Conductometric Acid Mixture Titration theory

Electrolytic conduction in aqueous electrolyte solutions arises from the directed migration of charged ionic species under the influence of an applied external electric field. Unlike metallic conduction, which is mediated by free electron transport through a crystalline lattice without mass transfer, electrolytic conduction involves the macroscopic convective and diffusive transport of solvated ionic matter accompanied by chemical oxidation and reduction reactions at electrode boundaries. The quantitative measure of an electrolyte's ability to conduct electricity is its electrolytic conductivity κ (expressed in Siemens per meter, S·m⁻¹, or mS·cm⁻¹), defined as the reciprocal of electrical resistivity ρ:



κ = 1 / ρ = G · (l / A) = G · K_cell



where G is measured electrolytic conductance (Siemens, S), l is the inter-electrode distance, A is the cross-sectional area of the conducting solution, and K_cell = l / A is the geometric cell constant (cm⁻¹).



Under the framework of Kohlrausch's law of independent migration of ions at infinite dilution, the limiting molar conductivity of an electrolyte Λ_m° is the additive sum of the individual limiting ionic molar conductivities λ_i° of its constituent cations and anions:



Λ_m° = ν_+ λ_+° + ν_- λ_-°



Each ionic molar conductivity is directly proportional to the fundamental ionic mobility u_i (defined as the drift velocity acquired per unit electric field strength, v_i / E, in m²·s⁻¹·V⁻¹):



λ_i° = z_i · F · u_i



where z_i is ionic charge and F is the Faraday constant (96,485.3 C·mol⁻¹). In conventional hydrodynamic migration, an ion travels through the viscous solvent medium subject to Stokes drag, yielding an effective hydrodynamic radius r_h governed by the Stokes-Einstein relation:



u_i = |z_i| e / (6 π η r_h)



For sodium ions (Na⁺) and chloride ions (Cl⁻), conduction occurs strictly via this hydrodynamic transport mechanism. Because of dielectric polarization and strong ion-dipole hydration, Na⁺ carries a tightly bound primary hydration shell (effective hydrodynamic radius r_h ≈ 0.184 nm), yielding a modest limiting ionic conductivity λ°(Na⁺) = 50.11 S·cm²·mol⁻¹ at 25 °C. Chloride ions, possessing a larger crystallographic radius and weaker surface charge density, exhibit less viscous hydration drag, yielding λ°(Cl⁻) = 76.35 S·cm²·mol⁻¹.



In striking contrast, the aqueous hydronium ion (H₃O⁺, or more realistically hydrated clusters such as the Zundel cation H₅O₂⁺ and Eigen cation H₉O₄⁺) exhibits an extraordinarily anomalous limiting ionic conductivity λ°(H⁺) = 349.8 S·cm²·mol⁻¹ at 25 °C—nearly seven times greater than that of sodium and five times greater than chloride. Similarly, the hydroxide ion (OH⁻) displays an exceptionally high limiting conductivity λ°(OH⁻) = 198.6 S·cm²·mol⁻¹. This remarkable phenomenon cannot be explained by classical Stokesian hydrodynamic migration. Instead, it is governed by the Grotthuss proton-hopping mechanism (structural diffusion), in which excess protons propagate through the liquid water hydrogen-bonded network via rapid concerted reorganization of covalent O–H bonds and intermolecular hydrogen bonds. A proton from a hydronium ion transfers across an existing hydrogen bond to an adjacent water molecule, effectively shifting the positive charge center across molecular distances in ~1.5 picoseconds without requiring physical hydrodynamic translation of the oxygen nucleus. This colossal disparity in ionic mobility between H⁺ (349.8 S·cm²·mol⁻¹) and Na⁺ (50.11 S·cm²·mol⁻¹) forms the fundamental physical basis for the dramatic, linear conductance drop observed during the neutralization of strong acids.

In any wet-lab volumetric titration, the progressive addition of titrant solution increases the total volume of the reaction mixture from the initial volume V_0 to (V_0 + V). Because electrolytic conductance is an extensive property proportional to the volumetric concentrations of all ionic charge carriers present in the solution cell, continuous volumetric dilution causes the observed conductance to systematically decrease independently of chemical reaction stoichiometry. If uncorrected raw conductance values G_raw are plotted directly against titrant volume V, the resulting titration curves exhibit pronounced downward curvature rather than linear segments, making the accurate graphical determination of equivalence point intersections via linear regression mathematically invalid.



