Metrological Assessment of pH_T in TRIS Buffers Within Artificial Seawater: Implications for High-Salinity Reference Materials

Abstract

Anthropogenic CO₂ emissions drive ocean acidification through changes in the carbonate system, lowering seawater pH. In contrast, salinity variations arise from physical processes such as freshwater fluxes and circulation. This study reports the preparation and Harned cell characterization of three equimolal TRIS buffer solutions (0.01 mol·kg⁻¹, 0.025 mol·kg⁻¹, and 0.04 mol·kg⁻¹) in artificial seawater (ASW) matrices with practical salinities of 35 and 50 and temperatures of 20 °C, 25 °C, and 30 °C. Determined pH_T values achieved expanded uncertainties ($U_{{\mathrm{p}\mathrm{H}}_{\mathrm{T}}}$ ≤ 0.006), meeting Global Ocean Acidification Observing Network (GOA-ON) “climate” quality standards. Absolute salinity (S_A) was concurrently measured via density (TEOS-10), revealing systematic deviations from practical salinity due to TRIS content. A nonlinear regression model was developed to predict pH_T as a function of salinity, temperature, and TRIS molality, with r² = 0.99998. These results provide a robust dataset for developing Certified Reference Materials (CRMs) for pH_T calibration under climate-relevant high-salinity environments at different temperature conditions, offering a practical tool for high-accuracy calibration in variable marine conditions.

1. Introduction

Since the industrial revolution, anthropogenic CO₂ emissions have significantly altered ocean carbonate chemistry, leading to increased acidification and measurable changes in seawater pH. In parallel, temperature and salinity variability are primarily governed by physical processes such as freshwater fluxes, evaporation, and ocean circulation. Monitoring these changes requires highly accurate and traceable measurements, especially for seawater pH, which is a critical indicator in global ocean acidification networks and climate models. However, accurate pH measurement in saline matrices presents substantial metrological challenges, primarily due to the influence of high ionic strength (I), liquid junction potentials, and the inadequacy of conventional buffer-based calibrations. These issues are particularly pronounced in environments subject to high evaporation rates and minimal freshwater input from fluvial or atmospheric sources [1,2,3,4,5,6,7].

pH is the key quantity to measure the acidity of any medium. According to the IUPAC 2002 definition [8], pH is linked with the molal activity of the hydrogen ion, a_H, in the following way:

$$\mathrm{p}\mathrm{H}={-\lg a}_{\mathrm{H}}=-\mathrm{l}\mathrm{g}\left(b_{\mathrm{H}}{\gamma }_{\mathrm{H}}\right)/b^0,$$

(1)

In Equation (1), b_H and γ_H are respectively the molality in mol·kg⁻¹ and the molal activity coefficient of the hydrogen ion (H⁺), and b⁰ is the standard molality equal to 1 mol·kg⁻¹.

pH is experimentally determined with the primary potentiometric method, namely the Harned cell. This pH scale is known as the IUPAC pH scale or activity pH scale, tra ceable to the International System of Units (SI) with validity until ionic strength (I) up to 0.1 mol·kg⁻¹. Seawater with a practical salinity (S) around 35 is a complex aqueous medium where the presence of salts as NaCl, Na₂SO₄, KCl, CaCl₂, MgCl₂, among others, makes its I higher than 0.7 mol·kg⁻¹ [9].

In oceanographic studies, several pH concentration scales are used in place of the activity-based pH scale, with the total pH scale (pH_T) being the most widely adopted due to its accuracy and ease of use. This scale considers the concentration of the free hydrogen ion [H⁺] and the [H⁺] linked to the sulphate ion [HSO₄²]⁻ in the following way:

$$\mathrm{p}{\mathrm{H}}_{\mathrm{T}}=-\mathrm{l}\mathrm{g}\left({\displaystyle \frac{\left[{\mathrm{H}}^{+}\right]+\left[{\mathrm{H}\mathrm{S}\mathrm{O}}_4^{-}\right]}{b^0}}\right).$$

(2)

According to Nemzer and Dickson (2005), this scale was first proposed in the works of Ramette et al. (1970) and Hansson (1973) [10,11,12]. The works of Dickson (1990) and DelValls and Dickson (1998) strongly contributed to the standardization of this scale by developing primary reference buffers (the so-called TRIS buffers) for the accurate measurement and calibration of pH equipment, ensuring that pH measurements made in different laboratories were consistent, accurate, and metrologically comparable [13,14].

Indeed, TRIS buffers are considered the highest-level standards for use in oceanic pH measurements and are based on the strong base Tris(hydroxymethyl)aminomethane and the conjugate acid (TRIS.HCl) in an artificial seawater matrix (ASW) adjusted to achieve the desired pH of seawater (between 7.9 and 8.1). The ASW medium is a solution that mimics the composition of standard seawater, containing all the major salts except bromides, which can make the response of the Ag/AgCl electrodes used in the Harned cell unstable, and fluoride, which is present in vestigial concentration and is not incorporated [13,15,16]. Other cations, such as strontium (Sr²⁺), are replaced by calcium (Ca²⁺) for convenience and simplicity of the formulation [11].

The preparation of these solutions is extensively described elsewhere [10,11,12,13,14,17,18,19,20], along with the 2008 recommendations of the International Association for the Physical Sciences of the Oceans (IAPSO).

The pH of reference buffer solutions is assigned by the Harned cell in the following configuration [13,14,18,19,20,21,22,23]:

$$\mathrm{P}\mathrm{t}\left|{\mathrm{H}}_2\left(\mathrm{g},1 \mathrm{a}\mathrm{t}\mathrm{m}\right)\right|\mathrm{A}\mathrm{S}\mathrm{W} \mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h} \mathrm{H}\mathrm{A} \mathrm{a}\mathrm{n}\mathrm{d} \mathrm{H}\mathrm{B}\left|\mathrm{A}\mathrm{g}\mathrm{C}\mathrm{l}\right|\mathrm{A}\mathrm{g},$$

where HA and HB are respectively the acidic and basic forms of the added buffer solution. In cell A, the measured potential, E, in volts (V), is related to the operational/molal pH, pH_b, through the Nernst equation according to:

$${\mathrm{p}\mathrm{H}}_b={\displaystyle \frac{\left(E-E^{0\ast }\right) F}{RT \mathrm{l}\mathrm{n}\left(10\right)}}+\mathrm{l}\mathrm{g}\left({\displaystyle \frac{b_{\mathrm{C}\mathrm{l}}}{b^0}}\right),$$

(3)

where E is the difference in the potential measured between the Pt/H₂ and Ag/AgCl electrodes immersed in the TRIS buffer solution; E⁰* (in V) is the standard potential of Ag/AgCl electrodes in the unbuffered ASW with HCl; b_Cl (in mol·kg⁻¹) is the total molality of the chloride ions present in the buffer solution; T, R, and F are respectively the absolute temperature (in K), the gas molar constant exactly equal to 8.314462618 J K⁻¹ mol⁻¹, and the Faraday constant exactly equal to 96,485.33212 C mol⁻¹.

