6.3: Relative Strengths of Acids and Bases

The larger the [latex]K_a[/latex] of an acid, the larger the concentration of [latex]\ce{H3O+}[/latex] and [latex]\ce{A^{−}}[/latex] relative to the concentration of the nonionized acid, [latex]\ce{HA}[/latex]. Thus a stronger acid has a larger ionization constant than does a weaker acid. The ionization constants increase as the strengths of the acids increase.

The following data on acid-ionization constants indicate the order of acid strength: [latex]\ce{CH3CO2H} < \ce{HNO2} < \ce{HSO4-}[/latex]

[latex]\begin{aligned} \ce{CH3CO2H}(aq) + \ce{H2O}(l) &⇌\ce{H3O+}(aq)+\ce{CH3CO2-}(aq) \quad &K_\ce{a}=1.8×10^{−5} \[latex]4pt] \ce{HNO2}(aq)+\ce{H2O}(l) &⇌\ce{H3O+}(aq)+\ce{NO2-}(aq) &K_\ce{a}=4.6×10^{-4} \[latex]4pt] \ce{HSO4-}(aq)+\ce{H2O}(l) &⇌\ce{H3O+}(aq)+\ce{SO4^2-}(aq) & K_\ce{a}=1.2×10^{−2} \end{aligned} \nonumber[/latex]

Another measure of the strength of an acid is its percent ionization. The percent ionization of a weak acid is the ratio of the concentration of the ionized acid to the initial acid concentration, times 100:

[latex]\% \:\ce{ionization}=\ce{\dfrac{[H3O+]_{eq}}{[HA]_0}}×100\% \label{PercentIon}[/latex]

Because the ratio includes the initial concentration, the percent ionization for a solution of a given weak acid varies depending on the original concentration of the acid, and actually decreases with increasing acid concentration.

We can rank the strengths of bases by their tendency to form hydroxide ions in aqueous solution. The reaction of a Brønsted-Lowry base with water is given by:

[latex]\ce{B}(aq)+\ce{H2O}(l)⇌\ce{HB+}(aq)+\ce{OH-}(aq) \nonumber[/latex]

Water is the acid that reacts with the base, [latex]\ce{HB^{+}}[/latex] is the conjugate acid of the base [latex]\ce{B}[/latex], and the hydroxide ion is the conjugate base of water. A strong base yields 100% (or very nearly so) of OH⁻ and HB⁺ when it reacts with water; Figure 6.3.1 lists several strong bases. A weak base yields a small proportion of hydroxide ions. Soluble ionic hydroxides such as NaOH are considered strong bases because they dissociate completely when dissolved in water.

As we did with acids, we can measure the relative strengths of bases by measuring their 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, [latex]\ce{B}[/latex]:

[latex]\ce{B}(aq)+\ce{H2O}(l)⇌\ce{HB+}(aq)+\ce{OH-}(aq), \nonumber[/latex]

we write the equation for the ionization constant as:

[latex]K_\ce{b}=\ce{\dfrac{[HB+][OH- ]}{[B]}} \nonumber[/latex]

where the concentrations are those at equilibrium. Again, we do not see water in the equation because water is the solvent and has an activity of 1. The chemical reactions and ionization constants of the three bases shown are:

[latex]\begin{aligned} \ce{NO2-}(aq)+\ce{H2O}(l) &⇌\ce{HNO2}(aq)+\ce{OH-}(aq) \quad &K_\ce{b}=2.17×10^{−11} \[latex]4pt] \ce{CH3CO2-}(aq)+\ce{H2O}(l) &⇌\ce{CH3CO2H}(aq)+\ce{OH-}(aq) &K_\ce{b}=5.6×10^{−10} \[latex]4pt] \ce{NH3}(aq)+\ce{H2O}(l) &⇌\ce{NH4+}(aq)+\ce{OH-}(aq) &K_\ce{b}=1.8×10^{−5} \end{aligned} \nonumber[/latex]

A table of ionization constants of weak bases appears in Table E2. As with acids, percent ionization can be measured for basic solutions, but will vary depending on the base ionization constant and the initial concentration of the solution.

Consider the ionization reactions for a conjugate acid-base pair, [latex]\ce{HA − A^{−}}[/latex]:

[latex]\ce{HA}(aq)+\ce{H2O}(l)⇌\ce{H3O+}(aq)+\ce{A-}(aq) \nonumber[/latex]

with [latex]K_\ce{a}=\ce{\dfrac{[H3O+][A- ]}{[HA]}}[/latex].

