The relationship between Ka, Kb, and Kw explains how acids and bases behave in water and how their strengths are connected. These three equilibrium constants describe different aspects of acid–base chemistry, but they are linked by a simple equation: Ka × Kb = Kw for a conjugate acid–base pair at the same temperature.
Ka measures how readily an acid donates a proton to water, Kb measures how readily a base accepts a proton from water, and Kw describes the self-ionization of water. Together, these constants help predict whether a substance behaves as an acid or a base, how strongly it reacts in solution, and how its conjugate partner will behave.
What Ka, Kb, and Kw mean
To understand the relationship among these constants, it helps to begin with the reactions they describe.
Ka: The acid dissociation constant
The acid dissociation constant, Ka, measures the extent to which an acid transfers a proton to water. A proton is a hydrogen ion, H⁺, and in aqueous solutions it is commonly associated with water molecules as hydronium ions, H₃O⁺.
For a general acid, HA, the reaction is:
HA + H₂O ⇌ H₃O⁺ + A⁻
Here, HA is the acid, and A⁻ is its conjugate base, the species left after the acid loses a proton.
The acid dissociation constant is expressed as:
Ka = /
The brackets represent the equilibrium concentrations of the dissolved species. Water is omitted from the expression because it is the solvent and its activity is treated as essentially constant in dilute aqueous solutions.
A larger Ka generally indicates a stronger acid because the equilibrium favors the formation of hydronium ions and the conjugate base. A smaller Ka indicates that the acid remains predominantly in its protonated form.
For example, hydrochloric acid is a strong acid in water and dissociates essentially completely under ordinary dilute conditions. Acetic acid, found in vinegar, is a weak acid that dissociates only partially. Its relatively small Ka reflects this behavior.
Kb: The base dissociation constant
The base dissociation constant, Kb, describes how readily a base accepts a proton from water. For a general base, B, the reaction is:
B + H₂O ⇌ BH⁺ + OH⁻
In this reaction, B accepts a proton and becomes its conjugate acid, BH⁺. Water donates the proton and produces a hydroxide ion, OH⁻.
The base dissociation constant is:
Kb = /
As with Ka, a larger Kb generally indicates a stronger base because the equilibrium favors the formation of the protonated base and hydroxide ions. A smaller Kb indicates a weaker base.
Ammonia, NH₃, is a familiar example of a weak base. In water, some ammonia molecules accept protons to form ammonium ions, NH₄⁺, and hydroxide ions. Because this reaction occurs only partially, ammonia has a relatively small Kb.
Ka and Kb therefore describe complementary processes: one measures proton donation, and the other measures proton acceptance.
Kw: The ionization constant of water
Water can act as both an acid and a base. One water molecule can donate a proton to another water molecule, producing hydronium and hydroxide ions. This process is called the self-ionization of water.
The reaction is:
2H₂O ⇌ H₃O⁺ + OH⁻
The ionization constant of water, Kw, is defined as:
Kw =
The concentration of liquid water is effectively constant in dilute solutions, so it is incorporated into the equilibrium constant.
At 25°C, Kw is approximately:
Kw = 1.0 × 10⁻¹⁴
This value is fundamental to aqueous acid–base chemistry. In pure water at 25°C, hydronium and hydroxide ions are present at equal concentrations:
= = 1.0 × 10⁻⁷ mol/L
This equality produces a pH of 7 under the usual dilute-solution convention. However, neutral water does not always have a pH of exactly 7 because Kw changes with temperature.
The relationship between Ka, Kb, and Kw
The central relationship among these constants is:
Ka × Kb = Kw
This equation applies to a conjugate acid–base pair in water at the same temperature, using consistent equilibrium conventions.
A conjugate acid–base pair consists of two species that differ by one proton. When an acid loses a proton, it forms its conjugate base. When that base accepts a proton, it regenerates the original acid.
Consider a weak acid, HA, and its conjugate base, A⁻.
The acid reaction is:
HA + H₂O ⇌ H₃O⁺ + A⁻
Its equilibrium constant is:
Ka = /
The conjugate base reacts with water in the opposite proton-transfer direction:
A⁻ + H₂O ⇌ HA + OH⁻
Its base dissociation constant is:
Kb = /
Multiplying the two expressions gives:
Ka × Kb = ( / ) × ( / )
The concentrations of HA and A⁻ cancel, leaving:
Ka × Kb =
Because the product on the right is Kw, the result is:
Ka × Kb = Kw
The equation follows directly from the equilibrium expressions. It is not an independent rule unrelated to the reactions; it reflects the fact that acid dissociation and conjugate-base proton acceptance are connected through the ionization of water.
What the relationship reveals about acid and base strength
The equation Ka × Kb = Kw reveals an important principle: the stronger an acid is, the weaker its conjugate base tends to be, and vice versa.
At a fixed temperature, Kw has a particular value. If Ka is large, Kb must be correspondingly small for the conjugate pair. If Ka is small, Kb must be larger.
For example, a strong acid transfers protons readily to water. Its conjugate base has little tendency to take those protons back from water, so it is a very weak base. Conversely, the conjugate base of a weak acid generally has a greater tendency to accept protons than the conjugate base of a strong acid.
This does not mean that every weak acid has a strong conjugate base in an absolute sense. Acid and base strengths exist on a spectrum, and the strength of a conjugate base depends on the strength of its corresponding acid.
The relationship also does not mean that a solution containing a weak acid must be basic. A solution’s acidity depends on the substances present, their concentrations, and the equilibria they establish. A weak acid can still produce an acidic solution because it generates more hydronium ions than are present in neutral water.