The pH_b in Equation (3) is obtained by measuring the electromotive force (e.m.f.) of solutions prepared in the molality base, mol·kg⁻¹ of H₂O, and is converted to pH_T of seawater, as defined by DelValls and Dickson (1998), based on the amount content (mol·kg⁻¹ of solution). While Pratt (2014) [20] and Capitaine et al. (2023) [21] used the water mass fraction (ω_H2O) to convert pH_b to pH_T in their work, the present study uses the salinity dependent correction to ensure direct comparability with the historical reference data [14,17,24], all of which employed the same conversion methodology. This is performed by relating S in the following way:

$${\mathrm{p}\mathrm{H}}_{\mathrm{T}}={\mathrm{p}\mathrm{H}}_b-\mathrm{l}\mathrm{g}\left(1-0.00106 S\right).$$

(4)

The standard potential of Ag/AgCl electrodes in ASW solutions, as a function of increasing HCl molality, assuming S and I are equivalent to those in TRIS buffer solutions, is determined as follows:

$$E^{0\ast }=\underset{b_{\mathrm{HCl}}\to 0}{\lim }{E}^{\prime }.$$

(5)

In Equation (5), E⁰* (in V) is the E′ value at zero HCl molality, b_HCl, of a second-order polynomial of E′ in terms of b_HCl. This polynomial expression of E′ is determined using the statistical least squares method [20,23]:

$$E^{\prime }=E_{\mathrm{A}\mathrm{S}\mathrm{W},\mathrm{H}\mathrm{C}\mathrm{l}}+{\displaystyle \frac{RT\mathrm{l}\mathrm{n}10}{F}}\mathrm{l}\mathrm{g}\left(b_{\mathrm{H}\mathrm{C}\mathrm{l}} b_{{\mathrm{C}\mathrm{l}}^{-}}\right),$$

(6)

where E_ASW,HCl, corrected for the hydrogen partial pressure of 101.325 kPa, is the potential difference between the electrodes immersed in ASW with HCl added at increasing molalities, b_HCl is the molality of the HCl added in mol·kg⁻¹, and b_Cl− is the molality of the chloride ions in mol·kg⁻¹. $E^{\prime }$ is determined with the Harned cell in the following configuration:

$$\mathrm{P}\mathrm{t}\left|{\mathrm{H}}_2\left(\mathrm{g},1 \mathrm{a}\mathrm{t}\mathrm{m}\right)\right|\mathrm{A}\mathrm{S}\mathrm{W} \mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h} \mathrm{H}\mathrm{C}\mathrm{l} \mathrm{a}\mathrm{d}\mathrm{d}\mathrm{e}\mathrm{d}\left|\mathrm{A}\mathrm{g}\mathrm{C}\mathrm{l}\right|\mathrm{A}\mathrm{g}$$

This work presents an advancement in the metrological characterization of TRIS buffer solutions in ASW at S = 50 and temperatures extending up to 30 °C. Existing datasets [14,17,21] have predominantly focused on lower salinities (S ≤ 40) and on temperatures ranging from the freezing point of water to 25 °C [24], highlighting the need for expanded characterization under more extreme conditions. In this context, two target salinities were chosen: S = 35 and S = 50. The choice of S = 35 (the reference salinity established by IAPSO [25], representing typical open-ocean conditions, ensures direct comparability with extensive existing pH_T literature [13,14,17,20,21,24,26,27], validation of our methodology, and continuity with GOA-ON measurement protocols, which predominantly use S = 35 reference materials. The selection of S = 50 addresses a critical gap in available data. While S up to 40 has been comprehensively characterized by DelValls and Dickson (1998) [14], Khoo et al. (1977) [22], and more recently Capitaine et al. (2025) [27], we chose to extend our measurements to S = 50 rather than replicating this well-established work. This decision maximizes the scientific contribution of our study by addressing the most significant data gap in the literature. Our regression model (Equation (11)), which incorporates and validates against existing S = 40 data, enables reliable interpolation to intermediate salinity conditions. In contrast, data at higher salinities remain scarce, with the notable exception of Papadimitriou et al. (2016) [24], who focused on hypersaline conditions at sub-zero temperatures and T = 25 °C.

Our work extends measurements to S = 50 at temperatures ranging from 20 °C to 30 °C, conditions relevant for tropical, subtropical, and hypersaline marine environments such as the Red Sea, Persian Gulf, and evaporative coastal systems. The S = 35/50 choice prioritizes filling the most critical data gap (high-salinity, elevated temperature), enabling accurate interpolation through robust regression, providing reference data for emerging oceanographic priorities (climate-impacted marginal seas), and efficient use of metrolo gical resources. This strategy enables the development of oceanographically relevant pH_T reference materials supporting improved traceability in regions subject to intense evaporation and reduced freshwater input.

Additionally, the study quantifies and analyzes the deviation between S_A and S due to TRIS base additions—an aspect rarely addressed in prior literature. Our findings confirm that while S_A varies due to TRIS contributions, pH_T remains consistent due to constant ionic strength and molal substitution protocols.

The development of a nonlinear model, based on the established models of DelValls and Dickson [14] and Papadimitriou et al. [24], incorporating salinity, temperature, and the approach of Müller et al., including buffer molality [17], further contributes to a practical tool for CRM design and pH calibration in diverse marine conditions such as high-salinity and warm-water environments.

Future studies should explore TRIS buffer behavior in natural seawater matrices and expand S_A determination across broader salinity regimes.

2. Materials and Methods

2.1. Artificial Seawater Composition

In this work, the composition and preparation of ASW was performed according to the “recipe” of Pratt, 2014, with the recommendations of the IAPSO reference compositions [20,25]. The concentration of ASW solutions is usually reported in molality rather than molarity due to its invariance with respect to temperature and pressure. This choice supports the preparation of buffer solutions with stable composition and traceable concentration values under varying experimental conditions. Furthermore, all reference data used for comparison in this study are expressed in molality, including key works on TRIS buffer characterization in synthetic seawater, thereby ensuring methodological consistency and compatibility [11,13,14,17,20,21,24,26,27].

In

Table 1. Composition of reference ASW with I = 0.72250 mol·kg⁻¹ and S = 35 (adapted from [20]).

