Buffer solutions: the Henderson-Hasselbalch equation and buffer capacity

Intuition Beginner

A buffer is a solution that fights pH change. Add a small amount of strong acid to pure water and the pH plunges. Add the same amount to a buffer, and the pH barely moves. The resistance comes from the buffer containing two species that can interconvert: a weak acid HA and its conjugate base A⁻.

When you add strong acid (H⁺), the A⁻ in the buffer grabs the proton and becomes HA. When you add strong base (OH⁻), the HA donates a proton to neutralise it and becomes A⁻. In both cases the added disturbance is absorbed by converting one buffer component into the other, leaving [H⁺] -- and hence the pH -- nearly unchanged.

The Henderson-Hasselbalch equation gives the pH of a buffer:

$$\text{pH}=\text{p}{K}_a+\log \frac{[{\text{A}}^{-}]}{[\text{HA}]}$$

The pH depends on the pKa of the weak acid and the ratio of conjugate base to acid. A buffer is most effective when $[{\text{A}}^{-}]\approx [\text{HA}]$, because both directions of conversion are equally available. This occurs when pH = pKa. The useful buffer range is approximately pKa ± 1.

Visual Beginner

A buffer capacity diagram plots pH on the horizontal axis and the moles of strong base needed per litre to shift the pH by one unit on the vertical axis. The curve peaks at pH = pKa and tapers to near zero several pH units away.

The diagram shows that outside the range pKa ± 1, the buffer capacity drops sharply. Within this range the ratio [A⁻]/[HA] stays between 0.1 and 10, and both species are present in sufficient concentration to absorb added acid or base. Beyond pKa + 1, almost all HA has been converted to A⁻ and the buffer cannot neutralise additional base. Below pKa − 1, almost all A⁻ has been protonated and the buffer cannot neutralise additional acid.

Worked example Beginner

You need to prepare 500 mL of a pH 5.00 buffer using propionic acid (CH₃CH₂COOH, pKa = 4.87) and sodium propionate. You have 0.10 mol of propionic acid available.

Step 1: Determine the ratio. Use Henderson-Hasselbalch:

$$5.00=4.87+\log \frac{[{\text{A}}^{-}]}{[\text{HA}]}$$ $$\log \frac{[{\text{A}}^{-}]}{[\text{HA}]}=0.13\frac{[{\text{A}}^{-}]}{[\text{HA}]}=10^{0.13}=1.35$$

Step 2: Calculate moles. Using all 0.10 mol of propionic acid as HA:

$$n_{{\text{A}}^{-}}=1.35\times 0.10=0.135\text{ mol sodium propionate}$$

Step 3: Prepare. Dissolve 0.10 mol propionic acid and 0.135 mol sodium propionate in water. Dilute to 500 mL. The concentrations are 0.20 M HA and 0.27 M A⁻.

Check. pH = 4.87 + log(0.27/0.20) = 4.87 + 0.13 = 5.00. The target falls within the effective range (3.87 to 5.87), so the buffer will function well.

Check your understanding Beginner

Formal definition Intermediate+

Henderson-Hasselbalch equation: derivation from Ka

The acid dissociation constant for a weak acid HA is:

$$K_a=\frac{[{\text{H}}^{+}][{\text{A}}^{-}]}{[\text{HA}]}$$

Rearrange to isolate [H⁺]:

$$[{\text{H}}^{+}]=K_a\cdot \frac{[\text{HA}]}{[{\text{A}}^{-}]}$$

Take −log of both sides:

$$-\log [{\text{H}}^{+}]=-\log K_a-\log \frac{[\text{HA}]}{[{\text{A}}^{-}]}$$ $$\boxed{\text{pH}=\text{p}{K}_a+\log \frac{[{\text{A}}^{-}]}{[\text{HA}]}}$$

The derivation is mathematically exact. The approximation enters when we replace the equilibrium concentrations [HA] and [A⁻] with the stoichiometric (added) concentrations. This is valid when the self-dissociation of HA and the hydrolysis of A⁻ contribute negligibly to the concentrations -- that is, when the buffer components dominate the equilibrium. This condition requires $c_T\gg [{\text{H}}^{+}]$ and $c_T\gg [{\text{OH}}^{-}]$.