[latex]\ce{A-}(aq)+\ce{H2O}(l)⇌\ce{OH-}(aq)+\ce{HA}(aq) \nonumber[/latex]

with [latex]K_\ce{b}=\ce{\dfrac{[HA][OH]}{[A- ]}}[/latex].

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

[latex]\begin{align*} \cancel{\ce{HA}(aq)}+\ce{H2O}(l)+\cancel{\ce{A-}(aq)}+\ce{H2O}(l) &⇌ \ce{H3O+}(aq)+\cancel{\ce{A-}(aq)}+\ce{OH-}(aq)+\cancel{\ce{HA}(aq)} \[latex]4pt] \ce{2H2O}(l) &⇌\ce{H3O+}(aq)+\ce{OH-}(aq) \end{align*} \nonumber[/latex]

As shown in the previous chapter on equilibrium, the [latex]K[/latex] expression for a chemical equation derived from adding two or more other equations is the mathematical product of the other equations’ [latex]K[/latex] expressions. Multiplying the mass-action expressions together and cancelling common terms, we see that:

[latex]K_\ce{a}×K_\ce{b}=\ce{\dfrac{[H3O+][A- ]}{[HA]}×\dfrac{[HA][OH- ]}{[A- ]}}=\ce{[H3O+][OH- ]}=K_\ce{w} \nonumber[/latex]

For example, the acid ionization constant of acetic acid (CH₃COOH) is 1.8 × 10⁻⁵, and the base ionization constant of its conjugate base, acetate ion ([latex]\ce{CH3COO-}[/latex]), is 5.6 × 10⁻¹⁰. The product of these two constants is indeed equal to [latex]K_w[/latex]:

[latex]K_\ce{a}×K_\ce{b}=(1.8×10^{−5})×(5.6×10^{−10})=1.0×10^{−14}=K_\ce{w} \nonumber[/latex]

The extent to which an acid, [latex]\ce{HA}[/latex], donates protons to water molecules depends on the strength of the conjugate base, [latex]\ce{A^{−}}[/latex], of the acid. If [latex]\ce{A^{−}}[/latex] is a strong base, any protons that are donated to water molecules are recaptured by [latex]\ce{A^{−}}[/latex]. Thus there is relatively little [latex]\ce{A^{−}}[/latex] and [latex]\ce{H3O+}[/latex] in solution, and the acid, [latex]\ce{HA}[/latex], is weak. If [latex]\ce{A^{−}}[/latex] is a weak base, water binds the protons more strongly, and the solution contains primarily [latex]\ce{A^{−}}[/latex] and [latex]\ce{H3O^{+}}[/latex]—the acid is strong. Strong acids form very weak conjugate bases, and weak acids form stronger conjugate bases (Figure 6.3.2).

Figure 6.3.2 lists a series of acids and bases in order of the decreasing strengths of the acids and the corresponding increasing strengths of the bases. The acid and base in a given row are conjugate to each other.

The first six acids in Figure 6.3.2 are the most common strong acids. These acids are completely dissociated in aqueous solution. The conjugate bases of these acids are weaker bases than water. When one of these acids dissolves in water, their protons are completely transferred to water, the stronger base.

Those acids that lie between the hydronium ion and water in Figure 6.3.2 form conjugate bases that can compete with water for possession of a proton. Both hydronium ions and nonionized acid molecules are present in equilibrium in a solution of one of these acids. Compounds that are weaker acids than water (those found below water in the column of acids) in Figure 6.3.2 exhibit no observable acidic behavior when dissolved in water. Their conjugate bases are stronger than the hydroxide ion, and if any conjugate base were formed, it would react with water to re-form the acid.

The extent to which a base forms hydroxide ion in aqueous solution depends on the strength of the base relative to that of the hydroxide ion, as shown in the last column in Figure 6.3.2. A strong base, such as one of those lying below hydroxide ion, accepts protons from water to yield 100% of the conjugate acid and hydroxide ion. Those bases lying between water and hydroxide ion accept protons from water, but a mixture of the hydroxide ion and the base results. Bases that are weaker than water (those that lie above water in the column of bases) show no observable basic behavior in aqueous solution.