Major Salt	Molality b/mol·kg−1	Amount Content ν/mol·kg−1
NaCl—b1	0.42753	0.41254
Na2SO4	0.02926	0.02824
KCl	0.01058	0.01021
MgCl2	0.05474	0.05282
CaCl2	0.01075	0.01037
HCl	b1	0
TRIS	b2	0

, the molalities of each salt (for S = 35 and S = 50) were first calculated from Equation (7) with the coefficients derived from the IUPAC 2013 atomic weights [17], which relates S and I according to the following equation:

$$I={\displaystyle \frac{19.919 S}{\left(1000-1.00192 S\right)}}.$$

(7)

It is essential to distinguish between practical salinity S (dimensionless, PSS-78) and absolute salinity S_A (in g·kg⁻¹, TEOS-10). The S is based on electrical conductivity and defines the target ionic strength during solution preparation. In this work, solutions were formulated to achieve S = 35 and S = 50 by adjusting ionic strength according to Equation (7). The S_A represents the total mass of dissolved solids per kg of solution, determined independently via density measurements (Section 2.3,

Table 2. S_A values (in g·kg⁻¹) deduced for the equimolal TRIS buffers, b (TRIS) = b (TRIS.HCl) = 0.01 mol·kg⁻¹, 0.025 mol·kg⁻¹, and 0.04 mol·kg⁻¹ at S = 35 and S = 50 and the expanded uncertainties U (k = 2) at the investigated temperatures.

S		TRIS 0.01/ mol·kg1		TRIS 0.025/ mol·kg1		TRIS 0.04/ mol·kg1
	T/K	SA/ g·kg−1	U/ g·kg−1	SA/ g·kg−1	U/ g·kg−1	SA/ g·kg−1	U/ g·kg−1
35	293.15	36.9	0.9	37.7	0.9	38.5	1.0
	298.15	37.0	1.0	37.7	1.0	38.6	1.1
	303.15	37.3	1.0	38.1	1.0	38.9	1.1
50	293.15	53.1	0.9	53.2	0.9	53.5	0.9
	298.15	53.1	0.9	53.2	0.9	53.5	0.9
	303.15	53.4	0.9	53.5	0.9	53.8	0.9

) using TEOS-10 algorithms [28].

For the preparation of solutions with S > 35, each salt’s molality was determined according to the following equation:

$$b_S={\displaystyle \frac{b_{35}·I_S}{I_{35}}},$$

(8)

where b_S is the molality at S > 35, b₃₅ is the molality at S = 35, I_S and I₃₅ are the ionic strengths at S > 35 and S = 35, respectively.

The concentration of each salt, in mol·kg⁻¹, in both solutions of S = 35 and S = 50 was determined by the following expression [23]:

$$b_i={\displaystyle \frac{{\mathrm{\nu }}_i}{\left(1-\frac{{\mathrm{\nu }}_i·M_i}{1000}-{\sum }_l\left(\frac{{\mathrm{\nu }}_j·M_j}{1000}\right)\right)}},$$

(9)

where b_i is the molality of NaCl, in mol·kg⁻¹, ν_i is the amount content of NaCl in mol·kg⁻¹, M_i is the NaCl molar mass in g·mol⁻¹, and ν_j and M_j are respectively the amount content and the molar masses of the remaining salts in the ASW solution.

The molalities of HCl (b₁) and TRIS (b₂) were independent of the chosen salinities [14,26,29]. The concentration of the solutions of HCl in ASW matrix at increasing molalities was calculated by subtracting the intended molality of HCl (b₁) from the molality of NaCl (0.42753 − b₁), and for the TRIS buffered solutions, b₂ = 2 b₁.

2.2. Preparation of ASW/HCl and TRIS Buffer Solutions

Before conducting the measurements in a saline matrix, the standard potential, E⁰_Ag/AgCl, in V, of Ag/AgCl electrodes was determined using a 0.01 mol·kg⁻¹ HCl solution. This solution was accurately characterized by potentiometric titration against TRIS SRM 723e primary standard, from the National Institute of Standards and Technology (NIST), with an expanded uncertainty U_HCl = 1.72 × 10⁻⁵ mol·kg⁻¹ (k = 2).

E⁰_Ag/ACl was measured at three temperatures, 20 °C, 25 °C, and 30 °C in a climatic chamber. These values were systematically compared with previously established reference standards, allowing for the traceability of E⁰* values [20].

The solutions for E⁰* determination were prepared with increasing molalities of HCl ranging from 0.005 mol·kg⁻¹ to 0.05 mol·kg⁻¹ for both S = 35 and S = 50. All preparations were conducted under environmentally controlled conditions of temperature, pressure, and relative humidity to enable buoyancy correction.

In this context, four stock solutions of the major salts constant in Table 1 were gravimetrically prepared in a Sartorius CCE2004 comparator with a resolution of 0.1 mg and diluted with Ultra-pure water (18.2 MΩ cm) provided by a MilliQ system. All the weights were corrected for air buoyancy.

The salts used in the preparation of stock solutions for measurements in the Harned cell must be of high purity or very well characterized. For this purpose, NaCl was purchased from Sigma-Aldrich (purity ≥ 99.999%), Na₂SO₄ and KCl, with purities of 99.999% and 99.880%, respectively, were both sent by the Physikalisch-Technische Bundesanstalt (PTB). The mass fraction of both salts was coulometrically characterized by the Slovak Metrology Institute (SMU) within the scope of the EMPIR project SapHTies. The MgCl₂ and CaCl₂ salts were both purchased from Merck. All the salts except MgCl₂ and CaCl_2, which are highly hygroscopic, were dried for 3 h at 110 °C before weighing. The MgCl₂ and CaCl₂ stock solutions with molalities around 1.3 mol·kg⁻¹ and 0.4 mol·kg⁻¹ were assayed by argentometric titration with AgNO₃ itself standardized against KCl from NIST (SRM 999b) with an expanded uncertainty U = 0.5% (k = 2). The TRIS ultrapure base was from Panreac Applichem with a purity of 99.9%, and HCl with the amount content of (0.106000 ± 0.000038) mol·kg⁻¹ was from the Danish Institute of Metrology (DFM).

The electrodes used in the Harned cell measurements were Ag/AgCl electrodes (thermal electrolytic type) and platinum hydrogen electrodes. Both types of electrodes were prepared according to the published literature [8,13,14,20,21,24].

Prior to the measurements, the climatic chamber is set to the target temperature, and the solutions are equilibrated for 30 min. Simultaneously, the water bath is adjusted to the measurement temperature. Before measurements, the cells are filled with buffer solution and equilibrated for an additional 30 min at the intended temperature in a Lauda ther mostated water bath with the temperature controlled within ±0.005 K with a platinum resistance thermometer. This 1 h equilibration protocol ensures thermal stability and minimizes temperature-dependent drift in TRIS dissociation, which is critical given the strong temperature dependence (∂pH_T/∂T ≈ −0.03 K⁻¹). Temperature was measured using a calibrated Pt100 platinum resistance thermometer with traceability to national standards (IPQ, Portugal) and ITS-90. The thermometer was calibrated within 12 months prior to measurements. The combined standard uncertainty in temperature measurement was determined according to [30] as u(T) = 0.021 K (k = 1), incorporating the following Type A and Type B uncertainty contributions: calibration uncertainty, u_cal = 0.0045 K (stated in the calibration certificate), repeatability, u_rep = 0.00025 K (standard deviation of repeated temperature readings under stable conditions, n ≥ 150), resolution, u_res = 0.0058 K (uncertainty contribution from the digital resolution of the temperature readout, estimated as Δ/(2√3) where Δ is the least significant digit), and residual systematic effects, u_resid = 0.0195 K (uncertainty from the residuals of the calibration curve fit, reflecting unmodeled deviations between the thermometer response and the calibration function).

