D24.3 Acid Constant and Base Constant

The relative strengths of Brønsted-Lowry acids and bases can be evaluated by comparing the equilibrium constants for their ionization reactions. For the reaction of a generic acid, HA, in water:

HA(aq) + H2O(ℓ) ⇌ A–(aq) + H3O+(aq)

we write the acid ionization equilibrium constant (Ka) expression as:

K_{\text{a}} = \dfrac{[\text{H}_3\text{O}^{+}][\text{A}^{-}]}{[\text{HA}]}

(Although water is a reactant in the reaction, it is also the solvent with its phase indicated as “ℓ”, so we do not include [H2O] in the expression.)

A stronger acid, which ionizes to a greater extent, would have a larger Ka compared to a weaker acid. In other words, compared to an acid with a smaller Ka, an acid with a larger Ka would have a higher concentration of H3O+ and A− relative to the concentration of the nonionized acid, HA, at equilibrium.

For example, these data on acid ionization constants:

CH3COOH(aq) + H2O(ℓ) ⇌ CH3COO–(aq) + H3O+(aq) Ka = 1.8 × 10-5
HNO2(aq) + H2O(ℓ) ⇌ NO2–(aq) + H3O+(aq) Ka = 7.4 × 10-4
HSO4–(aq) + H2O(ℓ) ⇌ SO42-(aq) + H3O+(aq) Ka = 1.1 × 10-2

indicate that the order of acid strength is: acetic acid (CH3COOH) is a weaker acid than nitrous acid (HNO2) which is a weaker acid than hydrogen sulfate ion (HSO4–).

We can consider the strength of a base (B) similarly by considering the extend that it will form hydroxide ions in an aqueous solution:

B(aq) + H2O(ℓ) ⇌ HB+(aq) + OH–(aq)

where the base ionization equilibrium constant (Kb) expression is:

K_{\text{b}} = \dfrac{[\text{HB}^{+}][\text{OH}^{-}]}{[\text{B}]}

A stronger base ionizes to a greater extent than a weaker base. Therefore, a stronger base has a larger Kb than a weaker base.

Notice that Ka and Kb provide a quantitative measure of acid and base strengths—significantly more accurate than qualitative descriptions of “strong acid” or “weak acid”.

Consider the ionization reactions for a conjugate acid base pair, HA and A−:

HA(aq) + H2O(ℓ) ⇌ A–(aq) + H3O+(aq)          K_{\text{a}} = \dfrac{[\text{H}_3\text{O}^{+}][\text{A}^{-}]}{[\text{HA}]}
A–(aq) + H2O(ℓ) ⇌ HA(aq) + OH–(aq)          K_{\text{b}} = \dfrac{[\text{HA}][\text{OH}^{-}]}{[\text{A}^{-}]}

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

HA(aq) + H2O(ℓ) + A–(aq) + H2O(ℓ) ⇌ A–(aq) + H3O+(aq) + HA(aq) + OH–(aq)

Therefore:

K_{\text{a}}\;\times\;K_{\text{b}} = \dfrac{[\text{H}_3\text{O}^{+}][\text{A}^{-}]}{[\text{HA}]}\;\times\;\dfrac{[\text{HA}][\text{OH}^{-}]}{[\text{A}^{-}]} = [\text{H}_3\text{O}^{+}][\text{OH}^{-}] = K_{\text{w}}

For example, at 25 °C, Ka of acetic acid (CH3COOH) is 1.8 × 10−5 M, and Kb of its conjugate base, acetate anion (CH3COO–), is 5.6 × 10−10 M. The product of these two equilibrium constants is indeed equal to Kw:

Ka × Kb = (1.8 × 10−5 M) × (5.6 × 10−10 M) = 1.0 × 10−14 = Kw

This relationship tells us that stronger acids form weaker conjugate bases, and weaker acids form stronger conjugate bases.