To restore linear stoichiometry, the raw conductance readings must be multiplied by the volume dilution correction factor:



G_corr = G_raw × (V_0 + V) / V_0



where V_0 is the initial volume of analyte solution in the conductometric cell prior to titrant addition (100.0 mL in our standardized protocol), and V is the cumulative volume of titrant dispensed from the burette (mL). Physically, G_corr represents the conductance that the reaction mixture would exhibit if all ions generated or consumed during the reaction were maintained at a constant total solution volume equal to V_0.



We can derive the rigorous mathematical equations governing the slope of each linear branch of the corrected conductometric titration curve. Prior to the first equivalence point (Branch 1: 0 ≤ V ≤ V_eq,1), standardized sodium hydroxide (concentration c_b) reacts with hydrochloric acid (initial concentration c_a1 in sample aliquot V_s):



H⁺(aq) + Cl⁻(aq) + Na⁺(aq) + OH⁻(aq) → Na⁺(aq) + Cl⁻(aq) + H₂O(l)



During this stage, each equivalent of added titrant consumes one equivalent of highly conducting hydronium ions (λ°_H = 349.8 S·cm²·mol⁻¹) and replaces them with an equivalent of lower-mobility sodium ions (λ°_Na = 50.11 S·cm²·mol⁻¹), while the concentration of chloride counterions remains unperturbed. Expressing the corrected electrolytic conductivity κ_corr as the sum of ionic contributions:



κ_corr(V) = (1 / 1000) · [ c_H(V) · λ°_H + c_Na(V) · λ°_Na + c_Cl · λ°_Cl ]



Substituting the stoichiometric relations c_H(V) = (c_a1 · V_s - c_b · V) / V_0 and c_Na(V) = (c_b · V) / V_0, and converting to conductance G_corr = κ_corr / K_cell:



G_corr(V) = G_0 - [ (c_b / (1000 · K_cell · V_0)) · (λ°_H - λ°_Na) ] · V



The slope of Branch 1 is therefore strictly negative:



Slope_1 = - (c_b / (1000 · K_cell · V_0)) · (λ°_H - λ°_Na)



Because (λ°_H - λ°_Na) = 349.8 - 50.11 = +299.69 S·cm²·mol⁻¹, Branch 1 exhibits a steep, highly linear negative slope. For c_b = 0.1000 M, V_0 = 100.0 mL, and K_cell = 1.000 cm⁻¹, the theoretical slope is Slope_1 ≈ -0.300 to -0.355 mS·mL⁻¹ (accounting for finite ionic strength activity coefficients).



Once all hydrochloric acid is consumed (V = V_eq,1), titration of the weak acetic acid begins (Branch 2: V_eq,1 ≤ V ≤ V_eq,2):



CH₃COOH(aq) + Na⁺(aq) + OH⁻(aq) → CH₃COO⁻(aq) + Na⁺(aq) + H₂O(l)



Because acetic acid is largely undissociated in the acetate buffer matrix, addition of titrant converts molecular CH₃COOH into sodium acetate (Na⁺ and CH₃COO⁻). Both ions contribute positively to solution conductivity:



Slope_2 = + (c_b / (1000 · K_cell · V_0)) · (λ°_Na + λ°_Ac)



Since λ°_Na = 50.11 and λ°_Ac = 40.90 S·cm²·mol⁻¹, the sum is (50.11 + 40.90) = +91.01 S·cm²·mol⁻¹. Consequently, Branch 2 exhibits a moderate positive slope (Slope_2 ≈ +0.091 mS·mL⁻¹), rising gently toward the second equivalence point.



Beyond the second equivalence point (Branch 3: V > V_eq,2), all acids have been completely neutralized. Further addition of titrant introduces unreacted sodium hydroxide:



Slope_3 = + (c_b / (1000 · K_cell · V_0)) · (λ°_Na + λ°_OH)



Because hydroxide ions are exceptionally mobile (λ°_OH = 198.6 S·cm²·mol⁻¹), the ionic sum increases dramatically to (50.11 + 198.6) = +248.71 S·cm²·mol⁻¹. Consequently, Branch 3 exhibits a sharp, steep positive slope (Slope_3 ≈ +0.249 mS·mL⁻¹). The abrupt slope discontinuities between Branches 1, 2, and 3 define two acute inflection vertices that allow unambiguous mathematical resolution of both equivalence points.

