To calculate the pH of a weak base, determine its base dissociation constant (Kᵦ), calculate the hydroxide ion concentration ([OH⁻]) at equilibrium, and use that concentration to find the pOH. Then convert the pOH to pH using the relationship pH + pOH = 14.00 at 25°C.
Unlike strong bases, which dissociate almost completely in water, weak bases react with water only partially. This incomplete reaction makes their pH calculation more involved because the hydroxide ion concentration depends on the position of chemical equilibrium.
The process becomes straightforward once you understand the equilibrium reaction, set up the appropriate expression, and solve for the hydroxide ion concentration.
Understand how a weak base behaves in water
A weak base is a substance that reacts incompletely with water to produce hydroxide ions (OH⁻). Ammonia (NH₃) is a common example. When ammonia dissolves in water, some of its molecules accept hydrogen ions from water, forming ammonium ions (NH₄⁺) and hydroxide ions.
The reaction is:
NH₃ + H₂O ⇌ NH₄⁺ + OH⁻
The double arrow indicates that the reaction is reversible. Ammonia reacts with water to form products, but the products can also react to regenerate ammonia and water. At equilibrium, the forward and reverse reaction rates are equal.
Because ammonia is a weak base, only a fraction of its dissolved molecules react with water. The solution nevertheless becomes basic because the reaction produces hydroxide ions.
The strength of a weak base is described by its base dissociation constant, Kᵦ. A larger Kᵦ generally indicates a stronger base because the equilibrium favors the formation of ions to a greater extent. A smaller Kᵦ indicates a weaker base.
For ammonia at 25°C, Kᵦ is approximately 1.8 × 10⁻⁵. This value allows you to calculate the equilibrium concentrations of the ions and determine the solution’s pH.
Learn the equations used to calculate pH
Three relationships are especially important when calculating the pH of a weak base.
The first is the base dissociation constant:
Kᵦ = [BH⁺][OH⁻] / [B]
Here, B represents the weak base, and BH⁺ represents its conjugate acid, the species formed when the base accepts a hydrogen ion. For ammonia, B is NH₃ and BH⁺ is NH₄⁺.
The brackets indicate equilibrium molar concentrations, measured in moles per liter (mol/L). In more rigorous chemical thermodynamics, equilibrium constants are defined using activities rather than raw concentrations, but concentration-based expressions are generally appropriate for introductory calculations involving dilute solutions.
The second relationship defines pOH:
pOH = −log₁₀[OH⁻]
This equation converts the hydroxide ion concentration into a logarithmic measure. A lower pOH corresponds to a higher hydroxide ion concentration.
The third relationship connects pH and pOH:
pH + pOH = 14.00
This relationship applies to aqueous solutions at 25°C, where the ion product of water, Kᵥ or K𝑤, is approximately 1.0 × 10⁻¹⁴.
Rearranging the equation gives:
pH = 14.00 − pOH
Therefore, the main challenge in calculating the pH of a weak base is finding its equilibrium hydroxide ion concentration.
Calculate the hydroxide ion concentration
Consider a weak base, B, dissolved in water at an initial concentration of C mol/L. Its reaction with water is:
B + H₂O ⇌ BH⁺ + OH⁻
Water is omitted from the Kᵦ expression because its concentration is effectively constant in dilute aqueous solutions.
Suppose x mol/L of the base reacts with water at equilibrium. Because the reaction produces one BH⁺ ion for every OH⁻ ion, both species have an equilibrium concentration of x mol/L, assuming the solution initially contains no appreciable amounts of either ion from other sources.
The equilibrium concentrations are therefore:
- [B] = C − x
- [BH⁺] = x
- [OH⁻] = x
Substituting these concentrations into the base dissociation expression gives:
Kᵦ = x² / (C − x)
Rearranging:
x² + Kᵦx − KᵦC = 0
Solving this quadratic equation gives the physically meaningful positive solution:
x = (−Kᵦ + √(Kᵦ² + 4KᵦC)) / 2
Since x represents the hydroxide ion concentration produced by the base, this solution gives:
[OH⁻] = (−Kᵦ + √(Kᵦ² + 4KᵦC)) / 2
This expression can be used when the approximation described below is not sufficiently accurate.