The potential measurements were obtained with a digital multimeter, Keithley 2700, and were corrected for the partial pressure of hydrogen, and the atmospheric pressure was measured using a digital barometer from Vaisala [17,20].

2.3. Absolute Salinity Determination S_A

The absolute salinities, S_A, in g·kg⁻¹, of all TRIS buffer solutions, with molalities b (TRIS) = b (TRIS.HCl) = 0.01 mol·kg⁻¹, 0.025 mol·kg⁻¹, and 0.04 mol·kg⁻¹, were determined from density measurements at temperatures 20 °C, 25 °C, and 30 °C using the approach described in [31,32,33].

The equipment used for the density measurements was a density meter from Anton Paar (DMA 5000) with a measuring range of 0–3000 kg·m⁻³, and an expanded uncertainty of 0.033 kg·m⁻³. S_A was calculated from the measured density ρ, in kg·m⁻³, using the relationship S_A = f(ρ, T, P) where the density–salinity conversion follows the algorithms established in the Thermodynamic Equation of Seawater 2010 (TEOS-10) [28], accounting for temperature (T) and pressure (P) effects on seawater density.

The uncertainties reported in Table 2 were obtained by standard propagation methods following [30]. The standard uncertainty in density measurements u_ρ = 0.017 kg·m⁻³ was propagated through the TEOS-10 algorithms to yield standard uncertainties u_SA ranging from 0.45 to 0.55 g·kg⁻¹, with expanded uncertainties U_SA = 0.9 to 1.1 g·kg⁻¹ (k = 2), accounting for temperature and pressure covariances as per the TEOS-10 GSW toolbox documentation [28]. The variation in the U_SA across different TRIS molalities and temperatures reflects the non-linear sensitivity of the density–salinity relationship under these conditions. The values obtained for each combination of S, S_A, b, and T are summarized in Table 2 and plotted in Figure 1.

As mentioned before, when preparing TRIS buffer solutions with increasing molalities of TRIS, HCl systematic substitutions had to be made. This was performed by replacing the amount of HCl added to the molality of NaCl, ensuring that both I and S remain constant across different TRIS molalities, allowing for direct comparison of pH measurements while maintaining identical ionic background conditions.

The results in Table 2 reveal several important trends that warrant detailed analysis: a systematic deviation from practical salinity confirmed by all measured S_A values, consistently exceeding their corresponding practical salinities (S = 35 and S = 50), with deviations ranging from 1.9 g·kg⁻¹ to 3.9 g·kg⁻¹ for S = 35 solutions and 3.1 g·kg⁻¹ to 3.8 g·kg⁻¹ for S = 50 solutions. This systematic increase reflects the fundamental difference between S (deduced from conductivity measurements) and S_A (total dissolved mass), particularly when components like TRIS are present. Another observation that can be made is the TRIS concentration dependence confirmed by S_A increasing progressively with TRIS buffer concentration, with the magnitude of increase being salinity-dependent, at S = 35, ΔS_A = 1.6 g·kg⁻¹ (from 0.01 mol·kg⁻¹ to 0.04 mol·kg⁻¹ TRIS), and S = 50, ΔS_A = 0.4 g·kg⁻¹ (from 0.01 mol·kg⁻¹ to 0.04 mol·kg⁻¹ TRIS). A trend is observed with S_A increasing with TRIS buffer concentration for both salinity levels, the effect being more pronounced at S = 35 (ΔS_A ≈ 1.6 g·kg⁻¹) compared to S = 50 (ΔS_A ≈ 0.4 g·kg⁻¹). This differential response can be attributed to the combined effects of proportional dilution, where the same absolute amount of added TRIS represents a smaller relative fraction of total dissolved solids in the higher salinity solution’s ionic activity, where at higher I (for S = 50, I ≈ 1.04 mol·kg⁻¹ and I ≈ 0.72 mol·kg⁻¹ for S = 35), the activity coefficients of all ionic species are reduced due to enhanced ion-ion interactions and competition for hydration sites. This results in a diminished effective contribution of the added TRIS components to solution density, and consequently to the calculated S_A and non-ideal solution behavior, where the more complex ionic environment at higher salinities leads to increased ion pairing and association phenomena, which partially offset the mass contribution of additional solutes to the overall solution density [23,25,34,35].

2.4. Harned Cell Measurements for the Standard Potential (E⁰*) Determination

For both salinity levels (S = 35 and S = 50), six solutions of ASW containing HCl at increasing molality were prepared following protocols outlined in Section 2.1 and Section 2.2. The HCl molalities investigated were 0.005 mol·kg⁻¹, 0.01 mol·kg⁻¹, 0.02 mol·kg⁻¹, 0.03 mol·kg⁻¹, 0.04 mol·kg⁻¹, and 0.05 mol·kg⁻¹. Electrochemical measurements were conducted using the Harned cell configuration B at three controlled temperatures: 293.15 K, 298.15 K, and 303.15 K. All potentials were corrected to the standard atmospheric pressure (101.325 kPa).

Table 3. The molalities of HCl and Cl⁻, the measured values of E (corrected to 101.325 kPa) at S = 35 for the temperatures investigated, and the corresponding standard uncertainty u_E (k = 1) in the synthetic matrices.

T/K	bHCl/mol·kg−1	bCl−/mol·kg−1	E/V	uE/V
293.15	0.005000	0.568843	0.396112	3.80 × 10−6
	0.009998	0.568798	0.378435	3.53 × 10−6
	0.020000	0.568734	0.360714	3.50 × 10−6
	0.029993	0.568668	0.349927	3.53 × 10−6
	0.039998	0.568752	0.342223	3.54 × 10−6
	0.049997	0.568824	0.336186	3.53 × 10−6
298.15	0.005000	0.568843	0.396817	4.37 × 10−6
	0.009998	0.568798	0.378687	3.53 × 10−6
	0.020000	0.568734	0.360689	4.08 × 10−6
	0.029993	0.568668	0.349560	3.51 × 10−6
	0.039998	0.568752	0.341809	3.48 × 10−6
	0.049997	0.568824	0.335578	3.47 × 10−6
303.15	0.005000	0.568843	0.397610	3.69 × 10−6
	0.009998	0.568798	0.378788	3.52 × 10−6
	0.020000	0.568734	0.360564	3.50 × 10−6
	0.029993	0.568668	0.349135	3.52 × 10−6
	0.039998	0.568752	0.341099	6.50 × 10−6
	0.049997	0.568824	0.334607	6.49 × 10−6

and

Table 4. The molalities of HCl and Cl⁻, the measured values of E (corrected to 101.325 kPa) at S = 50 for the temperatures investigated, and the corresponding standard uncertainty u_E (k = 1) in the synthetic matrices.