The accurate simultaneous conductometric determination of hydrochloric acid and acetic acid in a single sample aliquot relies upon the fundamental thermodynamic principle of common-ion suppression, governed by Le Chatelier's principle and chemical equilibrium speciation. Hydrochloric acid is a strong electrolyte that dissociates essentially 100% in dilute aqueous solution:



HCl(aq) + H₂O(l) → H₃O⁺(aq) + Cl⁻(aq) (K_a >> 10⁶)



In contrast, acetic acid is a weak organic carboxylic acid that participates in a reversible protolytic equilibrium:



CH₃COOH(aq) + H₂O(l) ⇄ H₃O⁺(aq) + CH₃COO⁻(aq) (K_a = 1.75 × 10⁻⁵, pK_a = 4.76 at 25 °C)



The thermodynamic acid dissociation constant expression is:



K_a = ( [H₃O⁺] · [CH₃COO⁻] · γ_H · γ_Ac ) / ( [CH₃COOH] · γ_HOAc )



In a pure 0.040 M acetic acid solution in the absence of strong acid, the degree of dissociation α is:



α ≈ √(K_a / c) = √(1.75 × 10⁻⁵ / 0.040) ≈ 0.0209 (2.09%)



yielding [H₃O⁺] ≈ 8.4 × 10⁻⁴ M (pH ≈ 3.08). However, in our binary acid mixture, the sample contains 0.0400 M HCl alongside 0.0600 M CH₃COOH. Upon 4-fold dilution in the conductometric vessel (25.00 mL diluted to 100.0 mL), the initial hydrochloric acid concentration is [HCl]_init = 0.0100 M, which completely ionizes to provide [H₃O⁺] ≈ 1.0 × 10⁻² M (pH ≈ 2.00).



Substituting this strong acid hydronium ion concentration into the acetic acid equilibrium expression:



[CH₃COO⁻] = K_a · [CH₃COOH] / [H₃O⁺] = (1.75 × 10⁻⁵) · (0.0150 M) / (0.0100 M) = 2.62 × 10⁻⁵ M



The degree of ionization of acetic acid is suppressed from 2.09% down to:



α' = [CH₃COO⁻] / [CH₃COOH]_total = (2.62 × 10⁻⁵) / (1.50 × 10⁻²) = 0.00175 (0.175%)



Over 99.8% of the acetic acid remains entirely in the unionized, neutral molecular form CH₃COOH. Because uncharged molecular species do not carry electric current, acetic acid contributes virtually zero conductance to the initial solution. The observed conductance is dictated entirely by the free H⁺ and Cl⁻ ions from hydrochloric acid.



Furthermore, when sodium hydroxide titrant is introduced, the hydroxide ions react preferentially with the strongest available proton donor in solution. Because hydronium ions have a proton affinity far greater than molecular acetic acid, the reaction:



H₃O⁺ + OH⁻ ⇄ 2 H₂O (K = 1 / K_w = 1.0 × 10¹⁴)



proceeds essentially to completion before hydroxide can react with the weak acid. Throughout Branch 1, as hydronium is neutralized, the common-ion suppression is gradually relaxed; however, even near the end of Branch 1 (at 90% neutralization of HCl, where [H⁺] ≈ 1.0 × 10⁻³ M), the ionization of acetic acid remains suppressed below 1.7%. Only when the hydrochloric acid concentration drops to negligible levels (within ~0.1 mL of V_eq,1) does the neutralization of molecular acetic acid commence.



This profound separation of thermodynamic acidities ensures that the titration curve does not blend into an unresolvable curved smear, but instead partitions cleanly into two distinct, sequential chemical regimes. Even though standard colorimetric indicators (such as phenolphthalein or methyl orange) fail completely to resolve mixtures of strong and weak acids with overlapping titration sigmoids, conductometry exploits the sharp change in sign of the slope—from negative (Slope_1 ≈ -0.355 mS·mL⁻¹) to positive (Slope_2 ≈ +0.091 mS·mL⁻¹)—to isolate V_eq,1 with precision better than ±0.03 mL.