Buffer capacity

The buffer capacity β quantifies how much strong base (or acid) a buffer can absorb per unit pH change:

$$\beta =\frac{dB}{d\text{pH}}$$

where B is moles of strong base added per litre. A larger β means the buffer is more resistant to pH change. Buffer capacity has units of mol L⁻¹ pH⁻¹.

For a monoprotic buffer system with total weak-acid concentration $c_T=[\text{HA}]+[{\text{A}}^{-}]$:

$$\beta =2.303\cdot \frac{c_T\cdot K_a\cdot [{\text{H}}^{+}]}{(K_a+[{\text{H}}^{+}]{)}^2}+{\beta }_w$$

where ${\beta }_w=2.303([{\text{H}}^{+}]+[{\text{OH}}^{-}])$ is the water contribution. The factor 2.303 converts between natural and base-10 logarithms ($\ln 10\approx 2.303$).

Effective buffer range

Setting the derivative of the buffer term with respect to [H⁺] equal to zero shows that β is maximised when [H⁺] = Ka (i.e., pH = pKa). The maximum buffer capacity is:

$${\beta }_{\max }=0.576c_T$$

The factor 0.576 is $2.303/4$. The capacity drops to roughly one-third of its maximum when the ratio [A⁻]/[HA] reaches 0.1 or 10, corresponding to pH = pKa ± 1. This is the conventional effective buffer range.

Dilution effects on pH and capacity

Diluting a buffer preserves the ratio [A⁻]/[HA] but reduces $c_T$. Because buffer capacity is proportional to $c_T$, dilution weakens the buffer. A buffer diluted tenfold has one-tenth the capacity. At very high dilution ($c_T<10^{-3}$ M), the Henderson-Hasselbalch approximation fails because [H⁺] and [OH⁻] become comparable to [HA] and [A⁻], and the pH drifts toward 7.

Counterexamples to common slips

• Buffer pH is not the same as buffer capacity. Two buffers can have identical pH but very different capacities. A 0.001 M acetate buffer at pH 4.76 and a 0.10 M acetate buffer at pH 4.76 have the same pH but the latter has 100 times the capacity.
• A strong acid plus its conjugate base does not make a buffer. HCl and Cl⁻ cannot form a buffer because Cl⁻ is such a weak base that it cannot recapture protons. Buffers require a weak acid and its conjugate base (or a weak base and its conjugate acid).
• pH = pKa does not mean the solution is neutral. An acetate buffer at pH 4.76 is acidic. The pKa merely determines the pH at which [HA] = [A⁻].

Key result Intermediate+

Theorem (Buffer preparation). To prepare a buffer at target pH with effective capacity, choose a weak acid whose pKa is within 1 unit of the target pH. Then set the ratio $[{\text{A}}^{-}]/[\text{HA}]=10^{\text{pH}-\text{p}{K}_a}$ and make the total concentration $c_T$ large enough for the desired buffer capacity.

Proof. Given a target pH, the required ratio follows from the Henderson-Hasselbalch equation:

$$\frac{[{\text{A}}^{-}]}{[\text{HA}]}=10^{\text{pH}-\text{p}{K}_a}$$

Combined with $c_T=[\text{HA}]+[{\text{A}}^{-}]$, the individual concentrations are:

$$[\text{HA}]=\frac{c_T}{1+10^{\text{pH}-\text{p}{K}_a}},[{\text{A}}^{-}]=\frac{c_T\cdot 10^{\text{pH}-\text{p}{K}_a}}{1+10^{\text{pH}-\text{p}{K}_a}}$$

The buffer capacity at this pH is $\beta =2.303\cdot c_T\cdot {\alpha }_0\cdot {\alpha }_1$, where ${\alpha }_0=[\text{HA}]/c_T$ and ${\alpha }_1=[{\text{A}}^{-}]/c_T$ are the fractional concentrations. The product ${\alpha }_0{\alpha }_1$ is maximised at ${\alpha }_0={\alpha }_1=0.5$ (pH = pKa) and decreases symmetrically as pH moves away from pKa. $\square$

Corollary (Choosing a buffer system). For each pH target, there exists a family of weak acids with appropriate pKa values. The practical selection also considers solubility, chemical inertness, temperature stability, absence of UV absorption, and compatibility with the system being studied.

