Weak-Acid Buffer Quadratic

Compare a stable acid-equilibrium quadratic with the buffer approximation and its singular boundary.

Weak-Acid Buffer Quadratic

Concept Overview

Weak acids only partially dissociate in water. When a weak acid (HA) is mixed with its conjugate base (A⁻), they form a buffer solution that resists changes in pH. The common Henderson-Hasselbalch approximation simplifies the calculation by assuming initial concentrations equal equilibrium concentrations. However, a more accurate model uses the mass action equation and the quadratic formula to find the exact equilibrium concentrations, especially useful when the approximation fails (like highly dilute solutions or strong acids).

Mathematical Definition

Equilibrium Expression

For a weak acid HA dissociating into H⁺ and A⁻, the acid dissociation constant (K_a) is given by:

K_a = [H⁺][A⁻] / [HA]

Let A be the initial concentration of HA, and B be the initial concentration of A⁻ (from a salt). At equilibrium, if x = [H⁺], then [A⁻] = B + x and [HA] = A - x. The mass action equation becomes:

K_a = x(B + x) / (A - x)

Quadratic Solution

Rearranging this gives a quadratic equation in x: x² + (B + K_a)x - K_aA = 0. We can solve for the positive root:

x = [H⁺] = 2K_aA / [(B + K_a) + sqrt((B + K_a)^2 + 4K_aA)]

Using this form of the quadratic formula avoids catastrophic cancellation that can occur when subtracting two close numbers.

limitations

The Henderson-Hasselbalch approximation breaks down when the acid is relatively strong (larger K_a), solutions are dilute, or the ratio of base to acid is extreme. Specifically, when pure acid is present (B = 0), the approximation is undefined, whereas the quadratic model still yields a valid positive root. Note that this simplified quadratic model neglects water auto-protolysis and activity corrections, so extreme dilution or highly concentrated solutions involving significant non-ideal ionic interactions may still invalidate it.

worked-example

Let us calculate the equilibrium [H⁺] and pH for a buffer with pK_a = 5, Initial acid M = 0.01, and Base acid ratio = 1 (so initial base is 0.01).

Calculate K_a = 10^-5 = 0.00001.
Identify A = 0.01 and B = 0.01.
Set up the quadratic: x² + (0.01 + 0.00001)x - (0.00001)(0.01) = 0.
x² + 0.01001x - 0.0000001 = 0.
Using the quadratic formula: x = 0.00000998005978 M.
Calculate pH = -log₁₀(x) = 5.000866857.

Comparing this to the Henderson-Hasselbalch approximation, where pH ≈ pK_a + log₁₀(1) = 5.0, we see a small difference due to the dissociation of the acid itself, which slightly increases the base concentration and decreases the acid concentration at equilibrium.

experiment

You can investigate the accuracy of the Henderson-Hasselbalch approximation by exploring the simulation Weak-Acid Buffer Quadratic module:

Set the pKa to exactly 5.
Set the Initial acid M to exactly 0.01.
Set the Base acid ratio to exactly 1. Observe the small difference between the equilibrium model pH and the Henderson Hasselbalch pH.
Change the Base acid ratio to exactly 0. Notice how the Henderson Hasselbalch pH becomes undefined (Unavailable), while the equilibrium model still correctly calculates the pH of the pure weak acid solution.

references

For more detailed information on weak acid equilibria and buffer solutions, consult the following resource:

Weak Acid Equilibria and Buffer Solutions (UMass Boston CH116)

Weak-Acid Buffer Quadratic