The diagram shows two horizontal bars. The first, labeled, “Relative acid strength,” at the top is red on the left and gradually changes to purple on the right. The red end at the left is labeled, “Stronger acids.” The purple end at the right is labeled, “Weaker acids.” Just outside the bar to the lower left is the label, “K subscript a.” The bar is marked off in increments with a specific acid listed above each increment. The first mark is at 1.0 with H subscript 3 O superscript positive sign. The second is ten raised to the negative two with H C l O subscript 2. The third is ten raised to the negative 4 with H F. The fourth is ten raised to the negative 6 with H subscript 2 C O subscript 3. The fifth is ten raised to a negative 8 with C H subscript 3 C O O H. The sixth is ten raised to the negative ten with N H subscript 4 superscript positive sign. The seventh is ten raised to a negative 12 with H P O subscript 4 superscript 2 negative sign. The eighth is ten raised to the negative 14 with H subscript 2 O. Similarly the second bar, which is labeled “Relative conjugate base strength,” is purple at the left end and gradually becomes blue at the right end. Outside the bar to the left is the label, “Weaker bases.” Outside the bar to the right is the label, “Stronger bases.” Below and to the left of the bar is the label, “K subscript b.” The bar is similarly marked at increments with bases listed above each increment. The first is at ten raised to the negative 14 with H subscript 2 O above it. The second is ten raised to the negative 12 C l O subscript 2 superscript negative sign. The third is ten raised to the negative ten with F superscript negative sign. The fourth is ten raised to a negative eight with H C O subscript 3 superscript negative sign. The fifth is ten raised to the negative 6 with C H subscript 3 C O O superscript negative sign. The sixth is ten raised to the negative 4 with N H subscript 3. The seventh is ten raised to the negative 2 with P O subscript 4 superscript three negative sign. The eighth is 1.0 with O H superscript negative sign.
Figure: acid-base strengths 1. This diagram shows the relative strengths of conjugate acid-base pairs, as indicated by their ionization constants in aqueous solution. Conjugate acids are above their conjugate bases.
This figure includes a table separated into a left half which is labeled “Acids” and a right half labeled “Bases.” A red arrow points up the left side, which is labeled “Increasing acid strength.” Similarly, a blue arrow points downward along the right side, which is labeled “Increasing base strength.” Names of acids and bases are listed next to each arrow toward the center of the table, followed by chemical formulas. Acids listed top to bottom are sulfuric acid, H subscript 2 S O subscript 4, hydrogen iodide, H I, hydrogen bromide, H B r, hydrogen chloride, H C l, nitric acid, H N O subscript 3, hydronium ion ( in pink text) H subscript 3 O superscript plus, hydrogen sulfate ion, H S O subscript 4 superscript negative, phosphoric acid, H subscript 3 P O subscript 4, hydrogen fluoride, H F, nitrous acid, H N O subscript 2, acetic acid, C H subscript 3 C O subscript 2 H, carbonic acid H subscript 2 C O subscript 3, hydrogen sulfide, H subscript 2 S, ammonium ion, N H subscript 4 superscript +, hydrogen cyanide, H C N, hydrogen carbonate ion, H C O subscript 3 superscript negative, water (shaded in beige) H subscript 2 O, hydrogen sulfide ion, H S superscript negative, ethanol, C subscript 2 H subscript 5 O H, ammonia, N H subscript 3, hydrogen, H subscript 2, methane, and C H subscript 4. The acids at the top of the listing from sulfuric acid through nitric acid are grouped with a bracket to the right labeled “Undergo complete acid ionization in water.” Similarly, the acids at the bottom from hydrogen sulfide ion through methane are grouped with a bracket and labeled, “Do not undergo acid ionization in water.” The right half of the figure lists bases and formulas. From top to bottom the bases listed are hydrogen sulfate ion, H S O subscript 4 superscript negative, iodide ion, I superscript negative, bromide ion, B r superscript negative, chloride ion, C l superscript negative, nitrate ion, N O subscript 3 superscript negative, water (shaded in beige), H subscript 2 O, sulfate ion, S O subscript 4 superscript 2 negative, dihydrogen phosphate ion, H subscript 2 P O subscript 4 superscript negative, fluoride ion, F superscript negative, nitrite ion, N O subscript 2 superscript negative, acetate ion, C H subscript 3 C O subscript 2 superscript negative, hydrogen carbonate ion, H C O subscript 3 superscript negative, hydrogen sulfide ion, H S superscript negative, ammonia, N H subscript 3, cyanide ion, C N superscript negative, carbonate ion, C O subscript 3 superscript 2 negative, hydroxide ion (in blue), O H superscript negative, sulfide ion, S superscript 2 negative, ethoxide ion, C subscript 2 H subscript 5 O superscript negative, amide ion N H subscript 2 superscript negative, hydride ion, H superscript negative, and methide ion C H subscript 3 superscript negative. The bases at the top, from perchlorate ion through nitrate ion are group with a bracket which is labeled “Do not undergo base ionization in water.” Similarly, the lower 5 in the listing, from sulfide ion through methide ion are grouped and labeled “Undergo complete base ionization in water.”
Figure: acid-base strengths 2. The chart shows the relative strengths of conjugate acid-base pairs. The acid and base in a given row are conjugate to each other.

Although “strong” and “weak” are relative terms, we generally refer to acids stronger than H3O+ as strong acids, and bases stronger than OH– as strong bases. Because strong acids and strong bases are completely ionized in aqueous solutions, the concentration of nonionized acid or base is essentially zero. For example, in a 0.10-M solution of HCl, [HCl]e = 0, [H3O+]e = 0.10 M, and [Cl−]e = 0.10 M.

A consequence of this complete ionization is that in aqueous solution there is no way to tell whether one strong acid is stronger than another: HCl, HBr, and HI all are completely ionized. This is known as the leveling effect of water. However, when dissolved in other solvents, these acids do not ionize completely. The extent of ionization increases in the order HCl < HBr < HI, and so HI is the strongest of these acids. Water exerts a similar leveling effect on strong bases.

Many acids and bases are considered “weak”. A solution of a weak acid in water is an equilibrium mixture of the nonionized acid, hydronium ion, and the conjugate base of the acid.

Exercise: Brønsted-Lowry Acids and Bases

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