How Ka, Kb, and Kw relate to pKa, pKb, and pKw
Because equilibrium constants can be very small or very large, chemists often express them using logarithmic scales.
The corresponding quantities are defined as:
pKa = −log₁₀(Ka)
pKb = −log₁₀(Kb)
pKw = −log₁₀(Kw)
Taking the negative logarithm of Ka × Kb = Kw gives:
pKa + pKb = pKw
At 25°C, pKw is approximately 14.00, so:
pKa + pKb ≈ 14.00
This relationship makes it easier to compare acid and base strengths. A lower pKa indicates a stronger acid, while a lower pKb indicates a stronger base.
For a conjugate acid–base pair at 25°C, if the acid has a pKa of 5, its conjugate base has a pKb of approximately 9. Their sum is 14.
The logarithmic relationship is particularly useful because it turns multiplication into addition. Instead of multiplying two small equilibrium constants, a chemist can subtract a pKa value from pKw to determine the corresponding pKb:
pKb = pKw − pKa
Likewise:
pKa = pKw − pKb
At temperatures other than 25°C, pKw is different, so the value 14 should not be used automatically.
How to calculate Ka or Kb using Kw
The relationship is useful when one equilibrium constant is known and the other must be calculated.
Suppose a weak acid has a Ka of 1.8 × 10⁻⁵ at 25°C. To determine the Kb of its conjugate base, use:
Kb = Kw / Ka
Substituting the values:
Kb = (1.0 × 10⁻¹⁴) / (1.8 × 10⁻⁵)
The result is approximately:
Kb = 5.6 × 10⁻¹⁰
This value describes the base strength of the acid’s conjugate base under the same conditions.
The reverse calculation works in exactly the same way. If Kb is known, the acid dissociation constant of the conjugate acid can be found using:
Ka = Kw / Kb
For instance, ammonia has a Kb of approximately 1.8 × 10⁻⁵ at 25°C. Its conjugate acid, ammonium, has a Ka of approximately:
Ka = (1.0 × 10⁻¹⁴) / (1.8 × 10⁻⁵)
Ka ≈ 5.6 × 10⁻¹⁰
These calculations show that ammonia is a weak base and ammonium is a weak acid. The two constants describe opposite proton-transfer reactions involving the same conjugate pair.
When performing these calculations, it is essential to use equilibrium constants measured or defined at the same temperature. It is also important to pair each acid with its actual conjugate base, rather than with an unrelated base.
Why the relationship matters for pH and chemical reactions
The connection between Ka, Kb, and Kw helps explain the pH of solutions containing weak acids, weak bases, and their salts.
A salt formed from a weak acid and a strong base may contain the weak acid’s conjugate base. That base can react with water to produce hydroxide ions, making the solution basic. For example, sodium acetate dissolves to produce sodium ions and acetate ions. The acetate ions can accept protons from water, producing acetic acid and hydroxide ions.
The extent of this reaction is governed by the Kb of acetate, which can be calculated from the Ka of acetic acid and Kw. Knowing this relationship helps chemists predict whether a salt solution will be acidic, basic, or approximately neutral.
Similarly, a salt containing the conjugate acid of a weak base may produce an acidic solution. Ammonium chloride, for example, releases ammonium ions into water. Ammonium can donate a proton to water, forming ammonia and hydronium ions. Its Ka determines how strongly this process occurs.
These principles also support the design and understanding of buffer solutions. A buffer contains a weak acid and its conjugate base, or a weak base and its conjugate acid. Such mixtures resist substantial pH changes when small amounts of acid or base are added. The relative amounts of the two forms and their equilibrium constants determine the pH at which the buffer works most effectively.
In analytical chemistry, environmental science, and biology, these relationships help explain how dissolved substances behave as conditions change. The acid–base equilibria of natural waters, for example, influence which chemical forms of dissolved compounds predominate. In biological systems, related equilibria contribute to the control of pH in fluids and cellular environments.
How temperature affects the relationship
The equation Ka × Kb = Kw remains valid for a conjugate acid–base pair at a given temperature, but the numerical value of Kw changes with temperature.
The self-ionization of water is temperature-dependent because the balance between water molecules and their ions changes as thermal energy changes. Consequently, neutral water does not have the same hydronium concentration at every temperature.
At 25°C, neutral water has a pH of approximately 7. At a different temperature, the pH of neutral water may be higher or lower than 7 while remaining neutral, because neutrality means that hydronium and hydroxide concentrations are equal.
The values of Ka and Kb can also change with temperature, depending on the thermodynamics of their respective reactions. Their product, however, must remain equal to Kw for the corresponding conjugate pair under the same conditions.
For this reason, calculations should use equilibrium constants appropriate to the temperature of interest. The familiar equation pKa + pKb ≈ 14.00 is specifically associated with the approximate value of pKw at 25°C, not a universal constant for all aqueous solutions.
The key principle connecting the three constants
Ka, Kb, and Kw describe different but interconnected aspects of proton transfer in water. Ka measures an acid’s tendency to donate a proton, Kb measures a base’s tendency to accept one, and Kw describes water’s self-ionization.
Their relationship is expressed by:
Ka × Kb = Kw
For the corresponding logarithmic quantities:
pKa + pKb = pKw
At 25°C, pKw is approximately 14.00, making pKa + pKb ≈ 14.00.
Together, these equations show why a strong acid has a weak conjugate base and why knowing one equilibrium constant makes it possible to calculate the other. They provide a consistent framework for understanding acid–base strength, predicting the behavior of dissolved salts, calculating pH, and interpreting chemical equilibria in water.