Buffer Calculator: Temperature, pH and Titrant Recipe

For each modeled dissociation, use the evaluated thermodynamic pKa, ionization enthalpy and heat-capacity change at 298.15 K from Goldberg, Kishore and Lennen (2002), attached source s1 pp. 130–131 (printed 359–360). Signs were checked against the PDF image. The temperature correction integrates the van't Hoff equation with constant ΔCp:

The local coefficient is $dpK_a/dT=-[\Delta H_r+\Delta C_p(T-T_r)]/(R\ln10\,T^2)$. This retains the temperature curvature that a single coefficient would omit.

For Davies (USGS PHREEQC 3 manual, SOLUTION_SPECIES; attached source s8), $\log_{10}\gamma_z=-A(T)z^2[\sqrt{I^*}/(1+\sqrt{I^*})-0.3I^*]$, where $I^*=I/(1\,\mathrm{mol/L})$. The selected total I is supplied by the user. $A(25\,^{\circ}\mathrm C)=0.509$; its temperature scaling is $(\epsilon_{25}T_r/\epsilon_T T)^{3/2}$, using the Malmberg–Maryott permittivity polynomial (s4 p. 1), at fixed water density. This is an approximate dilute-solution correction, not NIST's own activity model.

Mass action gives $c_j/c_{j-1}=10^{pH-pK_{a,j}}\gamma_{j-1}/\gamma_j$. Normalize the state weights to obtain fractions $\alpha_j$, mean deprotonation $\bar j$ and variance. Tris has charges +1, 0; the modeled Good buffers have charges 0, −1; phosphate and citrate have charges 0, −1, −2, −3.

Define $B=C\bar j+[OH^-]-[H^+]$. For reagent indices $j_a,j_b$, the two-form base fraction is $(B/C-j_a)/(j_b-j_a)$. From the acid form, NaOH amount is $V(B-Cj_a)$; from the base form, HCl amount is $V(Cj_b-B)$. Multiply reagent moles by the listed molar masses and divide titrant moles by stock molarity. These balances and the capacity relation are derived here from mass action and conservation.

Solve $B(pH,T_{prep})=B(pH_{target},T_{work})$ for preparation pH. Water uses the HSPF manual's $pK_w=4470.99/T-6.0875+0.01706T$ (Subroutine Group PHCARB, printed p. 258); $[H^+]=10^{-pH}/\gamma_1$ and $[OH^-]=10^{pH-pK_w}/\gamma_1$ in mol/L. Capacity at fixed I and volume is $\beta=\ln10\,[C\,\mathrm{Var}(j)+[H^+]+[OH^-]]$, reported as mmol/L per pH unit.

For each pH sweep state, retain all spectator ions in the selected target recipe and add the counterion from the further HCl or NaOH dose. Thus $C_{spect}(p)=C_{counter}+|B(p)-B_{target}|$ and $I_{need}(p)=\tfrac12[C\sum_j\alpha_j z_j^2+C_{spect}(p)+[H^+]+[OH^-]]$. In Davies mode, both pH charts retain only samples with $I_{need}\le I_{total}+0.000005\,\mathrm{mol/L}$. The grid includes the target pH. This allowance is half the 0.00001 mol/L reporting increment. The reported minimum rounds the unrounded recipe contribution upward; screening uses the unrounded value. Recheck the minimum after changing total ionic strength because it depends on the current activity estimate.

Route feasibility is directional: a two-form mixture requires $0\le x\le1$; acid + NaOH requires $x\ge0$; base + HCl requires $x\le1$. Strong titrant can supply free H⁺ or OH⁻ beyond the mixture's composition limits.

• Pure, fully soluble reagents in water; standardized HCl or NaOH; no purity correction. Check the exact reagent formula on the bottle.
• Phosphate uses anhydrous NaH₂PO₄ / Na₂HPO₄. Citrate uses citric acid monohydrate / trisodium citrate dihydrate. Tris uses anhydrous hydrochloride / free base. HEPES, MES and MOPS use anhydrous free acid / sodium salt.
• Concentrations refer to final solution volume; dissolution and titrant volumes are included when making up to that mark.
• pH denotes hydrogen-ion activity. Thermodynamic molality-based pKa values are used on a molarity basis as a dilute-solution approximation. No extra hydrogen-ion activity coefficient is inserted into the Henderson–Hasselbalch pH expression.
• Total ionic strength is held fixed during temperature changes and across the titration/capacity charts. The calculated recipe contribution includes buffer ions, water ions, reagent counterions and strong-titrant ions. The difference from entered total I must be supplied by the rest of the solution; no supporting-salt recipe is calculated.
• Density and volume expansion are neglected in the temperature correction. Davies A scales with water permittivity and absolute temperature at fixed density. Neutral-species activity coefficients equal one.
• Each pH-chart sample independently adds HCl or NaOH to the target recipe on a fixed final-volume basis, retaining all already-present sodium/chloride ions. Samples exceeding the entered ionic strength plus 0.000005 mol/L are omitted; negative supporting electrolyte is never assumed. The ionic diagnostic table stays visible when result cards are held.

Use as a starting recipe for 0–50 °C aqueous laboratory buffers. Davies is an approximate charge-only model; prefer I ≤ 0.1 mol/L and treat 0.1–0.5 mol/L cautiously, particularly for multivalent citrate and phosphate. Even lower-I predictions can differ from measurements because of ion-specific interactions and the operational pH scale.

Modeled target pH ranges: Tris 7–9; HEPES 6.5–8.8; phosphate 5.5–8.5; citrate 2.5–7.5; MES 4.8–7.8; MOPS 5.7–8.7. A warning identifies targets more than one pH unit from the nearest conditional pKa. The Good buffers use their near-neutral dissociation only; HEPES's lower, approximate pKa near 3 is excluded.

The signed titration-volume chart uses equivalents on a fixed final-volume basis, with the entered total ionic strength imposed throughout. Both pH charts omit states whose buffer, water and accumulated titrant counterions exceed that total, allowing 0.000005 mol/L for reporting/model precision. The diagnostic table gives the retained sampled pH range; no feasible samples gives an empty chart. The minimum rounds upward to 0.00001 mol/L and can therefore exceed an entry accepted within the unrounded allowance. For a real cumulative titration, changing dilution and ionic strength require a coupled model. Capacity is the differential fixed-I response, not a finite-dose guarantee.

No precipitation, metal complexation, specific ion interactions, mixed solvents, dissolved CO₂, proteins or multiple buffer families are modeled. Phosphate can precipitate with calcium; citrate and Good buffers can bind ions; check compatibility with the experiment. Temperature compensation on a pH meter corrects electrode response and does not remove the solution's chemical pH shift. Verify the final solution experimentally at working temperature.