T/K	bHCl/mol·kg−1	bCl−/mol·kg−1	E/V	u/V
293.15	0.005000	0.825812	0.386980	3.56 × 10−6
	0.010004	0.825488	0.369537	3.53 × 10−6
	0.020004	0.825367	0.351846	3.49 × 10−6
	0.030008	0.825301	0.341251	4.14 × 10−6
	0.040005	0.825451	0.333375	3.49 × 10−6
	0.050010	0.825275	0.327106	3.48 × 10−6
298.15	0.005000	0.825812	0.387602	3.80 × 10−6
	0.010004	0.825488	0.369650	3.50 × 10−6
	0.020004	0.825367	0.351477	3.48 × 10−6
	0.030008	0.825301	0.340646	3.47 × 10−6
	0.040005	0.825451	0.332670	3.54 × 10−6
	0.050010	0.825275	0.326528	3.46 × 10−6
303.15	0.005000	0.825812	0.388663	3.54 × 10−6
	0.010004	0.825488	0.370338	3.78 × 10−6
	0.020004	0.825367	0.351794	3.50 × 10−6
	0.030008	0.825301	0.340903	3.58 × 10−6
	0.040005	0.825451	0.332255	3.51 × 10−6
	0.050010	0.825275	0.325801	3.50 × 10−6

present the comprehensive electrochemical data, including HCl molality (b_HCl), total chloride molality ${(b}_{{\mathrm{C}\mathrm{l}}^{-}})$, measured cell potentials (E), and associated standard uncertainties (u_E, k = 1) for both salinity matrices across all the experimental temperatures. The standard uncertainties u_E = 3–6 μV (k = 1) arise from three principal sources: calibration, resolution, and repeatability of the Keithley 2700 multimeter (contributing 2 μV), pressure correction uncertainty from the Vaisala barometer (contributing 2 μV), and short-term electrode stability and reproducibility within a measurement series (contributing 3 μV). These components were combined according to the law of propagation of uncertainties. The variation in u_E across measurements reflects differences in electrode conditioning and solution stability.

Table 5. The standard potential E⁰* for the ASW unbuffered solutions of S = 35 and S = 50 with the U (k = 2), the three thermodynamic T investigated, and published references for the values of E⁰* and the differences obtained (ΔE⁰* = E⁰*meas. − E⁰*Ref.) between this study and the reference values.

S	T/K	E0*/V This Study	U/V	E0*/V Papadimitriou [24]	E0*/V Dickson [13]	E0*/V Müller [17]	ΔE0*/µV Papadimitriou [24]	ΔE0*/µV Dickson [13]	ΔE0*/µV Müller [17]
35	293.15	0.24818	2.14 × 10−4	0.24796 (a)	0.24800	0.24774	220	180	440
	298.15	0.24637	2.18 × 10−4	0.24642	0.24628	0.24604	−50.0	90.0	330
	303.15	0.24459	2.20 × 10−4		0.24459	0.24437		0.168	220
50	293.15	0.24831	2.11 × 10−4	0.24811 (a)			200
	298.15	0.24674	2.14 × 10−4	0.24664	0.24641 (b)		100	330
	303.15	0.245343	2.17 × 10−4

In Table 5, the values with the superscript ^(a) are derived from the model Equation (3) of Papadimitriou et al. [24] to determine E⁰* of ASW solutions within its interval of validity S ≥ 35 and for T from the freezing point of water up to 25 °C. Superscript ^(b) is the value from [24]. Good agreement was found between the values of this study and the reference literature for S = 35 and S = 50 (deviations ≤ 330 μV). This agreement is consistent with the deviations reported in the reference literature, indicating that the observed values fall within the expected range of experimental variability.

summarizes the determined standard potentials E⁰* for each salinity and temperature investigated, calculated using Equations (6) and (7), alongside comprehensive comparisons with established literature values and associated deviations plotted in Figure 2. The expanded uncertainties $U_{E^{0*}}$ = 2.1–2.2 × 10⁻⁴ V (k = 2) were determined by propagating uncertainties from two sources: regression intercept uncertainty from the second-order polynomial fit (Equations (5) and (6)), contributing 23% to the uncertainty budget, and propagated uncertainties from E measurements, temperature, and from molalities b_HCl and b_Cl− through the logarithmic term in Equation (6). The gravimetric preparation of solutions dominates the uncertainty budget, contributing around 58% via $b_{{\mathrm{C}\mathrm{l}}^{-}}$. The slight variation in $U_{E^{0*}}$ across conditions reflects differences in regression quality and temperature-dependent sensitivities. The uncertainty propagation pathway from raw measurements (E, T, p, weighing) to E⁰* with the corresponding data is discussed in Section 3.

Beyond ‘closeness,’ our results meet the metrological criterion of compatibility: the observed deviations in E⁰* (≤330 μV, maximum 440 μV at S = 35 and 293.15 K) are covered by the combined expanded uncertainties $U_{E^{0*}}$ ≈ 2.1–2.2 × 10⁻⁴ V (k = 2). When translated to pH_T via Equation (3), the maximum deviation corresponds to ≈ 0.002 pH units for b_TRIS = 0.01 mol·kg⁻¹, and the electrode-specific offset cancels in the difference E_TRIS − E⁰*, preserving pH_b and hence pH_T [20]. This is consistent with the E_n analysis reported in Section 3, where all comparisons to the literature remain below E_n < 1.

Figure 2 graphically represents the E′ values as a function of HCl molality obtained from Equation (6).

Figure 2 compares experimental data from this study with values reported by Dickson (1990) [13], Papadimitriou et al. [24], and Müller et al. [17] across different salinities (S = 35 and S = 50) and temperatures of 20 °C, 25 °C, and 30 °C. At S = 35, E′ increases slightly with HCl molality, with higher values observed at elevated temperatures. The dashed lines represent quadratic fits based on a second-order polynomial function, highlighting the non-linear behavior. For S = 50, the corrected data (panel d) shows strong agreement with Papadimitriou et al. [24], confirming the reliability of the applied correction. Overall, these trends indicate that both temperature and salinity exert a measurable influence on E′, which is consistent with previous findings.

All experimental datasets demonstrate good adherence to second-order polynomial functions with squared correlation coefficients ranging from 0.9744 to 0.9922 for S = 35 and from 0.9843 to 0.9970 for S = 50, validating both the theoretical framework and experimental methodology.

Standard uncertainties u (k = 1) and expanded uncertainties U (k = 2) reported in the tables are derived from instrument specifications, repeatability, weighing, temperature control/measurement, pressure corrections, density to S_A conversion (TEOS-10), and regression intercepts (E⁰*). The detailed propagation pathway from these inputs to the tabulated values is documented in the uncertainty notes accompanying Table 2, Table 3 and Table 4 and in Section 3.

