Buffer capacity ($\beta$)

Buffer capacity measures resistance to a change in pH: how much additional chemical acid or chemical base a chemical solution accommodates per small change in pH. Its differential form is a local derivative, rather than the total inventory available before a buffer solution is exhausted. Two buffer solutions can have identical pH and very different buffer capacities.

Choose a constant volume $V$ and fixed temperature. Let $n_B$ count added moles of fully dissociated, monovalent chemical base supplying hydroxide, and $n_A$ added moles of fully dissociated, monovalent strong acid supplying hydrogen ions. Neither addition supplies the weak acid component itself. The signed addition is $b=(n_B-n_A)/V$, with units $\mathrm{mol}{\mathrm{L}}^{-1}$, andThe last two forms describe separate base-only and acid-only additions. The buffer capacity is positive for the stable chemical equilibrium model below and is reported in $\mathrm{mol}{\mathrm{L}}^{-1}$ per unit of pH. Since pH is dimensionless, the dimensional unit is just $\mathrm{mol}{\mathrm{L}}^{-1}$; stating “per pH” makes the measurement convention clear. Here constant volume is an idealization, realized approximately by small additions of concentrated reagent.

For a sufficiently small addition, $\Delta \mathrm{pH}\approx \Delta b/\beta$. A large buffer capacity means a small local change of pH for a given addition. Over a finite interval, use the changing buffer capacity instead:This integral gives a finite dose in $\mathrm{mol}{\mathrm{L}}^{-1}$; it need not equal the initial buffer capacity times the eventual change in pH.

Assume an ideal dilute aqueous chemical solution with just one monoprotic acid pair $\mathrm{HA}/{\mathrm{A}}^{-}$, self-ionization of water, and other ions that balance electric charge without undergoing further chemical reactions. Hold the analytical concentration $C=[\mathrm{HA}]+[{\mathrm{A}}^{-}]$ constant. Keep the acid dissociation constant $K_a$ and the self-ionization of water constant $K_w$ dimensionless, consistently with pH:Thus $H$ denotes normalized hydrogen ion concentration, not a dimensional concentration. The chemical equilibrium and conserved analytical concentration giveAdding chemical base converts weak acid to conjugate base, increases free hydroxide, and decreases free hydrogen ion concentration. Balancing electric charge, including the added reagent's accompanying ions, gives the signed dose relative to any initial chemical equilibrium at $p_0$:Differentiate with respect to pH, using $\mathrm{d}H/\mathrm{d}p=-(\ln 10)H$. The result isThe first term is the monoprotic acid pair's contribution, ${\beta }_{\mathrm{pair}}$; the remaining terms are the free hydrogen ion and hydroxide contributions, ${\beta }_{\mathrm{water}}$. Each has units $\mathrm{mol}{\mathrm{L}}^{-1}$ per pH. Equivalently, with dimensional $H_c=[{\mathrm{H}}^{+}]$, $K_{a,c}=c^{\circ }{K}_a$ and $K_{w,c}=(c^{\circ }{)}^2{K}_w$, the expression is $(\ln 10)(C{K}_{a,c}{H}_c/(K_{a,c}+H_c{)}^2+H_c+K_{w,c}/H_c)$. Mixing the dimensional and dimensionless conventions would give incorrect units.

For $r=K_a/H=10^{\mathrm{pH}-\mathrm{p}{K}_a}$,because $(1+r{)}^2-4r=(1-r{)}^2\ge 0$. Equality occurs at $r=1$: equal weak acid and conjugate base concentrations, or pH equal to pKa. This is the exact maximum of the pair contribution in this model. The full buffer capacity need not have its maximum there: free hydrogen ions and hydroxide make it grow toward the acidic and alkaline extremes. When these contributions are small, the central peak lies close to pKa.

Stipulate pKa $5$ and $K_w=10^{-14}$; these are model inputs, not measurements of an identified substance. At pH $5$ the pair contribution is $0.575646\dots C$. For analytical concentrations $C=0.010$ and $0.100\mathrm{mol}{\mathrm{L}}^{-1}$, respectively, it isThe free hydrogen ion and hydroxide contribution is the same at this pH, ${\beta }_{\mathrm{water}}=(\ln 10)(10^{-5}+10^{-9})\mathrm{mol}{\mathrm{L}}^{-1}=0.0000230282\dots \mathrm{mol}{\mathrm{L}}^{-1}$ per pH. The full buffer capacities are therefore approximately $0.00577949$ and $0.0575877\mathrm{mol}{\mathrm{L}}^{-1}$ per pH. Identical pH fixes the free hydrogen ion thermodynamic activity; it does not fix the inventory of weak acid and conjugate base available for neutralization.