If direct current (DC) were applied across metallic electrodes immersed in an electrolytic solution to measure electrical conductance, two catastrophic electroanalytical phenomena would instantly corrupt the measurement: Faradaic electrolysis and capacitive electrode polarization.



When a DC voltage is applied, electrical charge is continuously transferred across the electrode-solution interface via Faradaic redox reactions. At the cathode, water or hydronium ions would undergo reduction to form gaseous hydrogen:



2 H⁺(aq) + 2 e⁻ → H₂(g)



while at the anode, water or chloride ions would undergo oxidation to form gaseous oxygen or chlorine:



2 H₂O(l) → O₂(g) + 4 H⁺(aq) + 4 e⁻

2 Cl⁻(aq) → Cl₂(g) + 2 e⁻



These Faradaic electrochemical reactions would rapidly alter the chemical composition and pH of the analyte solution, destroying the very ionic concentrations being measured. Furthermore, the generation of gas bubbles on the electrode surface would insulate the conducting metal plates, drastically reducing effective contact area.



Even below the thermodynamic decomposition voltage of the solvent, applying a direct current causes non-Faradaic electrical double-layer charging. Ions of opposite sign migrate toward the charged electrodes, forming an electrical double layer (Helmholtz and Gouy-Chapman layers) that behaves as an electrical capacitor in series with the solution resistance R_sol. This builds up a counter-electromotive force (back-EMF, known as polarization potential E_pol), which opposes the applied voltage and creates a severe, time-dependent apparent resistance increase:



V_apparent = I · R_sol + E_pol



To overcome both Faradaic electrolysis and electrode polarization, precision conductivity meters employ high-frequency alternating current (AC), typically operating at audio frequencies of 1.0 kHz to 3.0 kHz using a balanced Wheatstone impedance bridge. Under AC excitation, the polarity of each electrode reverses thousands of times per second. The duration of each half-cycle (0.5 ms at 1 kHz) is far too brief for net mass transport or Faradaic electrolysis to occur. During each half-cycle, ions merely oscillate over sub-nanometer distances in the solvent without undergoing interfacial electron transfer.



To further nullify the reactive impedance of the electrical double-layer capacitance C_dl, the platinum electrodes are platinized—electrochemically coated with an ultra-fine, velvety layer of colloidal platinum black by cathodic deposition from a chloroplatinic acid (H₂PtCl₆) solution containing lead acetate. Platinization transforms the smooth, reflective metal into an extraordinarily porous, high-surface-area nanostructure, increasing the microscopic effective surface area by a factor of 1,000 to 10,000 relative to the geometric plate area (1.0 cm²). Because double-layer capacitance is directly proportional to real interfacial surface area (C_dl = ε · A_real / d), platinization boosts C_dl to tens of millifarads. The capacitive reactance of the electrode-solution interface is:



X_C = 1 / (2 π f C_dl)



By maximizing both the excitation frequency f (1.0 kHz) and the interfacial capacitance C_dl, the capacitive reactance X_C is driven essentially to zero (X_C < 0.05 Ω). Consequently, the measured cell impedance Z = √(R_sol² + X_C²) collapses strictly to the pure ohmic solution resistance R_sol, enabling pure, unpolarized electrolytic conductance acquisition.

The measured electrolytic conductance G (in Siemens, S, or milliSiemens, mS) is not an intrinsic property of a chemical solution; rather, it is an extensive measurement that depends strictly upon the geometric dimensions and configuration of the conductivity cell. For an idealized conductivity cell consisting of two flat, parallel electrode plates of identical cross-sectional area A (cm²) separated by a uniform distance l (cm), the electrolytic resistance R and conductance G are related to the intrinsic electrolytic conductivity κ (mS·cm⁻¹) by Ohm's law:



R = ρ · (l / A) = (1 / κ) · (l / A)

G = 1 / R = κ · (A / l)



The geometric ratio of plate separation to electrode cross-sectional area is defined as the cell constant K_cell:



K_cell = l / A (with SI units of m⁻¹, commonly reported in physical chemistry as cm⁻¹)



Therefore, the fundamental equation converting observed conductance G into absolute conductivity κ is:



κ = G · K_cell



In practical laboratory apparatus, the electric field lines between real electrode plates are not perfectly parallel; significant fringe electric fields bow outward into the surrounding fluid, and non-uniform current density distributions arise near plate edges and vessel walls. Consequently, K_cell cannot be calculated accurately from physical caliper measurements of l and A. Instead, K_cell must be determined experimentally through secondary calibration using a standard electrolyte solution of certified specific conductivity.