3. Results

3.1. Harned Cell Measurements and pH_T Determination

In

Table 6. Measured potential (E_TRIS, in V) and pH_T values determined in duplicate for TRIS in ASW-buffered solutions at S = 35 and 50, with the respective expanded uncertainties ($U_{{\mathrm{p}\mathrm{H}}_{\mathrm{T}}}$), for the three thermodynamic temperatures and the studied molalities.

S	T/K	bTRIS/mol·kg−1	ETRIS/V		pHT		${\mathit{U}}_{{\mathbf{p}\mathbf{H}}_{\mathbf{T}}}$
35	293.15	0.01	0.74119	0.74132	8.247	8.249	0.005
		0.025	0.74137	0.74150	8.251	8.253	0.005
		0.04	0.74159	0.74162	8.254	8.255	0.005
	298.15	0.01	0.73822	0.73833	8.085	8.087	0.006
		0.025	0.73827	0.73843	8.086	8.089	0.006
		0.04	0.73852	0.73857	8.090	8.091	0.006
	303.15	0.01	0.73545	0.73556	7.932	7.934	0.006
		0.025	0.73578	0.73588	7.937	7.939	0.006
		0.04	0.73612	0.73599	7.943	7.941	0.006
50	293.15	0.01	0.73331	0.73338	8.278	8.279	0.005
		0.025	0.73349	0.73350	8.281	8.282	0.005
		0.04	0.73364	0.73367	8.284	8.284	0.005
	298.15	0.01	0.73053	0.73061	8.118	8.119	0.005
		0.025	0.73069	0.73067	8.121	8.120	0.005
		0.04	0.73077	0.73089	8.122	8.124	0.005
	303.15	0.01	0.72704	0.72691	7.948	7.946	0.005
		0.025	0.72719	0.72725	7.951	7.952	0.005
		0.04	0.72770	0.72771	7.959	7.959	0.005

, the results for pH_T determination of the three equimolal TRIS buffers at S = 35 and S = 50, and for the thermodynamic temperatures 293.15 K, 298.15 K, and 303.15 K, are summarized. It also presents the E_TRIS values obtained from measurements in the Harned cell, corrected for the standard pressure of 101.325 kPa. The pH_T values are given as amount of substance content on a molality basis, and Equation (4) was used to convert pH_b to pH_T. Along with the pH_T values of this study, the values for the expanded uncertainty $U_{{\mathrm{p}\mathrm{H}}_{\mathrm{T}}}$ (k = 2), calculated according to [30], are also presented.

3.2. Metrological Compatibility Analysis with Literature Data

According to the ISO/IEC 17025 standard [36], the recommendations of the Consultative Committee for Metrology in Chemistry and Biology (CCQM) [37], and the guidance of International Organization for Standardization (2022) [38], the normalized error (E_n) provides a standardized criterion for evaluating the metrological compatibility between measurement results and reference values, and is defined as follows:

$$E_n={\displaystyle \frac{\left|x_i-x_{Ref}\right|}{\sqrt{U_{x_i}^2+U_{x_{Ref}}^2}}},$$

(10)

where x_i and U_xi² are the pH_T values of this study and the expanded uncertainty (k = 2), and x_Ref and U_Ref.² are the reference values from the relevant literature and the expanded uncertainty U (k = 2), respectively.

The results in Table 7 present a quantitative comparison between pH_T values of this work and those reported in relevant literature sources [14,17,24] for identical experimental conditions. The agreement is within U ≤ 0.006 for pH_T values across all conditions:

-

DelValls and Dickson [14]: for S = 35, b_TRIS = 0.01 mol·kg⁻¹ at 20 °C; our value (8.248) differs by 0.001 pH units from Dickson’s reference value (8.249).

-

Müller et al. [17]: for S = 35, b_TRIS = 0.04 mol·kg⁻¹ at 25 °C; our value (8.091) differs by 0.004 units from Müller’s (8.095).

-

Papadimitriou et al. [24] for S = 50, b_TRIS = 0.04 mol·kg⁻¹ at 25 °C; our value (8.123) differs by 0.002 units from Papadimitriou’s (8.125).

Table 7. Mean pH_T values of this work, together with the available pH_T values of the reference literature and respective U (k = 2) at S = 35 and S = 50, with temperature ranging from 20 °C to 30 °C.

S	T/K	bTRIS/mol·kg−1	pHT This Study	U	pHT DelValls [14]	U	pHT Müller [17]	U	pHT Papadimitriou [24]	U
35	293.15	0.01	8.248	0.005	8.249	0.001	8.246	0.002	-
		0.04	8.255	0.005	8.251	0.001	8.253	0.002
	298.15	0.01	8.086	0.006	8.091	0.001	8.089	0.002	-	-
		0.04	8.091	0.006	8.094	0.001	8.095	0.002	8.093	0.001
	303.15	0.01	7.933	0.006	7.937	0.001	7.935	0.002	-
		0.04	7.942	0.006	7.939	0.001	7.941	0.002
50	298.15	0.04	8.123	0.005					8.125	0.001

Table 7. Mean pH_T values of this work, together with the available pH_T values of the reference literature and respective U (k = 2) at S = 35 and S = 50, with temperature ranging from 20 °C to 30 °C.

S	T/K	bTRIS/mol·kg−1	pHT This Study	U	pHT DelValls [14]	U	pHT Müller [17]	U	pHT Papadimitriou [24]	U
35	293.15	0.01	8.248	0.005	8.249	0.001	8.246	0.002	-
		0.04	8.255	0.005	8.251	0.001	8.253	0.002
	298.15	0.01	8.086	0.006	8.091	0.001	8.089	0.002	-	-
		0.04	8.091	0.006	8.094	0.001	8.095	0.002	8.093	0.001
	303.15	0.01	7.933	0.006	7.937	0.001	7.935	0.002	-
		0.04	7.942	0.006	7.939	0.001	7.941	0.002
50	298.15	0.04	8.123	0.005					8.125	0.001

In summary, all differences are well within the combined measurement uncertainties, validating our experimental methodology and demonstrating excellent inter-laboratory reproducibility.

Beyond the comparisons presented in Table 7, we conducted additional validation against Pratt (2014) [20] and Capitaine et al. (2023) [21], who employed the water mass fraction (ω_H2O) to convert pH_b to pH_T. For S = 35, T = 25 °C, and b_TRIS = 0.04 mol·kg⁻¹, Pratt reported pH_T = 8.0951 (mean of duplicate measurements: 8.0948 and 8.0954). Our corresponding value of pH_T = 8.0905 (mean of duplicates: 8.090 and 8.091) obtained using the salinity-dependent correction (Equation (4)) yields ΔpH_T = −0.005 with E_n = 0.73 < 1.0, confirming metrological compatibility. Capitaine reported for S = 35, T = 25 °C, and b_TRIS = 0.04 mol·kg⁻¹, with pH_T = 8.0915; and for S = 35, T = 30 °C, and b_TRIS = 0.04 mol·kg⁻¹, with pH_T = 7.9375. This yields ΔpH_T = −0.001 with E_n = 0.13 < 1.0 and ΔpH_T = −0.005 with E_n = 0.69 < 1.0, respectively. This agreement demonstrates the agreement of our pH_T values with the relevant published literature.