The Ionization of Weak Acids and Weak Bases

Many acids and bases are weak; that is, they do not ionize fully in aqueous solution. A solution of a weak acid in water is a mixture of the nonionized acid, hydronium ion, and the conjugate base of the acid, with the nonionized acid present in the greatest concentration. Thus, a weak acid increases the hydronium ion concentration in an aqueous solution (but not as much as the same amount of a strong acid).

Acetic acid ([latex]\ce{CH3CO2H}[/latex]) is a weak acid. When we add acetic acid to water, it ionizes to a small extent according to the equation:

[latex]\ce{CH3CO2H}(aq)+\ce{H2O}(l)⇌\ce{H3O+}(aq)+\ce{CH3CO2-}(aq) \nonumber[/latex]

giving an equilibrium mixture with most of the acid present in the nonionized (molecular) form. This equilibrium, like other equilibria, is dynamic; acetic acid molecules donate hydrogen ions to water molecules and form hydronium ions and acetate ions at the same rate that hydronium ions donate hydrogen ions to acetate ions to reform acetic acid molecules and water molecules. We can tell by measuring the pH of an aqueous solution of known concentration that only a fraction of the weak acid is ionized at any moment (Figure 6.3.4). The remaining weak acid is present in the nonionized form.

For acetic acid, at equilibrium:

[latex]K_\ce{a}=\ce{\dfrac{[H3O+][CH3CO2- ]}{[CH3CO2H]}}=1.8 \times 10^{−5} \nonumber[/latex]

Table 6.3.1 gives the ionization constants for several weak acids; additional ionization constants can be found in Table E1.

At equilibrium, a solution of a weak base in water is a mixture of the nonionized base, the conjugate acid of the weak base, and hydroxide ion with the nonionized base present in the greatest concentration. Thus, a weak base increases the hydroxide ion concentration in an aqueous solution (but not as much as the same amount of a strong base).

For example, a solution of the weak base trimethylamine, (CH₃)₃N, in water reacts according to the equation:

[latex]\ce{(CH3)3N}(aq)+\ce{H2O}(l)⇌\ce{(CH3)3NH+}(aq)+\ce{OH-}(aq) \nonumber[/latex]

This gives an equilibrium mixture with most of the base present as the nonionized amine. This equilibrium is analogous to that described for weak acids.

We can confirm by measuring the pH of an aqueous solution of a weak base of known concentration that only a fraction of the base reacts with water (Figure 14.4.5). The remaining weak base is present as the unreacted form. The equilibrium constant for the ionization of a weak base, [latex]K_b[/latex], is called the ionization constant of the weak base, and is equal to the reaction quotient when the reaction is at equilibrium. For trimethylamine, at equilibrium:

[latex]K_\ce{b}=\ce{\dfrac{[(CH3)3NH+][OH- ]}{[(CH3)3N]}} \nonumber[/latex]

The ionization constants of several weak bases are given in Table 6.3.2 and Table E2.

Example 6.3.3: Determination of K_a from Equilibrium Concentrations

Acetic acid is the principal ingredient in vinegar; that's why it tastes sour. At equilibrium, a solution contains [CH₃CO₂H] = 0.0787 M and [latex]\ce{[H3O+]}=\ce{[CH3CO2- ]}=0.00118\:M[/latex]. What is the value of [latex]K_a[/latex] for acetic acid?

Solution

We are asked to calculate an equilibrium constant from equilibrium concentrations. At equilibrium, the value of the equilibrium constant is equal to the reaction quotient for the reaction:

[latex]\ce{CH3CO2H}(aq)+\ce{H2O}(l)⇌\ce{H3O+}(aq)+\ce{CH3CO2-}(aq) \nonumber[/latex]

[latex]\begin{align*} K_\ce{a} &=\ce{\dfrac{[H3O+][CH3CO2- ]}{[CH3CO2H]}} \[latex]4pt] &=\dfrac{(0.00118)(0.00118)}{0.0787} \[latex]4pt] &=1.77×10^{−5} \end{align*} \nonumber[/latex]

Example 6.3.6: Equilibrium Concentrations in a Solution of a Weak Acid

Formic acid, HCO₂H, is the irritant that causes the body’s reaction to ant stings.

What is the concentration of hydronium ion and the pH in a 0.534-M solution of formic acid?

