louis Gay-Lussac concluded that acids are substances that can neutralize bases and that these two classes of substances can be defined only in terms of each other. The significance of hydrogen was reemphasized in 1884 when Svante Arrhenius defined an acid as a compound that dissolves in water to yield hydrogen cations (now recognized to be hydronium ions) and a base as a compound that dissolves in water to yield hydroxide anions.

Johannes Brønsted and Thomas Lowry proposed a more general description in 1923 in which acids and bases were defined in terms of the transfer of hydrogen ions, H⁺. (Note that these hydrogen ions are often referred to simply as protons, since that subatomic particle is the only component of cations derived from the most abundant hydrogen isotope, ¹H.) A compound that donates a proton to another compound is called a Brønsted-Lowry acid, and a compound that accepts a proton is called a Brønsted-Lowry base. An acid-base reaction is, thus, the transfer of a proton from a donor (acid) to an acceptor (base).

The concept of conjugate pairs is useful in describing Brønsted-Lowry acid-base reactions (and other reversible reactions, as well). When an acid donates H⁺, the species that remains is called the conjugate base of the acid because it reacts as a proton acceptor in the reverse reaction. Likewise, when a base accepts H⁺, it is converted to its conjugate acid. The reaction between water and ammonia illustrates this idea. In the forward direction, water acts as an acid by donating a proton to ammonia and subsequently becoming a hydroxide ion, OH⁻, the conjugate base of water. The ammonia acts as a base in accepting this proton, becoming an ammonium ion, NH₄⁺, the conjugate acid of ammonia. In the reverse direction, a hydroxide ion acts as a base in accepting a proton from ammonium ion, which acts as an acid.

The reaction between a Brønsted-Lowry acid and water is called acid ionization. For example, when hydrogen fluoride dissolves in water and ionizes, protons are transferred from hydrogen fluoride molecules to water molecules, yielding hydronium ions and fluoride ions:

Base ionization of a species occurs when it accepts protons from water molecules. In the example below, pyridine molecules, C₅NH₅, undergo base ionization when dissolved in water, yielding hydroxide and pyridinium ions:

The preceding ionization reactions suggest that water may function as both a base (as in its reaction with hydrogen fluoride) and an acid (as in its reaction with ammonia). Species capable of either donating or accepting protons are called amphiprotic, or more generally, amphoteric, a term that may be used for acids and bases per definitions other than the Brønsted-Lowry one. The equations below show the two possible acid-base reactions for two amphiprotic species, bicarbonate ion and water:

$${\text{HCO}}_3{}^{–}(aq)+{\text{H}}_3{\text{O}}^{+}(aq)+{\text{H}}_2\text{O}(l)\rightleftharpoons {\text{CO}}_3{}^{2–}(aq)+{\text{H}}_3{\text{O}}^{+}(aq)$$

In the liquid state, molecules of an amphiprotic substance can react with one another as illustrated for water in the equations below:

The process in which like molecules react to yield ions is called autoionization. Liquid water undergoes autoionization to a very slight extent; at 25 °C, approximately two out of every billion water molecules are ionized. The extent of the water autoionization process is reflected in the value of its equilibrium constant, the ion-product constant for water, K_w:

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 1.6.1.1.

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 H₃O⁺ 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 (Table 1.6.1.1) of ionization constants for weak acids is provided at the bottom of the page.

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 < HSO₄⁻:

$${\text{CH}}_3{\text{CO}}_2\text{H}(aq)+{\text{H}}_2\text{O}(l)\rightleftharpoons {\text{H}}_3{\text{O}}^{\text{+}}(aq)+{\text{CH}}_3{\text{CO}}_2{}^{\text{−}}(aq)K_{\text{a}}=1.8\times {10}^{\mathrm{-5}}$$ $${\text{HNO}}_2(aq)+{\text{H}}_2\text{O}(l)\rightleftharpoons {\text{H}}_3{\text{O}}^{\text{+}}(aq)+{\text{NO}}_2{}^{\text{−}}(aq)K_{\text{a}}=4.6\times {10}^{\mathrm{-4}}$$ $${\text{HSO}}_4{}^{\text{−}}(aq)+{\text{H}}_2\text{O}(aq)\rightleftharpoons {\text{H}}_3{\text{O}}^{\text{+}}(aq)+{\text{SO}}_4{}^{\mathrm{2-}}(aq)K_{\text{a}}=1.2\times {10}^{\mathrm{-2}}$$