Worked example: buffer capacity calculation

A buffer contains 0.20 M acetic acid (pKa = 4.76) and 0.30 M sodium acetate. Calculate the buffer capacity at this pH.

First find the pH: pH = 4.76 + log(0.30/0.20) = 4.76 + 0.176 = 4.94.

The fractional concentrations: ${\alpha }_0=0.20/0.50=0.40$, ${\alpha }_1=0.30/0.50=0.60$.

$$\beta =2.303\times 0.50\times 0.40\times 0.60=2.303\times 0.50\times 0.24=0.277{\text{ mol L}}^{-1}{\text{ pH}}^{-1}$$

The maximum capacity at this $c_T$ would be $0.576\times 0.50=0.288$ mol L⁻¹ pH⁻¹, achieved when [HA] = [A⁻] = 0.25 M. The buffer at pH 4.94 retains 96% of the maximum capacity for this total concentration because the pH is close to pKa.

Bridge. Buffer preparation and capacity connect to 14.10.03 titrations, where the buffer region of a titration curve is the plateau during which the buffer resists pH change. The same Henderson-Hasselbalch equation governs both the static buffer and the dynamic titration curve. In 14.11.01 electrochemistry, buffer solutions of precisely known pH calibrate glass electrodes. In 17.02.02 membrane transport, biological buffer systems maintain the pH gradients essential for ATP synthesis.

Exercises Intermediate+

Exact buffer capacity and dilution Master

Derivation of the buffer-capacity formula

The buffer capacity is defined as $\beta =dB/d\text{pH}$, where B is moles of strong base added per litre. In a solution containing a weak acid HA at total concentration $c_T$, the charge balance gives:

$$[{\text{H}}^{+}]+B=[{\text{A}}^{-}]+[{\text{OH}}^{-}]$$

From the definition of $K_a$ and the mass balance $c_T=[\text{HA}]+[{\text{A}}^{-}]$:

$$[{\text{A}}^{-}]=\frac{c_T{K}_a}{K_a+[{\text{H}}^{+}]}$$

Differentiating the charge balance with respect to pH and using $d[{\text{H}}^{+}]/d\text{pH}=-2.303[{\text{H}}^{+}]$:

$$\beta =\frac{d[{\text{A}}^{-}]}{d\text{pH}}+\frac{d[{\text{OH}}^{-}]}{d\text{pH}}-\frac{d[{\text{H}}^{+}]}{d\text{pH}}$$

The buffer term, after applying the quotient rule and substituting the derivative:

$$\frac{d[{\text{A}}^{-}]}{d\text{pH}}=2.303\cdot \frac{c_T{K}_a[{\text{H}}^{+}]}{(K_a+[{\text{H}}^{+}]{)}^2}$$

Rewrite in terms of fractional concentrations. Let ${\alpha }_0=[\text{HA}]/c_T=[{\text{H}}^{+}]/(K_a+[{\text{H}}^{+}])$ and ${\alpha }_1=[{\text{A}}^{-}]/c_T=K_a/(K_a+[{\text{H}}^{+}])$. Then:

$$\beta =2.303\cdot c_T\cdot {\alpha }_0\cdot {\alpha }_1+2.303([{\text{H}}^{+}]+[{\text{OH}}^{-}])$$

The product ${\alpha }_0{\alpha }_1$ is maximised when ${\alpha }_0={\alpha }_1=0.5$ (pH = pKa), giving ${\beta }_{\max }=2.303\times c_T/4=0.576c_T$.

