4.1 Buffer solutions and the Henderson-Hasselbalch equation

Buffer Solutions

Buffer Solutions and pH Stability

A buffer solution is an aqueous solution that resists changes in pH when small amounts of acid or base are added. This resistance is what makes buffers so important across chemistry and biology.

Buffers are composed of two components working together:

• A weak acid and its conjugate base (e.g., acetic acid and acetate)
• Or a weak base and its conjugate acid (e.g., ammonia and ammonium)

When a small amount of strong acid is introduced, the conjugate base neutralizes it. When a small amount of strong base is introduced, the weak acid neutralizes it. The key word here is small: buffers can only handle limited additions before the pH starts shifting significantly.

Buffers maintain stable pH in a wide range of systems:

• Biological systems: Blood stays in the narrow range of pH 7.35–7.45, largely thanks to the carbonic acid–bicarbonate buffer. Enzymes and cellular processes depend on this tight control.
• Chemical reactions: Many synthetic and analytical procedures require a constant pH to produce reliable results.
• Environmental systems: The carbonate buffer system in oceans helps stabilize seawater pH despite inputs like acid rain and dissolved $CO_2$.

Henderson-Hasselbalch Equation for Buffers

The Henderson-Hasselbalch equation relates a buffer's pH to the strength of its weak acid and the ratio of conjugate base to weak acid:

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

• $pK_a$ is the negative logarithm of the acid dissociation constant ($K_a$) of the weak acid
• $[A^-]$ is the molar concentration of the conjugate base
• $[HA]$ is the molar concentration of the weak acid

This equation is derived from the $K_a$ expression by taking the negative log of both sides. It only applies when the weak acid/conjugate base pair is actually present in solution and the concentrations haven't been overwhelmed by added acid or base.

Calculating the pH of a buffer, step by step:

1. Identify the weak acid and its conjugate base in the buffer (e.g., acetic acid, $CH_3COOH$, and acetate, $CH_3COO^-$).
2. Look up or calculate the $pK_a$ of the weak acid ($pK_a = 4.76$ for acetic acid).
3. Determine the concentrations of the weak acid and conjugate base (say 0.1 M acetic acid and 0.2 M acetate).
4. Plug into the equation and solve:

$pH = 4.76 + \log\frac{0.2}{0.1} = 4.76 + \log(2) = 4.76 + 0.30 = 5.06$

Notice that because the conjugate base concentration is higher than the acid concentration, the pH sits above the $pK_a$. If the acid were in excess instead, the pH would sit below the $pK_a$.

Buffer Capacity and Range

Buffer Capacity and Range

Buffer capacity refers to the amount of strong acid or strong base a buffer can neutralize before its pH changes significantly. Two things control it:

• Total concentration of buffer components. A 1.0 M buffer can neutralize far more added acid or base than a 0.1 M buffer of the same composition, because there are simply more molecules available to react.
• The ratio of $[A^-]$ to $[HA]$. Buffer capacity is greatest when this ratio is 1:1, meaning $[A^-] = [HA]$. At that point, the buffer is equally prepared to handle added acid or added base, and the pH equals the $pK_a$.

Buffer range is the pH window over which the buffer works effectively. As a rule of thumb, a buffer is effective within approximately ±1 pH unit of the $pK_a$. For an acetic acid–acetate buffer ($pK_a = 4.76$), the useful range is roughly pH 3.76 to 5.76. Outside this range, one component is so depleted relative to the other that the buffer can no longer resist pH changes well.

So when you're choosing a buffer for a particular application, pick a weak acid whose $pK_a$ is close to the target pH. Then adjust concentrations to get the capacity you need.

Effects of Acids and Bases on Buffers

When a small amount of strong acid is added (e.g., $HCl$):

• The added $H^+$ reacts with the conjugate base ($A^-$), converting it into the weak acid ($HA$).
• $[A^-]$ decreases and $[HA]$ increases.
• The $\frac{[A^-]}{[HA]}$ ratio drops, so pH decreases slightly.

When a small amount of strong base is added (e.g., $NaOH$):

• The added $OH^-$ reacts with the weak acid ($HA$), converting it into the conjugate base ($A^-$).
• $[HA]$ decreases and $[A^-]$ increases.
• The $\frac{[A^-]}{[HA]}$ ratio rises, so pH increases slightly.

In both cases, the pH change is minimal as long as the amount of acid or base added does not exceed the buffer capacity. Once you add more moles of strong acid than there are moles of $A^-$ (or more moles of strong base than there are moles of $HA$), the buffer is overwhelmed and the pH will change dramatically.

You can use the Henderson-Hasselbalch equation to calculate the new pH after an addition: just update the concentrations of $[A^-]$ and $[HA]$ using an ICE-style approach (subtract moles consumed, add moles produced), then plug the new values back in.