The international metrological primary reference standard for conductivity cell calibration is pure potassium chloride (KCl) dissolved in ultrapure water (specific resistance > 18.2 MΩ·cm), as established by the International Union of Pure and Applied Chemistry (IUPAC) and the International Organization of Legal Metrology (OIML). At 25.00 °C, standard solutions of KCl exhibit rigorously characterized electrolytic conductivities:

- 1.000 M KCl: κ = 111.34 mS·cm⁻¹

- 0.1000 M KCl: κ = 12.856 mS·cm⁻¹

- 0.0100 M KCl: κ = 1.4087 mS·cm⁻¹



In our experimental protocol, a certified 0.0100 M KCl standard solution equilibrated at 25.00 ± 0.05 °C is employed. The clean, platinized dip cell is immersed in the standard, and its conductance G_std is logged. The cell constant is computed via:



K_cell = κ_std / G_std = (1.4087 mS·cm⁻¹) / G_std



For a standard commercial dip-type cell with nominal 1.0 cm × 1.0 cm plates separated by 1.0 cm, K_cell is calibrated to K_cell = 1.000 ± 0.005 cm⁻¹.



Proper maintenance of the cell constant requires meticulous operational care. The platinum black coating must remain fully hydrated; drying causes platinum nanoparticles to agglomerate and flake off, altering the effective surface area and cell constant. When not in use, the electrode must be stored immersed in deionized water. The plates must never be touched with fingers, spatulas, or abrasive surfaces. Periodic recalibration against certified KCl ensures long-term metrological traceability and measurement accuracy.

Electrolytic conductivity exhibits an exceptionally large, positive temperature coefficient, typically ranging from +1.5% to +2.5% per degree Celsius for common aqueous electrolyte solutions at room temperature. This is fundamentally opposite to metallic conductors, whose conductivity decreases with increasing temperature due to enhanced electron-phonon scattering. In electrolytic solutions, elevated temperature enhances conductivity primarily through two coupled thermodynamic and hydrodynamic mechanisms: the reduction of solvent dynamic viscosity and the thermal enhancement of ionic hydration cluster hopping.



According to the Stokes-Einstein relation, the ionic mobility u_i of a solvated ion is inversely proportional to the dynamic shear viscosity η of the solvent:



u_i = |z_i| e / (6 π η r_h)



The viscosity of liquid water decreases sharply with temperature, following an Arrhenius-type exponential decay:



η(T) = A_η · exp(E_visc / (R · T))



where E_visc ≈ 15.0 kJ·mol⁻¹ is the activation energy for viscous water shear. As temperature rises from 20 °C to 25 °C, water viscosity drops from 1.002 mPa·s to 0.890 mPa·s—a decrease of over 11% (~2.2% per °C). This drastic reduction in solvent hydrodynamic drag allows ions to migrate significantly faster under an applied electric field.



Furthermore, for hydronium and hydroxide ions propagating via the Grotthuss proton-hopping mechanism, the rate-limiting step is the thermal rupture and reorientation of solvent hydrogen bonds in the water coordination cluster. The activation energy for Grotthuss proton hopping is E_hop ≈ 10.5 kJ·mol⁻¹. Consequently, the limiting ionic conductivities of H⁺ and OH⁻ also increase substantially with temperature (dλ°_H / dT ≈ +4.8 S·cm²·mol⁻¹·K⁻¹).



The temperature dependence of electrolytic conductivity is conventionally expressed using a linear temperature coefficient α_25:



κ_T = κ_25 · [ 1 + α_25 · (T - 25.00) ]



For a typical hydrochloric acid and acetic acid mixture in water, α_25 ≈ 0.018 K⁻¹ (+1.8% per °C). If the titration cell temperature were allowed to drift uncontrolled by as little as ±1.0 °C during the course of a 30-minute titration, the measured conductance would undergo an artificial drift of ±1.8%. In Branch 1, where the total conductance drop from V = 0 to V_eq,1 is from ~5.2 mS to ~1.65 mS (a net change of 3.55 mS), a 1.8% temperature fluctuation corresponds to an error of ±0.09 mS. This represents an unacceptable distortion of over 2.5% of the entire titration signal, which would induce substantial curvature in the linear branches and introduce a systematic error of ±0.3 mL into the determined equivalence point V_eq,1.