At one unit of pH on either side of pKa, $r=10$ or $0.1$, giving ${\beta }_{\mathrm{pair}}/{\beta }_{\mathrm{pair},\max }=40/121\approx 0.331$. The partners still form a buffer solution, but its local resistance is only about one third of the maximum for that analytical concentration. At fixed pH, the pair contribution scales exactly with $C$; the full buffer capacity contains the additional $C$-independent water contribution. Consequently, extreme dilution does not justify dropping self-ionization of water.

Prepare the same ideal monoprotic acid model at initial pH $5$, with appropriate accompanying ions to balance electric charge. To predict the new pH after a finite signed addition $\Delta b$, solve $Q({\mathrm{pH}}_1)-Q(5)=\Delta b$ using the chemical equilibrium expression above. For $\Delta b=0.002\mathrm{mol}{\mathrm{L}}^{-1}$ of added chemical base, the calculated final pH is $5.366798$ at $C=0.010\mathrm{mol}{\mathrm{L}}^{-1}$, but $5.034749$ at $C=0.100\mathrm{mol}{\mathrm{L}}^{-1}$. The initial local approximation gives changes of about $0.34605$ and $0.034730$, respectively; its accuracy decreases as the partner balance changes appreciably.

For the dilute example, additions beyond approximately $0.005\mathrm{mol}{\mathrm{L}}^{-1}$ in either direction exceed the initial inventory of one partner. The full chemical equilibrium calculation still produces a finite pH; an inventory-only Henderson-Hasselbalch equation calculation would attempt to use a negative partner quantity. At $+0.006\mathrm{mol}{\mathrm{L}}^{-1}$ the dilute example reaches pH $10.99564$, whereas the concentrated example reaches only $5.104698$. This demonstrates a finite operating range, not an abrupt cessation of all buffer capacity.

A proposed modern measurement extends the weak acid dissociation measurement by estimating the local slope of an acid-base titration. Transfer a measured sample with a pipette into a stirred vessel; clamp a combination glass electrode/reference electrode clear of the stirring magnet and place a temperature probe nearby. A calibrated burette delivers a chemical base of independently standardized concentration $c_T$, such as sodium hydroxide. Record the initial sample volume $V_0$, cumulative delivered volume $v$, stable pH, and temperature after each small addition. Use a separate fresh portion for an acidward series. These are proposed measurement conditions, not reported experimental results.

For the instrument calibration, use fresh standard buffer solutions bracketing the measured pH range, with values appropriate to the actual temperature. The Hanna manual specifies separate rinse and calibration portions, gentle stirring, and approximately $4\mathrm{cm}$ immersion of the probe tip. It calls for stable readings; the sensing bulb and reference electrode junction must remain immersed. Rinse between standards and samples and avoid carrying rinse water into the measured portion.^[ref]

The readout pairs each added amount of substance with the measured pH. With negligible added volume, neighboring readings give the finite difference estimateassigned to the interval's midpoint. High buffer capacity appears as a small change of pH for a measured reagent dose. Choose increments small enough to resolve changing buffer capacity, but large enough that the change of pH exceeds measurement noise. Repeat with a fresh portion and different increment sizes; remeasure a standard afterward to check drift. A water blank can reveal contamination, but its pH must not be subtracted from the sample pH. These controls are proposed tests of the measurement and model assumptions.

If additions appreciably increase volume, even the analytical concentration changes: $C(v)=C_0{V}_0/(V_0+v)$. Merely dividing the added moles by the current volume does not restore the constant-volume derivative. One alternative is explicitly to report a changing-volume convention. For $c(v)=c_{T}v/(V_0+v)$, defineThis definition includes the reagent's changing concentration and differs from the constant-volume buffer capacity defined above.^[ref]

To compare with the constant-volume theory, fit the changing-volume chemical equilibrium model instead of treating the observed slope as $\beta$. In the same ideal monoprotic acid system, define $S_0=[{\mathrm{A}}^{-}{]}_0+[{\mathrm{OH}}^{-}{]}_0-[{\mathrm{H}}^{+}{]}_0$. For baseward additions, the electric charge balance at each reading isUse a minus sign before $c_{T}v$ for the corresponding acidward series. From the model parameters, evaluate the constant-volume buffer capacity at the chosen analytical concentration and pH. The fit also tests whether one acid dissociation constant describes the observations.

Limitations include incomplete mixing, slow glass electrode response, uncertain reagent concentration, and changes in temperature or ionic strength. Dissolved atmospheric carbon dioxide introduces an additional chemical acid system; comparable exposure and freshly prepared samples help distinguish it from the intended monoprotic acid response. Additional reacting chemical species and nonideal activity coefficients invalidate the simple formula. Especially when pH increments are tiny, measurement noise can dominate the inferred derivative; reporting the dose, pH interval, volume convention, and repeatability is part of reporting a meaningful buffer capacity.