For many introductory chemistry problems, however, the calculation can be simplified. If the base reacts only slightly, then x is small compared with C. Under that condition, C − x is approximately equal to C, giving:
Kᵦ ≈ x² / C
Solving for x produces the commonly used approximation:
[OH⁻] ≈ √(Kᵦ × C)
This simplified equation makes weak-base pH calculations much easier. However, it should be used only when the assumption that x is small compared with C is reasonable.
Work through an example using ammonia
Suppose you have a 0.10 M ammonia solution at 25°C and want to calculate its pH. The base dissociation constant for ammonia is 1.8 × 10⁻⁵.
Step 1: Identify the initial concentration and Kᵦ.
The initial ammonia concentration is:
C = 0.10 M
The base dissociation constant is:
Kᵦ = 1.8 × 10⁻⁵
Step 2: Calculate the hydroxide ion concentration.
Using the approximation:
[OH⁻] ≈ √(Kᵦ × C)
Substitute the values:
[OH⁻] ≈ √((1.8 × 10⁻⁵)(0.10))
Multiplying the values inside the square root gives:
[OH⁻] ≈ √(1.8 × 10⁻⁶)
Taking the square root:
[OH⁻] ≈ 1.34 × 10⁻³ M
This is the approximate equilibrium hydroxide ion concentration.
Step 3: Calculate the pOH.
Use the equation:
pOH = −log₁₀[OH⁻]
Substitute the calculated concentration:
pOH = −log₁₀(1.34 × 10⁻³)
Therefore:
pOH ≈ 2.87
Step 4: Calculate the pH.
At 25°C:
pH = 14.00 − pOH
Substituting the result:
pH = 14.00 − 2.87
The final result is:
pH ≈ 11.13
The ammonia solution is basic because its pH is greater than 7 at 25°C. Its relatively high pH results from the hydroxide ions produced when ammonia reacts with water.
The approximation is reasonable in this example because the calculated hydroxide concentration is small compared with the initial ammonia concentration. The fraction of ammonia that reacts is approximately 1.34 × 10⁻³ / 0.10, or 1.34%, which is below the commonly used 5% guideline for introductory equilibrium calculations.
Know when the approximation is valid
The simplified equation [OH⁻] ≈ √(Kᵦ × C) is convenient, but it relies on an assumption: the amount of base that reacts with water must be small compared with the initial base concentration.
The usual introductory chemistry guideline is that the approximation is acceptable when the calculated change is no more than about 5% of the initial concentration.
The percentage change can be calculated as:
Percent ionization = (x / C) × 100%
If the result is approximately 5% or less, the approximation is generally considered acceptable for a typical introductory problem. If the result is larger, use the quadratic equation or another appropriate equilibrium method.
For example, consider a weak base with Kᵦ = 1.0 × 10⁻⁵ and an initial concentration of 1.0 × 10⁻⁴ M.
Using the approximation gives:
[OH⁻] ≈ √((1.0 × 10⁻⁵)(1.0 × 10⁻⁴))
[OH⁻] ≈ 3.16 × 10⁻⁵ M
The estimated percentage change is:
Percent ionization ≈ (3.16 × 10⁻⁵ / 1.0 × 10⁻⁴) × 100%
Percent ionization ≈ 31.6%
Because this value exceeds 5%, the approximation is not sufficiently accurate under the usual guideline.
The quadratic equation should be used instead:
x = (−Kᵦ + √(Kᵦ² + 4KᵦC)) / 2
Substituting the values:
x = (−1.0 × 10⁻⁵ + √((1.0 × 10⁻⁵)² + 4(1.0 × 10⁻⁵)(1.0 × 10⁻⁴))) / 2
This gives:
[OH⁻] ≈ 2.70 × 10⁻⁵ M
The resulting pOH is approximately 4.57, so the pH at 25°C is approximately 9.43.
This example illustrates why checking the approximation matters. When the initial base concentration is low relative to Kᵦ, the amount of base that reacts may no longer be negligible.
Understand how base strength and concentration affect pH
The pH of a weak base solution depends on both the base’s strength and its initial concentration. These factors are related but are not interchangeable.
The base dissociation constant describes how strongly the base reacts with water at equilibrium. For solutions with comparable initial concentrations, a larger Kᵦ generally produces a greater hydroxide ion concentration and a higher pH.
