Buffers and the Henderson-Hasselbalch Equation

Clinical blood-gas analyzers report pH, $P_{\text{CO}_2}$, and calculated $[\text{HCO}_3^-]$ using exactly the bicarbonate Henderson–Hasselbalch relationship from Example 2, and physicians classify acid-base disorders (respiratory acidosis, metabolic alkalosis, and their compensations) by tracking how these three variables move together. Pharmaceutical formulation scientists use the equation to predict a drug's ionization state at physiological pH — critical for oral bioavailability, since only the neutral (un-ionized) form of a weak-acid or weak-base drug crosses lipid membranes efficiently, and the fraction ionized at any pH follows directly from Henderson–Hasselbalch. Molecular biology and biochemistry labs rely on the equation daily to prepare Tris, phosphate, HEPES, and citrate buffers for protein purification, PCR, and enzyme assays, choosing a buffering agent whose p$K_a$ lies close to the desired working pH because that is exactly where the logarithmic buffering effect (and buffer capacity, treated quantitatively elsewhere) is strongest. In analytical chemistry, buffers prepared via Henderson–Hasselbalch control the ionization state of analytes in capillary electrophoresis and HPLC mobile phases, determine the shape of complexometric and precipitation titration curves by fixing metal-ion side reactions, and set the pH windows used in gravimetric and electrochemical (ISE, glass electrode) analyses. Environmental chemists apply the same framework to the carbonate buffering of seawater and its sensitivity to rising atmospheric CO₂ — ocean acidification is, quantitatively, a slow shift of the marine bicarbonate/carbonate Henderson–Hasselbalch ratio.

:::quiz[Blood plasma has $[\text{HCO}_3^-] = 24\,\text{mmol/L}$ and dissolved CO₂ = 1.2 mmol/L, with apparent p$K_a'$ = 6.10. What is the pH?]

• 6.10
• 7.10
• 7.40 ✓
• 7.70

Ask the tutor: How does the buffer capacity β formally quantify the "flatness" of the titration curve near pH = pKa, and how is it derived from the Henderson–Hasselbalch framework?

Picture a weak acid HA dissolved alongside its conjugate base A⁻, supplied for instance as the sodium salt NaA. Both species are present in bulk, at concentrations far exceeding the tiny amount of H₃O⁺ that equilibrium chemistry alone would produce. The acid-dissociation equilibrium

is always straining toward its equilibrium constant $K_a$. If a chemist adds a small increment of strong acid, the incoming H₃O⁺ does not accumulate freely — it is consumed by the abundant A⁻ reservoir, converting some A⁻ into HA. If strong base is added instead, the OH⁻ deprotonates some HA, converting it into A⁻. In both cases the perturbation is absorbed by shifting the HA/A⁻ ratio slightly, and because pH depends on the logarithm of that ratio, even a substantial fractional shift in composition produces only a small pH change. This logarithmic damping is the entire secret of buffering. Consider an acetate buffer at pH = p$K_a$, where [A⁻] = [HA]. Adding enough strong acid to convert 10% of the A⁻ into HA changes the ratio from 1.00 to about 0.82, and $\log(0.82) = -0.086$ — under a tenth of a pH unit. Compare this to unbuffered water, where adding acid changes [H₃O⁺] directly and linearly, so the same absolute amount of acid can move the pH by several units. The buffer's flat response near pH = p$K_a$ is visible as the near-horizontal plateau in a titration curve, a signature you can trace directly in acid-base titration curves. The Henderson–Hasselbalch equation is nothing more than the equilibrium-constant expression for HA, rearranged and logarithmically transformed. Because $K_a$ is fixed by temperature, the pH is dictated entirely by the ratio [A⁻]/[HA] — a compositional variable the chemist controls by choice of stoichiometry. This is why buffer pH is, to first approximation, independent of dilution: diluting the solution scales both [HA] and [A⁻] by the same factor, leaving their ratio — and hence the pH — essentially unchanged, provided dilution does not become so extreme that water's autoionization or incomplete dissociation begins to matter.