Dilution and the limits of the Henderson-Hasselbalch approximation

The Henderson-Hasselbalch equation assumes that the stoichiometric concentrations of HA and A⁻ are good approximations to their equilibrium concentrations. This holds when $c_T\gg [{\text{H}}^{+}]$ and $c_T\gg [{\text{OH}}^{-}]$. As the buffer is diluted, $c_T$ decreases but $[{\text{H}}^{+}]$ and $[{\text{OH}}^{-}]$ do not change proportionally -- they approach the values dictated by the acid-base equilibrium alone.

The exact pH of a buffer at any dilution follows from the charge-balance equation:

$$[{\text{H}}^{+}]+c_b=\frac{c_T{K}_a}{K_a+[{\text{H}}^{+}]}+\frac{K_w}{[{\text{H}}^{+}]}$$

where $c_b$ is the formal concentration of the conjugate-base salt. This cubic equation in [H⁺] reduces to the Henderson-Hasselbalch equation when the terms involving $K_w$ and $[{\text{H}}^{+}{]}^2$ are neglected. As $c_T\to 0$, the solution approaches $[{\text{H}}^{+}]=\sqrt{K_w}$ (pH 7.00) regardless of the initial buffer ratio.

The practical consequence is that dilution shifts buffer pH toward 7. The shift is negligible for $c_T>10^{-3}$ M but becomes substantial at lower concentrations. For a $10^{-3}$ M acetate buffer at pH 4.76, tenfold dilution to $c_T=10^{-4}$ M shifts the pH from 4.76 to approximately 5.1 -- a nontrivial deviation that invalidates the Henderson-Hasselbalch approximation.

Temperature effects on buffer pH

The pKa of most weak acids is temperature-dependent. The van 't Hoff equation gives the temperature coefficient:

$$\frac{d\text{p}{K}_a}{dT}=-\frac{\Delta H^{\circ }}{2.303R{T}^2}$$

For acetic acid, $\Delta H^{\circ }\approx -0.4$ kJ/mol (slightly exothermic dissociation), so pKa is nearly temperature-independent. For Tris buffer (tris(hydroxymethyl)aminomethane, pKa = 8.06 at 25 °C), $\Delta H^{\circ }\approx +47.5$ kJ/mol, so pKa decreases by approximately 0.028 units per degree Celsius. A Tris buffer adjusted to pH 8.06 at 25 °C will read pH 7.50 at 37 °C -- a shift of 0.56 units large enough to affect enzyme activity.

Buffers with low $\Delta H^{\circ }$ values (and hence small temperature coefficients) are called isohydric buffers. Phosphate ($\Delta H^{\circ }\approx +3.7$ kJ/mol, dpKa/dT ≈ −0.0028/°C) is far more temperature-stable than Tris and is preferred when temperature varies during an experiment.

Biological buffer systems Master

Three buffer systems maintain physiological pH in mammals. Each operates in a different compartment and at a different effective pH.

The bicarbonate buffer system

The primary extracellular buffer is the ${\text{CO}}_2/{\text{HCO}}_3^{-}$ system. Dissolved CO₂ is in equilibrium with carbonic acid:

$${\text{CO}}_2(\text{aq})+{\text{H}}_2\text{O}\rightleftharpoons {\text{H}}_2{\text{CO}}_3\rightleftharpoons {\text{H}}^{+}+{\text{HCO}}_3^{-}$$