The expanded uncertainties for the pH_T values reported in [14,24] are not explicitly stated, despite these values being considered the most reliable reference data currently available.

For the purposes of this study, to assess the accuracy and reliability of pH_T measurements throughout E_n deduction, these uncertainties were conservatively estimated (values in italic in Table 7). The estimation was based on the nominal resolution expected for pH_T measurements obtained using the primary potentiometric method, following standard metrological guidance [30,36]. Typically, pH_T values determined via the Harned cell are reported with a resolution of 0.001, corresponding to a high level of accuracy [30,36,37,38,39]. Although estimating uncertainty from the corresponding instrumental resolution may appear approximate, it remains a justifiable and traceable approach when no explicit uncertainty is provided. This approach provides a realistic assessment of measurement compatibility while acknowledging inherent methodological limitations.

Given the use of Harned cells with comparable gravimetric procedures, our analysis highlights weighing steps as the primary contributor to uncertainty, accounting for approximately 50% of the total. We estimated uncertainties in the literature data to fall within 0.001, although procedural differences may account for some variability.

Indeed, the major source of uncertainty in pH_T determination is the calculation of E⁰*, the standard potential of the Ag/AgCl electrode in ASW medium, for different salinities and temperatures as the intercept of a second-order polynomial, in terms of HCl molalities, b_HCl, derived from Equations (6) and (7) [17]. This calculation relies on E measurements in a Harned cell, of solutions of unbuffered ASW with increasing molalities of HCl. These solutions are gravimetrically prepared by adding ultrapure water to a mixture of major seawater salts (NaCl, KCl, Na₂SO₄, CaCl₂, MgCl₂) and HCl in concentrations ran ging from 0.005 mol·kg⁻¹ to 0.05 mol·kg⁻¹ for the studied salinities.

The major salts are typically prepared from stock solutions, which are gravimetrically prepared from high-purity salts and ultrapure water, and characterized by coulometric or argentometric titration, which are primary methods to assess their content. In summary, the primary source of uncertainty arises from successive weighing during solution pre paration, making this process the dominant contributor to the total uncertainty associated with E⁰* estimation, accounting for approximately 50% of the total uncertainty budget and around 63% in pH_T determination [17].

Table 8. Most relevant parameters for the determination of E⁰* u (k = 1) of a solution of unbuffered ASW in HCl (b = 0.005 mol·kg⁻¹), S = 35, and T = 25 °C.

Parameter/qi	Units	xi	ui	ci	(ui × ci)2	U/%
E	V	0.396817	4.37 × 10−6	1	1.91 × 10−11	0.086
T	K	298.15	2.08 × 10−2	−5.05 × 10−4	1.10 × 10−10	0.499
bH+	mol·kg−1	0.004999	1.90 × 10−5	5.14	9.53 × 10−9	43.1
bCl−	mol·kg−1	0.568843	5.61 × 10−4	4.52 × 10−2	6.41 × 10−10	2.90
pH2	Pa	97,383	1.01 × 101	−1.32 × 10−7	1.77 × 10−12	0.008
E′	V	0.246193	1.02 × 10−4	1	1.03 × 10−8	46.6
ΔEAg/AgCl	V	0	3.39 × 10−5	1	1.15 × 10−9	5.20
I	V	0.24637	1.90 × 10−5	1	3.61 × 10−10	1.63

u(E⁰*) = 1.09 × 10⁻⁴ V. U = [(u_i × c_i)²/u²(E⁰)]/%.

and

Table 9. Most relevant parameters for the determination of $U_{{\mathrm{p}\mathrm{H}}_{\mathrm{T}}}$ (k = 2) of a solution of TRIS buffer (b = 0.04 mol·kg−1), S = 35, and T = 25 °C.

Parameter/qi	Units	xi	ui	ci	(ui × ci)2	U/%
E0*	V	0.24637	1.09 × 10−4	−16.9	3.80 × 10−6	33.8
E	V	0.738518	4.71 × 10−6	16.90	6.34 × 10−9	0.06
SDAg/AgCl	V	0	6.40 × 10−5	1	4.10 × 10−9	0.04
T	K	298.15	2.08 × 10−2	−2.79 × 10−2	3.37 × 10−7	3.38
R	J·mol−1·K−1	8.314462618	0	−1.00	0	0
F	C·mol−1	96,485.33212	0	8.62 × 10−5	0	0
pH2	Pa	98,161	0.814 × 101	−1.31 × 10−7	1.13 × 10−12	1.14 × 10−5
bCl−	mol·kg−1	0.568725	3.48 × 10−4	−7.18	6.26 × 10−6	62.7

U (pH_T) = 0.006. U = [(u_i × c_i)²/u²(E⁰)]/%.

summarize the influencing quantities for the deduction of E⁰* and pH_T.

As summarized in Table 9, the $u_{E^{0*}}$ propagates into pH_T through Equation (3) with the sensitivity coefficient −F/(RT ln 10), contributing around 34% of $U_{{\mathrm{p}\mathrm{H}}_{\mathrm{T}}}$ at S = 35 and 298.15 K, while $b_{{\mathrm{C}\mathrm{l}}^{-}}$ contributes around 63%, and T contributes 3–4%.

The E_n values calculated for the pH_T results of this work using Equation (10) and the assumptions outlined above were all below 1, indicating acceptable agreement.

The highest values, E_n = 0.82 and E_n = 0.79 (but still within E_n < 1), were observed for the TRIS buffer with b = 0.01 mol·kg⁻¹, S = 35, and T = 25 °C (compared with DelValls and Dickson values) and for TRIS buffer with b = 0.04 mol·kg⁻¹, S = 35, and T = 25 °C (compared with Müller values), respectively.

3.3. Fitting Function for pH_T Results (20 ≤ S ≤ 50) from Combined Results

To extend the predictive capability and thermodynamic representativeness of the model, the pH_T data obtained in this work were combined with experimental results from DelValls and Dickson [14], Müller et al. [17], and Papadimitriou et al. [24]. This composite dataset spans a range of salinities S = 20 to S = 50 and temperatures from –2.795 °C to 45 °C, enabling the construction of a generalized nonlinear regression model as a function of S, T (in K), and b_TRIS (in mol·kg⁻¹). The model was fitted using multivariate regression tools in Excel, yielding an excellent fit (r² = 0.99998) with residuals within the uncertainty bounds of ±0.006 pH units, even under extreme conditions. The fit equation is based on Equation (18) of Del Valls and Dickson, derived from it by including b_TRIS, as in [17].