:::svg[Buffer action: added acid or base shifts the HA/A⁻ ratio, not free [H₃O⁺]]

The acid-dissociation equilibrium constant for a weak acid HA is

where: $K_a$ is the thermodynamic (or, in practice, concentration) dissociation constant at a given temperature and ionic strength; the bracketed terms are equilibrium concentrations, not formal (analytical) concentrations. Taking the negative logarithm of both sides and rearranging gives the Henderson–Hasselbalch equation:

where: p$K_a = -\log K_a$; [A⁻] and [HA] are the equilibrium concentrations of conjugate base and conjugate acid. Reading it: pH equals p$K_a$ when the two species are equal in concentration, and moves one full unit for every tenfold change in their ratio — a direct consequence of the logarithmic scale of pH itself. For a weak base B with conjugate acid BH⁺, the analogous relation is often written using p$K_a$ of the conjugate acid:

where: p$K_b = -\log K_b$, and p$K_a(\text{BH}^+) = \text{p}K_w - \text{p}K_b$ at 25 °C ($K_w = 1.0\times10^{-14}$). Reading it: the same logarithmic-ratio law governs base buffers, just referenced to the conjugate acid's $K_a$. The approximation buried in ordinary use is that equilibrium concentrations [HA] and [A⁻] are replaced by their formal (stoichiometric, pre-equilibrium) concentrations $C_{\text{HA}}$ and $C_{\text{A}^-}$:

where: $C_{\text{A}^-}$, $C_{\text{HA}}$ are moles of base/acid form added per liter, before any equilibrium shift. Reading it: this is valid only when the amount of HA that dissociates (or A⁻ that hydrolyzes) is negligible compared to $C_{\text{HA}}$ and $C_{\text{A}^-}$ themselves — the working definition of "buffer region."

Example 1 — Foundational: preparing an acetate buffer at a target pH. A student must prepare 1.00 L of an acetate buffer at pH = 5.00 using acetic acid ($K_a = 1.8\times10^{-5}$, p$K_a = 4.745$) and sodium acetate, with a total acetate-species concentration (HA + A⁻) fixed at 0.200 M by the amount of stock solution available.

Step 1. Apply Henderson–Hasselbalch to find the required ratio:

Step 2. Use the total-concentration constraint $[\text{HA}] + [\text{A}^-] = 0.200\ \text{M}$. Let $[\text{HA}] = x$; then $[\text{A}^-] = 1.80x$:

Step 3. Convert to masses for 1.00 L. Acetic acid, $M = 60.05\ \text{g/mol}$: mass $= 0.0714\ \text{mol} \times 60.05\ \text{g/mol} = 4.29\ \text{g}$. Sodium acetate (anhydrous), $M = 82.03\ \text{g/mol}$: mass $= 0.1286\ \text{mol} \times 82.03\ \text{g/mol} = 10.55\ \text{g}$.

Step 4 — Validity check. Formal concentrations (0.0714 M and 0.129 M) are both roughly 4,000-fold larger than the expected $[\text{H}_3\text{O}^+] \approx 10^{-5}\ \text{M}$, easily satisfying the $\gtrsim 100\times$ criterion from the dilution derivation — the Henderson–Hasselbalch approximation is excellent here, good to better than ±0.01 pH unit from approximation error alone (real bench uncertainty, from balance and glassware tolerances, is typically ±0.02–0.05 pH unit).

Limiting-case check. If the target pH had been set exactly to p$K_a = 4.745$, the ratio would collapse to 1.00 and the two masses would be closer (roughly equal molar amounts), consistent with the "maximum buffering at pH = p$K_a$" principle developed physically above.

Example 2 — Applied: the bicarbonate buffer of blood plasma. Human blood plasma is buffered chiefly by the CO₂/HCO₃⁻ system:

Using the apparent p$K_a'$ = 6.10 (which lumps dissolved CO₂ and true carbonic acid together, a standard clinical convention), typical plasma values are $[\text{HCO}_3^-] = 24.0\ \text{mmol/L}$ and dissolved $\text{CO}_2 = 1.2\ \text{mmol/L}$ (this is set by the partial pressure of CO₂ in the lungs via Henry's law, $[\text{CO}_2] = \alpha_{\text{CO}_2}\,P_{\text{CO}_2}$, with $\alpha_{\text{CO}_2} = 0.0306\ \text{mmol L}^{-1}\text{mmHg}^{-1}$ at 37 °C and normal $P_{\text{CO}_2} = 40\ \text{mmHg}$).

Step 1 — Compute normal plasma pH.

This matches the physiological reference value precisely, confirming the apparent p$K_a'$ convention.

Step 2 — Metabolic perturbation. Suppose a patient in metabolic acidosis has $[\text{HCO}_3^-]$ fall to 14.0 mmol/L while respiratory compensation holds $P_{\text{CO}_2}$ at 32 mmHg (partial compensation, lowering CO₂ to reduce the denominator too). Then

This is a clinically realistic compensated acidosis (pH depressed from 7.40 but not to the dangerous sub-7.20 range), and it illustrates how two variables — bicarbonate (renal, metabolic) and $P_{\text{CO}_2}$ (respiratory) — jointly control blood pH through the same logarithmic-ratio law.