The combined first dissociation constant (using dissolved CO₂ rather than H₂CO₃ as the acid species) gives an effective $\text{p}{K}_a^{\prime }=6.10$ at 37 °C and physiological ionic strength. The Henderson-Hasselbalch equation for this system is:

$$\text{pH}=6.10+\log \frac{[{\text{HCO}}_3^{-}]}{0.0301\times {\text{PCO}}_2}$$

where ${\text{PCO}}_2$ is in mmHg and 0.0301 is the solubility coefficient of CO₂ in plasma (mmol L⁻¹ mmHg⁻¹). In arterial blood: $[{\text{HCO}}_3^{-}]\approx 24$ mM, ${\text{PCO}}_2\approx 40$ mmHg:

$$\text{pH}=6.10+\log \frac{24}{1.20}=6.10+1.30=7.40$$

The bicarbonate system is an open-system buffer: excess CO₂ is removed by ventilation and bicarbonate is regulated by renal excretion. The open-system buffer capacity far exceeds the closed-system value because the acid component (dissolved CO₂) can be expelled. This is why the bicarbonate system, despite its pKa being 1.3 units below physiological pH (where the closed-system capacity would be only ~18% of maximum), is the dominant extracellular buffer.

Clinical acid-base disturbance is classified using the Henderson-Hasselbalch framework. Metabolic acidosis (decreased $[{\text{HCO}}_3^{-}]$) and respiratory acidosis (increased ${\text{PCO}}_2$) both lower pH. Metabolic alkalosis (increased $[{\text{HCO}}_3^{-}]$) and respiratory alkalosis (decreased ${\text{PCO}}_2$) both raise pH. Compensatory mechanisms adjust the other variable to restore pH toward 7.40.

The phosphate buffer system

Intracellular pH is buffered primarily by ${\text{H}}_2{\text{PO}}_4^{-}/{\text{HPO}}_4^{2-}$ (pKa₂ = 7.20). At physiological pH 7.40:

$$\frac{[{\text{HPO}}_4^{2-}]}{[{\text{H}}_2{\text{PO}}_4^{-}]}=10^{7.40-7.20}=1.58$$

Phosphate is a closed-system buffer within the cell. Its pKa is within 0.2 units of physiological pH, giving 96% of maximum capacity per unit concentration. Total inorganic phosphate in cytoplasm is approximately 1 mM, providing modest but well-positioned buffering. Phosphate also serves as the primary urinary buffer, where its concentration is much higher.

Protein buffering

The ionisable side chains of amino acids in proteins provide a distributed buffer system. The most important at physiological pH is the imidazole side chain of histidine (pKa ≈ 6.0-6.8 depending on the protein environment). Haemoglobin contributes substantially to blood buffering: each tetramer contains 38 histidine residues, and haemoglobin accounts for roughly 60% of non-bicarbonate blood buffering.

The buffer capacity of whole blood (bicarbonate + haemoglobin + plasma proteins + phosphate) at pH 7.40 is approximately $\beta \approx 58$ mM/pH unit -- much larger than the bicarbonate system alone in a closed container, because haemoglobin and the open-system behaviour of CO₂ both contribute.

Buffer selection: Good's buffers Master

In 1966, Good, Winget, Winter, and Connolly published criteria for selecting biological buffers and introduced a series of zwitterionic sulfonic acid buffers (HEPES, MES, MOPS, PIPES, and others) that satisfy them ^{[Good et al. 1966]}. The criteria are:

1. pKa between 6 and 8 (physiological range)
2. High solubility in water
3. Minimal permeability through biological membranes
4. No significant complexation with metal ions
5. Chemical stability and enzymatic non-reactivity
6. Minimal UV absorption above 240 nm (for spectrophotometric assays)
7. Low temperature coefficient of pKa
8. Well-defined purity and commercial availability

HEPES (pKa = 7.50 at 25 °C, 7.31 at 37 °C) is the most widely used buffer in cell culture. Its pKa at 37 °C is within 0.1 units of physiological pH, giving near-maximum capacity. However, HEPES produces free radicals under fluorescent lighting in the presence of riboflavin -- a limitation not shared by MOPS (pKa = 7.20 at 25 °C).

The selection of an appropriate buffer for a given application requires balancing these criteria. For enzyme kinetics at 37 °C, the effective pKa at the working temperature must be used, not the tabulated 25 °C value. For metalloenzyme studies, buffers that chelate the metal cofactor (phosphate binds Ca²⁺ and Mg²⁺; Tris binds Cu²⁺ and Ni²⁺) must be avoided.