The initial concentration also matters. For a weak base when the small-change approximation is valid:
[OH⁻] ≈ √(Kᵦ × C)
Taking the negative logarithm gives:
pOH ≈ −½ log₁₀(Kᵦ × C)
The pH at 25°C can therefore be estimated using:
pH ≈ 14.00 + ½ log₁₀(Kᵦ × C)
This expression is a convenient alternative when the approximation is valid and the base concentration and Kᵦ are known.
It also reveals how concentration affects pH. If the initial concentration increases by a factor of 100 while Kᵦ remains unchanged, the approximate hydroxide ion concentration increases by a factor of 10. The pOH decreases by 1 unit, and the pH increases by approximately 1 unit.
Conversely, diluting a weak base generally lowers its pH toward neutrality at 25°C. Dilution decreases the hydroxide ion concentration, even though the fraction of base molecules that ionize may increase.
This distinction is important: a weak base does not necessarily have a low pH, and a concentrated solution of a weak base can be strongly basic. The term weak describes the extent of its reaction with water, not simply the pH of its solution.
Account for conjugate acids and other substances in the solution
The basic calculation assumes that the solution contains only the weak base and water before the equilibrium reaction occurs. In practice, other substances can affect the hydroxide ion concentration and change the pH.
For example, ammonia produces ammonium ions when it reacts with water. If a solution already contains substantial ammonium ions, the equilibrium is affected by the presence of this conjugate acid. The reaction may produce less additional hydroxide than it would in a solution containing ammonia alone.
A solution containing a weak base and its conjugate acid can form a buffer. A buffer resists large changes in pH when small amounts of acid or base are added. Calculating the pH of such a solution often requires both the base and conjugate acid concentrations rather than the weak-base equation alone.
For a buffer containing a weak base B and its conjugate acid BH⁺, the Henderson–Hasselbalch relationship can be written as:
pH = pKₐ + log₁₀([B] / [BH⁺])
Here, pKₐ is the negative base-10 logarithm of the acid dissociation constant of BH⁺. This equation is useful when the buffer contains appreciable amounts of both species and their equilibrium concentrations can be appropriately approximated.
The relationship between the acid and base dissociation constants is:
Kₐ × Kᵦ = K𝑤
At 25°C, this relationship allows Kₐ for a conjugate acid to be calculated from the corresponding Kᵦ of its weak base.
If the solution contains other acids, bases, salts, or substances that react with the weak base, a more complete equilibrium analysis may be necessary. The simple square-root equation should not be applied without checking whether its assumptions fit the solution.
Avoid common mistakes when calculating weak-base pH
One frequent mistake is treating a weak base like a strong base. A strong base such as sodium hydroxide dissociates essentially completely in dilute aqueous solution, so its hydroxide concentration can often be determined directly from its concentration and stoichiometry. A weak base reacts only partially with water, so its equilibrium constant must be considered.
Another common error is using the initial base concentration as the hydroxide ion concentration. For a weak base, these quantities are not equal. The hydroxide concentration must be calculated from the equilibrium reaction.
It is also important to use Kᵦ rather than Kₐ in the weak-base equilibrium expression. Kₐ describes acid dissociation, whereas Kᵦ describes the base’s reaction with water. The two constants are related through the ion product of water, but they are not interchangeable.
Rounding too early can introduce small numerical errors, particularly when working with logarithms. Keep several digits during intermediate calculations and round the final pH according to the precision of the given data.
Finally, remember that pH + pOH = 14.00 applies at 25°C. At other temperatures, the ion product of water changes, so the corresponding relationship becomes:
pH + pOH = pK𝑤
where:
pK𝑤 = −log₁₀K𝑤
For most introductory problems, the temperature is assumed to be 25°C unless another temperature is specified.
Apply the calculation consistently
Calculating the pH of a weak base follows a reliable sequence. First, identify the base’s initial concentration and its Kᵦ. Next, write the reaction with water and use an equilibrium expression to determine the hydroxide ion concentration. If the change in concentration is sufficiently small, use the square-root approximation; otherwise, solve the quadratic equation.
Once [OH⁻] is known, calculate pOH using the negative base-10 logarithm of the hydroxide concentration. Then subtract pOH from 14.00 to obtain pH at 25°C.
The central principle is that a weak base establishes an equilibrium with water rather than reacting completely. Understanding that equilibrium explains why the base dissociation constant is essential and provides a consistent method for calculating pH across a wide range of weak-base solutions.