[latex]\ce{HCO2H}(aq)+\ce{H2O}(l)⇌\ce{H3O+}(aq)+\ce{HCO2-}(aq) \hspace{20px} K_\ce{a}=1.8×10^{−4} \nonumber[/latex]

Solution

1. Determine x and equilibrium concentrations. The equilibrium expression is:

[latex]\ce{HCO2H}(aq)+\ce{H2O}(l)⇌\ce{H3O+}(aq)+\ce{HCO2-}(aq) \nonumber[/latex]

Because water is the solvent, it has a fixed activity equal to 1. Any small amount of water produced or used up during the reaction will not change water's role as the solvent, so the value of its activity remains equal to 1 throughout the reaction so we do not need to consider it when setting up the ICE table.

The table shows initial concentrations (concentrations before the acid ionizes), changes in concentration, and equilibrium concentrations follows (the data given in the problem appear in color):

2. Solve for [latex]x[/latex] and the equilibrium concentrations. At equilibrium:

[latex]\begin{align*} K_\ce{a} &=1.8×10^{−4}=\ce{\dfrac{[H3O+][HCO2- ]}{[HCO2H]}} \[latex]4pt] &=\dfrac{(x)(x)}{0.534−x}=1.8×10^{−4} \end{align*} \nonumber[/latex]

Now solve for [latex]x[/latex]. Because the initial concentration of acid is reasonably large and [latex]K_a[/latex] is very small, we assume that [latex]x << 0.534[/latex], which permits us to simplify the denominator term as [latex](0.534 − x) = 0.534[/latex]. This gives:

[latex]K_\ce{a}=1.8×10^{−4}=\dfrac{x^{2}}{0.534} \nonumber[/latex]

Solve for [latex]x[/latex] as follows:

[latex]\begin{align*} x^2 &=0.534×(1.8×10^{−4}) \[latex]4pt] &=9.6×10^{−5} \[latex]4pt] x &=\sqrt{9.6×10^{−5}} \[latex]4pt] &=9.8×10^{−3} \end{align*} \nonumber[/latex]

To check the assumption that [latex]x[/latex] is small compared to 0.534, we calculate:

[latex]\begin{align*} \dfrac{x}{0.534} &=\dfrac{9.8×10^{−3}}{0.534} \[latex]4pt] &=1.8×10^{−2} \, \textrm{(1.8% of 0.534)} \end{align*} \nonumber[/latex]

[latex]x[/latex] is less than 5% of the initial concentration; the assumption is valid.

We find the equilibrium concentration of hydronium ion in this formic acid solution from its initial concentration and the change in that concentration as indicated in the last line of the table:

[latex]\begin{align*} \ce{[H3O+]} &=~0+x=0+9.8×10^{−3}\:M. \[latex]4pt] &=9.8×10^{−3}\:M \end{align*} \nonumber[/latex]

The pH of the solution can be found by taking the negative log of the [latex]\ce{[H3O+]}[/latex], so:

[latex]pH = −\log(9.8×10^{−3})=2.01 \nonumber[/latex]

The following example shows that the concentration of products produced by the ionization of a weak base can be determined by the same series of steps used with a weak acid.

Some weak acids and weak bases ionize to such an extent that the simplifying assumption that x is small relative to the initial concentration of the acid or base is inappropriate. As we solve for the equilibrium concentrations in such cases, we will see that we cannot neglect the change in the initial concentration of the acid or base, and we must solve the equilibrium equations by using the quadratic equation.

The Relative Strengths of Strong Acids and Bases

Strong acids, such as [latex]\ce{HCl}[/latex], [latex]\ce{HBr}[/latex], and [latex]\ce{HI}[/latex], all exhibit the same strength in water. The water molecule is such a strong base compared to the conjugate bases Cl⁻, Br⁻, and I⁻ that ionization of these strong acids is essentially complete in aqueous solutions. In solvents less basic than water, we find [latex]\ce{HCl}[/latex], [latex]\ce{HBr}[/latex], and [latex]\ce{HI}[/latex] differ markedly in their tendency to give up a proton to the solvent. For example, when dissolved in ethanol (a weaker base than water), the extent of ionization increases in the order [latex]\ce{HCl < HBr < HI}[/latex], and so [latex]\ce{HI}[/latex] is demonstrated to be the strongest of these acids. The inability to discern differences in strength among strong acids dissolved in water is known as the leveling effect of water.