Another measure of the strength of an acid is its percent ionization. The percent ionization of a weak acid is defined in terms of the composition of an equilibrium mixture:

$$\text{\% ionization}=\frac{{[{\text{H}}_3{\text{O}}^{\text{+}}]}_{\text{eq}}}{{[\text{HA]}}_0}\times 100$$where the numerator is equivalent to the concentration of the acid’s conjugate base (per stoichiometry, [A⁻] = [H₃O⁺]). Unlike the K_a value, the percent ionization of a weak acid varies with the initial concentration of acid, typically decreasing as concentration increases. Equilibrium calculations of the sort described later in this chapter can be used to confirm this behavior.

EXAMPLE 1.6.1.4

Calculation of Percent Ionization from pH

Calculate the percent ionization of a 0.125-M solution of nitrous acid (a weak acid), with a pH of 2.09.

Solution

The percent ionization for an acid is:

$$\frac{{[{\text{H}}_3{\text{O}}^{\text{+}}]}_{\text{eq}}}{{[{\text{HNO}}_2]}_0}\times 100$$ 

Converting the provided pH to hydronium ion molarity yields

$$[{\text{H}}_3{\text{O}}^{+}]={10}^{-2.09}=0.0081M$$ 

Substituting this value and the provided initial acid concentration into the percent ionization equation gives

$$\frac{8.1\times {10}^{\mathrm{-3}}}{0.125}\times 100=6.5\%$$ 

(Recall the provided pH value of 2.09 is logarithmic, and so it contains just two significant digits, limiting the certainty of the computed percent ionization.)

Check Your Learning

Calculate the percent ionization of a 0.10-M solution of acetic acid with a pH of 2.89.

ANSWER:

1.3% ionized

LINK TO LEARNING

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

Just as for acids, the relative strength of a base is reflected in the magnitude of its base-ionization constant (K_b) in aqueous solutions. In solutions of the same concentration, stronger bases ionize to a greater extent and so yield higher hydroxide ion concentrations than do weaker bases. A stronger base has a larger ionization constant than does a weaker base. For the reaction of a base, B:

$$\text{B}(aq)+{\text{H}}_2\text{O}(l)\rightleftharpoons {\text{HB}}^{\text{+}}(aq)+{\text{OH}}^{\text{−}}(aq),$$ the ionization constant is written as

$$K_{\text{b}}=\frac{[{\text{HB}}^{\text{+}}]\left[{\text{OH}}^{\text{−}}\right]}{\left[\text{B}\right]}$$Inspection of the data for three weak bases presented below shows the base strength increases in the order

$$\begin{array}{ccccc}{\text{NO}}_2{}^{\text{−}}(aq)+{\text{H}}_2\text{O}(l)\rightleftharpoons {\text{HNO}}_2(aq)+{\text{OH}}^{\text{−}}(aq)\hfill & & \hfill K_{\text{b}}& =\hfill & 2.17\times {10}^{\mathrm{-11}}\hfill \\ {\text{CH}}_3{\text{CO}}_2{}^{\text{−}}(aq)+{\text{H}}_2\text{O}(l)\rightleftharpoons {\text{CH}}_3{\text{CO}}_2\text{H}(aq)+{\text{OH}}^{\text{−}}(aq)\hfill & & \hfill K_{\text{b}}& =\hfill & 5.6\times {10}^{\mathrm{-10}}\hfill \\ {\text{NH}}_3(aq)+{\text{H}}_2\text{O}(l)\rightleftharpoons {\text{NH}}_4{}^{\text{+}}(aq)+{\text{OH}}^{\text{−}}(aq)\hfill & & \hfill K_{\text{b}}& =\hfill & 1.8\times {10}^{\mathrm{-5}}\hfill \end{array}$$

A table of ionization constants for weak bases appears in Table 1.6.1.2 at the end of this chapter.