To achieve analytical research accuracy (R² ≥ 0.9999 and titrimetric accuracy within ±0.2%), the titration cell must be enclosed in a double-walled glass jacket connected to a circulating constant-temperature water bath regulated at 25.00 ± 0.05 °C. Holding temperature variations within ±0.05 °C restricts temperature-induced conductance drift to less than ±0.09% (less than ±0.005 mS), ensuring that observed conductance changes reflect purely the chemical stoichiometry of neutralization.

The quantitative evaluation of conductometric titration data for binary strong and weak acid mixtures requires multi-segment linear least-squares regression followed by analytical line-line intersection computation. In contrast to indicator titrations, where the endpoint is identified from a single fleeting color change, conductometry utilizes the collective information of multiple data points distributed along each linear chemical branch, greatly suppressing random measurement noise.



Let the corrected titration data points (V_i, G_corr,i) be partitioned into three distinct subsets corresponding to the three chemical regimes:

1. Branch 1 (Strong Acid Neutralization): V_min,1 ≤ V ≤ V_max,1 (e.g., 1.00 mL to 8.50 mL), where G_corr decreases linearly.

2. Branch 2 (Weak Acid Neutralization): V_min,2 ≤ V ≤ V_max,2 (e.g., 12.00 mL to 23.00 mL), where G_corr increases moderately.

3. Branch 3 (Excess Hydroxide Base): V_min,3 ≤ V ≤ V_max,3 (e.g., 27.00 mL to 35.00 mL), where G_corr increases steeply.



For each branch k ∈ {1, 2, 3}, ordinary linear least-squares regression is applied to fit the equation:



G_corr(V) = m_k · V + b_k



where slope m_k and Y-intercept b_k are calculated using the standard closed-form regression formulas:



m_k = [ N ∑(V_i G_corr,i) - (∑V_i)(∑G_corr,i) ] / [ N ∑(V_i²) - (∑V_i)² ]

b_k = [ ∑G_corr,i - m_k ∑V_i ] / N



The first equivalence point V_eq,1, corresponding to the complete stoichiometric neutralization of hydrochloric acid, is determined from the mathematical intersection of Branch 1 and Branch 2:



m_1 · V_eq,1 + b_1 = m_2 · V_eq,1 + b_2

V_eq,1 = (b_2 - b_1) / (m_1 - m_2)



Alternatively, if Branch 1 is extrapolated to the conductance minimum defined by the baseline conductance of pure sodium chloride in the acetate matrix G_min:



V_eq,1 = (b_1 - G_min) / |m_1|



The second equivalence point V_eq,2, corresponding to the complete neutralization of both hydrochloric acid and acetic acid (total acidity), is determined from the mathematical intersection of Branch 2 and Branch 3:



m_2 · V_eq,2 + b_2 = m_3 · V_eq,2 + b_3

V_eq,2 = (b_3 - b_2) / (m_2 - m_3)



The volume of standardized sodium hydroxide consumed solely by the weak acetic acid is the differential volume:



ΔV_HOAc = V_eq,2 - V_eq,1



From the fundamental stoichiometry of the 1:1 neutralization reactions (HCl + NaOH → NaCl + H₂O and CH₃COOH + NaOH → CH₃COONa + H₂O), the molar concentration of hydrochloric acid in the original analyte sample aliquot (volume V_aliquot = 25.00 mL) is:



c_HCl = (V_eq,1 · c_NaOH) / V_aliquot



and the molar concentration of acetic acid is:



c_HOAc = (ΔV_HOAc · c_NaOH) / V_aliquot = [ (V_eq,2 - V_eq,1) · c_NaOH ] / V_aliquot



Applying standard error propagation to the linear intersection coordinates, the variance in V_eq,1 is given by:



σ²(V_eq,1) = [ σ²(b_1) + σ²(b_2) + V_eq,1² (σ²(m_1) + σ²(m_2)) ] / (m_1 - m_2)²



Because |m_1 - m_2| is large (~0.44 mS·mL⁻¹), the denominator strongly attenuates experimental regression uncertainties, yielding equivalence point precisions with expanded uncertainties U(V_eq,1) < 0.04 mL and U(c_HCl) < 0.15 mM at the 95% confidence level.