Water also exerts a leveling effect on the strengths of strong bases. For example, the oxide ion, O²⁻, and the amide ion, [latex]\ce{NH2-}[/latex], are such strong bases that they react completely with water:

[latex]\ce{O^2-}(aq)+\ce{H2O}(l)⟶\ce{OH-}(aq)+\ce{OH-}(aq) \nonumber[/latex]

[latex]\ce{NH2-}(aq)+\ce{H2O}(l)⟶\ce{NH3}(aq)+\ce{OH-}(aq) \nonumber[/latex]

Thus, O²⁻ and [latex]\ce{NH2-}[/latex] appear to have the same base strength in water; they both give a 100% yield of hydroxide ion.

In the absence of any leveling effect, 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 [latex]\ce{HF < HCl < HBr < HI}[/latex]. Likewise, for group 16, the order of increasing acid strength is H₂O < H₂S < H₂Se < H₂Te. 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 [latex]\ce{CH4 < NH3 < H2O < HF}[/latex]; across the third row, it is [latex]\ce{SiH4 < PH3 < H2S < HCl}[/latex] (see Figure 6.3.6).

Compounds containing oxygen and one or more hydroxyl (OH) groups can be acidic, basic, or amphoteric, depending on the position in the periodic table of the central atom E, the atom bonded to the hydroxyl group. Such compounds have the general formula O_nE(OH)_m, and include sulfuric acid, [latex]\ce{O2S(OH)2}[/latex], sulfurous acid, [latex]\ce{OS(OH)2}[/latex], nitric acid, [latex]\ce{O2NOH}[/latex], perchloric acid, [latex]\ce{O3ClOH}[/latex], aluminum hydroxide, [latex]\ce{Al(OH)3}[/latex], calcium hydroxide, [latex]\ce{Ca(OH)2}[/latex], and potassium hydroxide, [latex]\ce{KOH}[/latex]:

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 6.3.7).

Hydroxy compounds of elements with intermediate electronegativities and relatively high oxidation numbers (for example, elements near the diagonal line separating the metals from the nonmetals in the periodic table) are usually amphoteric. This means that the hydroxy compounds act as acids when they react with strong bases and as bases when they react with strong acids. The amphoterism of aluminum hydroxide, which commonly exists as the hydrate [latex]\ce{Al(H2O)3(OH)3}[/latex], is reflected in its solubility in both strong acids and strong bases. In strong bases, the relatively insoluble hydrated aluminum hydroxide, [latex]\ce{Al(H2O)3(OH)3}[/latex], is converted into the soluble ion, [latex]\ce{[Al(H2O)2(OH)4]-}[/latex], by reaction with hydroxide ion:

[latex][\ce{Al(H2O)3(OH)3}](aq)+\ce{OH-}(aq)⇌\ce{H2O}(l)+\ce{[Al(H2O)2(OH)4]-}(aq) \nonumber[/latex]

In this reaction, a proton is transferred from one of the aluminum-bound H₂O molecules to a hydroxide ion in solution. The [latex]\ce{Al(H2O)3(OH)3}[/latex] compound thus acts as an acid under these conditions. On the other hand, when dissolved in strong acids, it is converted to the soluble ion [latex]\ce{[Al(H2O)6]^3+}[/latex] by reaction with hydronium ion:

[latex]\ce{3H3O+}(aq)+\ce{Al(H2O)3(OH)3}(aq)⇌\ce{Al(H2O)6^3+}(aq)+\ce{3H2O}(l) \nonumber[/latex]

In this case, protons are transferred from hydronium ions in solution to [latex]\ce{Al(H2O)3(OH)3}[/latex], and the compound functions as a base.

Summary

The strengths of Brønsted-Lowry acids and bases in aqueous solutions can be determined by their acid or base ionization constants. Stronger acids form weaker conjugate bases, and weaker acids form stronger conjugate bases. Thus strong acids are completely ionized in aqueous solution because their conjugate bases are weaker bases than water. Weak acids are only partially ionized because their conjugate bases are strong enough to compete successfully with water for possession of protons. Strong bases react with water to quantitatively form hydroxide ions. Weak bases give only small amounts of hydroxide ion. The strengths of the binary acids increase from left to right across a period of the periodic table (CH₄ < NH₃ < H₂O < HF), and they increase down a group (HF < HCl < HBr < HI). The strengths of oxyacids that contain the same central element increase as the oxidation number of the element increases (H₂SO₃ < H₂SO₄). The strengths of oxyacids also increase as the electronegativity of the central element increases [H₂SeO₄ < H₂SO₄].