Henderson-Hasselbalch Equation and Buffer Solutions

Henderson-Hasselbalch Equation and Buffer Solutions

Henderson-Hasselbalch Equation

The Henderson-Hasselbalch equation is a useful tool in acid-base chemistry for calculating the pH of buffer solutions without the need for complex equilibrium calculations. It is especially applicable to solutions containing a weak acid and its conjugate base (or a weak base and its conjugate acid).

• Equation:

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

• Where:

  • $[\mathrm{A}^-]$ = concentration of the conjugate base

  • $[\mathrm{HA}]$ = concentration of the weak acid

  • $\mathrm{p}K_a = -\log K_a$ (acid dissociation constant)

• Only applies to buffer systems of conjugate acid-base pairs.

Example Calculation

Example: Calculate the pH of a solution containing 2.0 M nitrous acid (HNO2) and 1.48 M lithium nitrite (LiNO2), $K_a = 4.5 \times 10^{-4}$.

• First, calculate $\mathrm{p}K_a$:

$\mathrm{p}K_a = -\log(4.5 \times 10^{-4}) = 3.35$

• Apply the Henderson-Hasselbalch equation:

$\mathrm{pH} = 3.35 + \log \left( \frac{1.48}{2.0} \right) = 3.35 + (-0.13) = 3.22$

Buffer Solutions and Buffer Range

A buffer solution is a solution that resists changes in pH when small amounts of acid or base are added. Buffers are most effective when the concentrations of the weak acid and its conjugate base are similar.

• Buffer Range: The effective buffering range is typically $\mathrm{p}K_a \pm 1$.

• At $[\mathrm{A}^-] = [\mathrm{HA}]$, $\mathrm{pH} = \mathrm{p}K_a$.

$\mathrm{pH} = \mathrm{p}K_a + \log(1) = \mathrm{p}K_a$

• Buffers resist pH change best within this range.

Example: Buffer Range Calculation

Example: Determine the buffering range of a solution containing lactic acid ($K_a = 1.4 \times 10^{-4}$) and sodium lactate.

• Calculate $\mathrm{p}K_a$:

$\mathrm{p}K_a = -\log(1.4 \times 10^{-4}) = 3.85$

• Buffering range: $2.85$ to $4.85$

Practice Problems

• Practice 1: The $K_a$ of CH3NH3+ (methylammonium ion) is $3.6 \times 10^{-10}$. Determine the pH of a buffer solution made up of 500 mL of 1.4 M CH3NH3Cl and 220 mL of 1.2 M CH3NH2.

• Practice 2: Determine the buffer component concentration ratio ($[\mathrm{CB}]/[\mathrm{WA}]$) for a buffer with a pH of 4.7. $K_a$ of boric acid (H3BO3) is $5.4 \times 10^{-10}$.

• Practice 3: Calculate the mass of NaO2H that must be added to 1.0 L of 0.35 M HBrO2 in order to make a buffer with a pH of 6.5. $K_a$ of hypobromous acid is $1.8 \times 10^{-6}$.

• Practice 4: Which of the following weak acid–conjugate base combinations would result in an ideal buffer solution with a pH of 7.5?

  • a) nitrous acid (HNO2) and sodium nitrite ($K_a = 4.5 \times 10^{-4}$)

  • b) acetic acid (CH3COOH) and potassium acetate ($K_a = 1.8 \times 10^{-5}$)

  • c) benzoic acid (C6H5COOH) and sodium benzoate ($K_a = 6.3 \times 10^{-5}$)

  • d) acetic acid (HOCl) and sodium lactate ($K_a = 1.4 \times 10^{-4}$)

Summary Table: Henderson-Hasselbalch Equation Applications

Application	Equation	Key Variables	Notes
Calculate pH of buffer	$\mathrm{pH} = \mathrm{p}K_a + \log \left( \frac{[\mathrm{A}^-]}{[\mathrm{HA}]} \right)$	$[\mathrm{A}^-]$, $[\mathrm{HA}]$, $K_a$	Use for weak acid/conjugate base pairs
Find buffer ratio for target pH	$\frac{[\mathrm{A}^-]}{[\mathrm{HA}]} = 10^{\mathrm{pH} - \mathrm{p}K_a}$	Target pH, $K_a$	Rearrange Henderson-Hasselbalch equation
Buffering range	$\mathrm{p}K_a \pm 1$	$K_a$	Buffer is most effective within this range

Additional info: The notes provide practice problems and examples to reinforce the application of the Henderson-Hasselbalch equation in buffer calculations, which is a key topic in acid-base equilibria (Ch.16-17).