 

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

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

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

$$K_{\text{a}}\times K_{\text{b}}=\frac{[{\text{H}}_3{\text{O}}^{\text{+}}][{\text{A}}^{\text{−}}]}{\text{[HA]}}\times \frac{\text{[HA]}\left[{\text{OH}}^{\text{−}}\right]}{\left[{\text{A}}^{\text{−}}\right]}=[{\text{H}}_3{\text{O}}^{\text{+}}]\left[{\text{OH}}^{\text{−}}\right]=K_{\text{w}}$$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 1.5.1.2 illustrates this relation for several conjugate acid-base pairs.

The listing of conjugate acid–base pairs shown in Figure 1.6.1.3 is arranged to show the relative strength of each species as compared with water, whose entries are highlighted in each of the table’s columns. In the acid column, those species listed below water are weaker acids than water. These species do not undergo acid ionization in water; they are not Bronsted-Lowry acids. All the species listed above water are stronger acids, transferring protons to water to some extent when dissolved in an aqueous solution to generate hydronium ions. Species above water but below hydronium ion are weak acids, undergoing partial acid ionization, wheres those above hydronium ion are strong acids that are completely ionized in aqueous solution.

If all these strong acids are completely ionized in water, why does the column indicate they vary in strength, with nitric acid being the weakest and perchloric acid the strongest? Notice that the sole acid species present in an aqueous solution of any strong acid is H₃O⁺(aq), meaning that hydronium ion is the strongest acid that may exist in water; any stronger acid will react completely with water to generate hydronium ions. This limit on the acid strength of solutes in a solution is called a leveling effect. To measure the differences in acid strength for “strong” acids, the acids must be dissolved in a solvent that is less basic than water. In such solvents, the acids will be “weak,” and so any differences in the extent of their ionization can be determined. For example, the binary hydrogen halides HCl, HBr, and HI are strong acids in water but weak acids in ethanol (strength increasing HCl < HBr < HI).

The right column of Figure 1.6.1.3 lists a number of substances in order of increasing base strength from top to bottom. Following the same logic as for the left column, species listed above water are weaker bases and so they don’t undergo base ionization when dissolved in water. Species listed between water and its conjugate base, hydroxide ion, are weak bases that partially ionize. Species listed below hydroxide ion are strong bases that completely ionize in water to yield hydroxide ions (i.e., they are leveled to hydroxide). A comparison of the acid and base columns in this table supports the reciprocal relation between the strengths of conjugate acid-base pairs. For example, the conjugate bases of the strong acids (top of table) are all of negligible strength. A strong acid exhibits an immeasurably large K_a, and so its conjugate base will exhibit a K_b that is essentially zero:

$\begin{array}{l}\text{strong}\text{acid}:K_{\text{a}}\approx \infty \\ \text{conjugate}\text{base}:K_{\text{b}}=K_{\text{w}}\text{/}{K}_{\text{a}}=K_{\text{w}}\text{/}\infty \approx 0\end{array}$

A similar approach can be used to support the observation that conjugate acids of strong bases (K_b ≈ ∞) are of negligible strength (K_a ≈ 0).

Acid-Base Equilibrium Calculations

EXAMPLE 1.6.1.5

Determination of K_a from Equilibrium Concentrations

Acetic acid is the principal ingredient in vinegar (Figure 1.6.1.4) that provides its sour taste. At equilibrium, a solution contains [CH₃COOH] = 0.0787 M and [H₃O⁺] = [CH₃COO^–] = 0.00118 M. What is the value of K_a for acetic acid?