Step 3 — Check the buffer-validity criterion. $[\text{HCO}_3^-]$ and $[\text{CO}_2]$ here are in the millimolar range, while $[\text{H}_3\text{O}^+]$ at pH 7.3–7.4 is about $4$–$5\times10^{-8}\ \text{M}$ — five orders of magnitude smaller than the formal concentrations, so the approximation is essentially exact for physiological purposes; the open, CO₂-exchanging nature of this buffer (constantly replenished by respiration) is what makes it far more powerful than a closed acetate buffer of similar formal concentration, a nuance developed further under buffer capacity.

Example 3 — Exam/competition level: buffer with significant self-dissociation correction. A very dilute buffer is prepared by mixing $1.00\times10^{-3}$ mol of a weak acid HA (p$K_a = 5.00$, so $K_a = 1.00\times10^{-5}$) with $2.00\times10^{-4}$ mol of its sodium salt NaA in exactly 1.00 L. Naively applying Henderson–Hasselbalch:

But formal $C_{\text{A}^-} = 2.00\times10^{-4}\ \text{M}$ is uncomfortably close to the expected $[\text{H}_3\text{O}^+] \sim 10^{-4.3}\ \text{M} = 5.0\times10^{-5}\ \text{M}$ — only about a factor of 4, not the required $\gtrsim 100$. The naive approximation is suspect and the exact quadratic must be solved.

Step 1 — Write the exact equilibrium equation. From the derivation above,

Since pH is expected near 4.3, $[\text{OH}^-] = K_w/[\text{H}_3\text{O}^+] \approx 10^{-14}/10^{-4.3} \approx 2\times10^{-10}\ \text{M}$ — utterly negligible compared to $[\text{H}_3\text{O}^+] \sim 5\times10^{-5}\ \text{M}$. Drop $[\text{OH}^-]$ but retain the $[\text{H}_3\text{O}^+]$ correction terms.

Step 2 — Let $h = [\text{H}_3\text{O}^+]$. The equation becomes

Step 3 — Expand into a quadratic.

Step 4 — Solve the quadratic. Using the quadratic formula with $a=1$, $b = 2.10\times10^{-4}$, $c = -1.00\times10^{-8}$:

Step 5 — Convert to pH.

Step 6 — Compare. The naive Henderson–Hasselbalch value was 4.30; the exact treatment gives 4.40 — a discrepancy of 0.10 pH units, well outside typical analytical tolerance (±0.02–0.05), confirming that ignoring the self-ionization/depletion correction was not justified for this dilute, base-poor mixture. Physically, HA dissociates enough on its own that the effective [A⁻] exceeds its formal value by a non-negligible fraction ($h = 4.0\times10^{-5}$ compared to $C_{\text{A}^-} = 2.0\times10^{-4}$, a 20% correction), explaining why the true pH is higher (less acidic) than naively predicted.

Limiting-case check. If instead $C_{\text{HA}} = 1.00\ \text{M}$ and $C_{\text{A}^-} = 0.200\ \text{M}$ (same ratio, 1000-fold higher concentration), $h$ from the naive formula would be $5.0\times10^{-5}\ \text{M}$, now utterly negligible next to both formal concentrations (ratio $> 10^4$), and the quadratic correction shrinks to well under 0.001 pH units — recovering the ordinary Henderson–Hasselbalch regime and confirming the correction term vanishes as formal concentrations grow relative to $[\text{H}_3\text{O}^+]$.

The equation bears two names for good reason. Lawrence Joseph Henderson, an American physiologist and biochemist working at Harvard in the first decade of the twentieth century, was investigating how blood maintains a nearly constant pH despite continuous production of CO₂ and lactic acid by metabolism. In 1908 he derived the mathematical relationship between the concentrations of carbonic acid and bicarbonate and the hydrogen-ion concentration of blood, framing it as a mass-action equilibrium — but he expressed it in terms of concentration ratios directly, without logarithms, because the pH concept itself did not yet exist in its modern form. The logarithmic recasting came from Karl Albert Hasselbalch, a Danish biochemist and physician working at the Rigshospitalet in Copenhagen, who in 1916–1917 was extending Søren Sørensen's newly introduced pH scale (1909) to physiological measurements of blood. Hasselbalch rewrote Henderson's mass-action relationship in logarithmic form, producing the equation essentially as used today, and applied it to interpret clinical blood-gas and acid-base disturbances — work that laid the foundation for modern blood-gas analysis and acid-base nomenclature (respiratory versus metabolic acidosis/alkalosis) still used at every hospital bedside.