Polyprotic buffer systems

Polyprotic acids provide multiple buffering regions. Phosphoric acid (pKa₁ = 2.15, pKa₂ = 7.20, pKa₃ = 12.35) buffers effectively at three pH ranges: 1.15-3.15, 6.20-8.20, and 11.35-13.35. When two pKa values are within 2-3 units of each other, the buffer capacities of adjacent pairs overlap. Citric acid (pKa₁ = 3.13, pKa₂ = 4.76, pKa₃ = 6.40) has overlapping regions that provide continuous buffering from pH 2 to 7.5, though the capacity at any single pH is lower than a monoprotic buffer at the same total concentration.

The total buffer capacity of a polyprotic system is the sum of the individual buffer capacities from each conjugate pair:

$${\beta }_{\text{total}}=\sum _{i=1}^{n}2.303\cdot c_T\cdot {\alpha }_{i-1}\cdot {\alpha }_i+{\beta }_w$$

where ${\alpha }_i$ are the fractional compositions for the (n+1) species. Each successive pair contributes a peak at pH = pKa,i, and the peaks blend smoothly when the pKa values are closely spaced.

Connections Master

• Acid-base equilibria 14.10.01. This unit applies the Henderson-Hasselbalch equation and $K_a$ expression developed in 14.10.01 to the specific problem of pH regulation. The buffer-ratio theorem (pH depends on ratio, not absolute concentration) is the foundation of all buffer preparation.

• Titrations 14.10.03. The buffer region of a titration curve is governed by the same equations derived here. Buffer capacity determines the slope of the titration curve: high β gives a flat (well-buffered) region, and low β gives a steep rise near the equivalence point.

• Electrochemistry 14.11.01. Standard buffer solutions (pH 4.01, 6.86, 7.41, 10.01 at 25 °C) calibrate glass electrodes. The accuracy of pH measurement depends on the accuracy of these standard buffers, prepared from primary standard salts (potassium hydrogen phthalate, potassium dihydrogen phosphate/disodium hydrogen phosphate, borax).

• Membrane transport and bioenergetics 17.02.02. The proton motive force across mitochondrial and bacterial membranes is an acid-base gradient. The bicarbonate and phosphate buffer systems maintain the pH stability required for enzyme function and proton-coupled ATP synthesis.

Historical notes Master

Lawrence Joseph Henderson derived the non-logarithmic form of the buffer equation in 1908 while studying the transport of CO₂ and O₂ in blood ^{[Henderson 1908]}. His equation expressed the relationship between bicarbonate concentration, dissolved CO₂, and hydrogen-ion concentration as a proportionality, without logarithms. Henderson was a physiological chemist at Harvard, and his interest was the blood buffer system, not a general equation for all buffers.

Karl Albert Hasselbalch, a Danish physician and chemist, adapted Henderson's equation into its modern logarithmic form in 1916 ^{[Hasselbalch 1916]}. He was motivated by the need for a clinically convenient formula for blood pH. By expressing the equation in terms of pH (Sorensen's logarithmic scale, introduced in 1909), he produced a form usable directly with pH measurements from the recently invented glass electrode.

Donald Dexter Van Slyke derived the quantitative theory of buffer capacity in 1922, introducing the symbol β and showing that buffer capacity is proportional to the product of fractional concentrations of acid and conjugate base ^{[Van Slyke 1922]}. Van Slyke's monograph on blood acid-base chemistry established the clinical framework still used in modern blood-gas analysis.

Norman Good and colleagues at Michigan State University published their systematic study of biological buffers in 1966 ^{[Good et al. 1966]}. Prior to Good's work, biochemists relied on a small number of available buffers (phosphate, Tris, carbonate), each with significant limitations. Good's criteria for biological buffers and the introduction of zwitterionic sulfonic acid buffers gave biochemists a rational basis for buffer selection and transformed the practice of enzyme kinetics and cell culture.

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