Solution

The relevant equilibrium equation and its equilibrium constant expression are shown below. Substitution of the provided equilibrium concentrations permits a straightforward calculation of the K_a for acetic acid.

$${\text{CH}}_3{\text{COO}}_{}\text{H}(aq)+{\text{H}}_2\text{O}(l)\rightleftharpoons {\text{H}}_3{\text{O}}^{\text{+}}(aq)+{\text{CH}}_3{\text{COO}}_{\mathrm{}}{}^{\text{−}}(aq)$$ $$K_{\text{a}}=\frac{[{\text{H}}_3{\text{O}}^{\text{+}}][{\text{CH}}_3{\text{COO}}_{}{}^{\text{–}}]}{[{\text{CH}}_3{\text{COO}}_{}\text{H}]}=\frac{(0.00118)(0.00118)}{0.0787}=1.77\times {10}^{\mathrm{-5}}$$

Check Your Learning

The HSO₄⁻ ion, weak acid used in some household cleansers:

$${\text{HSO}}_4{}^{\text{−}}(aq)+{\text{H}}_2\text{O}(l)\rightleftharpoons {\text{H}}_3{\text{O}}^{\text{+}}(aq)+{\text{SO}}_4{}^{\mathrm{2-}}(aq)$$ 

What is the acid ionization constant for this weak acid if an equilibrium mixture has the following composition:

$$ $[{\text{H}}_3{\text{O}}^{\text{+}}]$ $=0.027M;$ $[{\text{HSO}}_4{}^{\text{−}}]=0.29M;$ and

$[{\text{SO}}_4{}^{\mathrm{2-}}]=0.13M?$ 

ANSWER:

K_a for HSO₄^– = 1.2 × 10⁻²

Table 1.6.1.1 Ionization Constants of Weak Acids

Acid	Formula	Ka at 25 °C	Lewis Structure
acetic	CH3COOH	1.8 × 10−5
carbonic	H2CO3	4.3 × 10−7
	HCO3–	4.7 × 10−11
cyanic	HCNO	2 × 10−4
formic	HCOOH	1.8 × 10−4
hydrofluoric	HF	6.4 × 10−4
hydrogen peroxide	H2O2	2.4 × 10−12
hydrogen sulfate ion	HSO4–	1.2 × 10−2
hydrogen sulfide	H2S	8.9 × 10−8
	HS–	1.0 × 10−19
hypochlorous	HClO	2.9 × 10−8
phosphoric	H3PO4	7.5 × 10−3
	H2PO4–	6.2 × 10−8
	HPO42-	4.2 × 10−13
phosphorous	H3PO3	5 × 10−2
	H2PO3–	2.0 × 10−7
sulfurous	H2SO3	1.6 × 10−2
	HSO3–	6.4 × 10−8

Table 1.6.1.2 Ionization Constants of Weak Bases

Base	Lewis Structure	Kb at 25 °C
ammonia		1.8 × 10−5
dimethylamine		5.9 × 10−4
methylamine		4.4 × 10−4
phenylamine (aniline)		4.3 × 10−10
trimethylamine		6.3 × 10−5

Section Summary

• There are several theories about acids and bases.
• According to Brønsted-Lowry theory:
  • An acid is a compound that donates a proton to another compound.
  • A base is a compound that accepts a proton.
  • An acid-base reaction is the transfer of a proton from a donor (acid) to an acceptor (base).
• Water is capable of either donating or accepting protons and thus is called amphiprotic.
  • the ion-product constant for water, K_w, which describes the extent of the water autoionization process is 1.0 × 10⁻¹⁴ at 25 °C.
• The equilibrium constant for an acid is called the acid-ionization constant, K_a.
  • A strong acid is completely ionized and has a very large K_a.
  • A weak acid is partially ionized and has a small K_a.
• Similarly, the relative strength of a base is reflected in the magnitude of its base-ionization constant (K_b).