8.9: Henderson-Hasselbalch Equation

Introduction

The Henderson-Hasselbalch equation is a useful tool for calculating the pH of a Buffer Solutions. It provides a quick method for determining the pH of a solution containing a weak acid and its conjugate base (or a weak base and its conjugate acid).

The Equation

The Henderson-Hasselbalch equation is given by:

$$ pH = pK_a + log \frac{[A^-]}{[HA]} $$
Where:

• $ pH $ is the measure of acidity.
• $ pK_a $ is the negative logarithm of the Acid Dissociation Constant ( $ K_a $ ) of the weak acid. $$ pK_a = -log(K_a) $$ * $ [A^-] $ is the concentration of the conjugate base.
• $ [HA] $ is the concentration of the weak acid.

Alternatively, for a weak base and its conjugate acid:

$$ pOH = pK_b + log \frac{[HB^+]}{[B]} $$
Where:

• $ pOH $ is the measure of basicity.
• $ pK_b $ is the negative logarithm of the Base Dissociation Constant ( $ K_b $ ) of the weak base. $$ pK_b = -log(K_b) $$ * $ [HB^+] $ is the concentration of the conjugate acid.
• $ [B] $ is the concentration of the weak base.

Derivation of the Henderson-Hasselbalch Equation

The Henderson-Hasselbalch equation is derived from the acid dissociation equilibrium expression. Consider the equilibrium for a weak acid, $ HA $ :

$$ HA(aq) \rightleftharpoons H^+(aq) + A^-(aq) $$
The Acid Dissociation Constant ( $ K_a $ ) is:

$$ K_a = \frac{[H^+][A^-]}{[HA]} $$
Taking the negative logarithm of both sides:

$$ -log(K_a) = -log\left(\frac{[H^+][A^-]}{[HA]}\right) $$
$$ -log(K_a) = -log[H^+] - log\left(\frac{[A^-]}{[HA]}\right) $$
Since $ pK_a = -log(K_a) $ and $ pH = -log[H^+] $ :

$$ pK_a = pH - log\left(\frac{[A^-]}{[HA]}\right) $$
Rearranging to solve for pH yields the Henderson-Hasselbalch equation:

$$ pH = pK_a + log\left(\frac{[A^-]}{[HA]}\right) $$

Using the Henderson-Hasselbalch Equation

1. Identify the weak acid/base and its conjugate pair.
2. Determine the $ K_a $ (or $ K_b $ ) value. If given $ K_a $ , calculate $ pK_a $ ( $ pK_a = -log(K_a) $ ). If given $ K_b $ , calculate $ pK_b $ ( $ pK_b = -log(K_b) $ ).
3. Determine the concentrations of the weak acid/base and its conjugate.
4. Plug the values into the appropriate equation and solve for pH or pOH.
5. If you calculated pOH, find pH using the relationship: $$ pH + pOH = 14 $$

Important Considerations

• Validity: The Henderson-Hasselbalch equation is most accurate when the concentrations of the acid and its conjugate base are relatively high and close to each other. It is less accurate when either $ [HA] $ or $ [A^-] $ is very small compared to the other. It is most reliable when the ratio of $ [A^-]/[HA] $ is between 0.1 and 10.
• Approximations: The equation assumes that the Equilibrium Concentrations of the acid and base are approximately equal to their initial concentrations. This assumption is valid when the acid is weak and the concentrations are sufficiently high.
• Buffer Capacity: The Henderson-Hasselbalch equation describes the pH of a buffer, but it doesn’t tell us about the buffer capacity, which is the amount of acid or base a buffer can neutralize before significant pH change occurs.

Applications

• Calculating the pH of buffer solutions: Determining the pH of a solution containing a weak acid and its conjugate base, or a weak base and its conjugate acid.
• Preparing buffer solutions: Determining the amounts of acid and base needed to create a buffer with a specific pH.
• Estimating pH changes after adding acid or base to a buffer: While ICE Tables would be more accurate, the Henderson-Hasselbalch equation can provide an estimate.