The output coefficients for Equation (11) are summarized in

Table 10. Coefficients and respective values for the calculation of the pH_T model Equation (11).

Coefficient	Value
a0	1841.815785
a1	−158.7621782
a2	2.781023769
a3	0.551733008
a4	−0.047854353
a5	−44929.99834
a6	−324.2751098
a7	−71.62354127
a8	0.000835236
a9	4082.029463
a10	27.96623154
a11	−0.489648562
a12	0.156471814

:

$$\begin{array}{ll}{\mathrm{p}\mathrm{H}}_{\mathrm{T}}=a_0+a_1& · S+a_2·S^2+a_3·T+a_4·\left(S·T\right)+a_5·T^{-1}+a_6·\mathit{ln}T\\ & + a_7·\left({\displaystyle \frac{S^2}{T}}\right)+a_8·T·S^2+a_9·\left({\displaystyle \frac{S}{T}}\right)+a_{10}·S·\mathit{ln}T+a_{11}\\ & · S^2·\mathit{ln}T+a_{12}·b_{\mathrm{T}\mathrm{R}\mathrm{I}\mathrm{S}}\end{array}$$

(11)

To assess the statistical significance of the regression model, an Analysis of Variance (ANOVA) was performed.

Table 11. Analysis of Variance (ANOVA) for the pH_T fitting equation (Equation (11)).

Source of Variation	Sum of Squares	Degrees of Freedom	Mean Squares	F-Statistic	p-Value
Regression	21.714	12	1.809	441,123	<2.259 × 10−230
Residual	4.10 × 10−4	100	4.10 × 10−6	—	—
Total	21.714	112	—	—	—

F-critical (α = 0.05, df₁ = 12, df₂ = 100) = 1.91.

presents the ANOVA analysis parameters for the multivariate regression model described in Equation (11). The F-test assesses whether the explained variance (due to regression) is significantly greater than the unexplained variance (residual), thereby validating the inclusion of the predictor variables in the model [38,40].

The high F-statistic (441,123) with 12 and 100 degrees of freedom, far exceeding the critical F-value of 1.91 at α = 0.05, confirms the regression model’s statistical significance (p < 2.26 × 10⁻²³⁰). This indicates that the predictor variables (S, T, and b_TRIS) collectively explain 99.998% of the total variance in pH_T values, thereby validating the functional form of all 12 terms in Equation (11).

The fit equation has validity for temperatures from 270.36 K to 303.15 K and 20 ≤ S ≤ 50 with the quality of the fit given by r² = 0.99998, p ≤ 1.1 × 10⁻¹⁴³ (very close to zero), and s_Fit = 0.001 (standard error). The fit equation, tested for the b_TRIS = 0.04 mol·kg⁻¹, at S = 50, T = 270.36 K, yields a pH_T = 9.082; at S = 40, T = 318.15 K, yields a pH_T = 7.500; at S = 35, T = 298.15 K, the pH_T = 8.094; and at S = 20, T = 273.15 K, the pH_T = 8.893. The fitted residuals are represented in Figure 3.

All the ΔpH_T values are below 0.006 pH units from the pH measured values in Table 6, which is the expanded uncertainty for pH_T values [17,21], except for TRIS buffer with b = 0.01 mol·kg⁻¹), S = 50, and T = 303.15 K. A possible explanation would be that at higher salinities, the buffering capacity of TRIS is less pronounced for lower molalities of the base (0.02 mol·kg⁻¹) because it depends on the protonation equilibrium of TRIS, which is influenced by ionic strength and the presence of salts in the solution [10]. This model stands out in the current literature by combining moderate complexity, wide applicability, and the ability to resolve TRIS buffer molality effects. Unlike Papadimitriou et al. [24], whose model for equimolal TRIS assumes a fixed concentration (b = 0.04 mol·kg⁻¹) and does not distinguish pH_T variations across equimolal formulations, our equation incorporates molality as an explicit variable, allowing it to predict pH_T differences between 0.01 mol·kg⁻¹, 0.025 mol·kg⁻¹, and 0.04 mol·kg⁻¹ buffers. Müller et al. [17] include a molality term, and their model can capture molality effects; however, it is more complex—comprising numerous interaction terms—and its validation range is limited to S ≤ 40. The present model, therefore, offers a more practical and extensible solution for marine chemistry applications, particularly in high-salinity environments. This fitting equation prioritizes the primary thermodynamic and empirical effects relevant to the dataset used, capturing the major influences of salinity, temperature, and buffer concentration on pH_T without losing accuracy. This indicates that the model captures the aggregate effects of complex interactions effectively within the range of the experimental data provided, leading to a robust, accurate, and interpretable model without overfitting.

4. Discussion

This study aimed to characterize TRIS buffers in artificial seawater (ASW) across salinities (S = 35 and 50), temperatures (20 °C, 25 °C, and 30 °C), and molalities (0.01 mol·kg⁻¹ to 0.04 mol·kg⁻¹) by focusing on the dependence of these variables and their effects on pH_T. By analyzing these relationships, the research provides valuable insights into the thermodynamic behavior of TRIS buffers in marine environments.

The characterization was performed using the primary potentiometric method (Harned cell), the only method that allows accurate pH_T measurements. These findings are critical for understanding seawater’s acid-base chemistry and buffering capacity.

A critical evaluation against the complete published literature reveals excellent agreement across multiple independent high-quality studies spanning 27 years [14,17,20,21,24,27] and this work. All normalized errors E_n < 1.0 confirm metrological compatibility according to international standards. Importantly, our results agree with studies employing two different pH_b to pH_T conversion methods: (a) the salinity-dependent correction (Equation (4)) used by DelValls & Dickson, Papadimitriou, Müller, and this work; and (b) the water mass fraction (ω_H2O) method employed by Pratt and Capitaine. The negligible differences between results obtained with these approaches (ΔpH_T ≤ 0.005) confirm that both are metrologically sound when properly applied. We chose the salinity-based method specifically to maintain direct comparability with the historical reference database [14,17,24] that forms the foundation of oceanographic pH measurements. The systematic multi-laboratory agreement at S = 35, where validation data from five independent sources now exist, provides strong confidence in our novel measurements at S = 50 and T = 30 °C, where prior high-quality data remain scarce.

The results align closely with existing literature, with an experimental expanded uncertainty up to 0.006 corresponding to a maximum standard uncertainty of 0.003 and consequently meeting the GOA-ON “climate” quality objective [27].

One of the key contributions of this work is a detailed examination of the relationship between pH_T, salinity, temperature, and molality. The findings reveal complex, nonlinear interactions. For instance, as salinity increases, pH_T deviates from linear behavior, an effect magnified by temperature changes. Higher temperatures generally shift pH_T values more significantly, particularly at higher buffer molalities [17,24]. Buffer molality itself plays a significant role in stabilizing pH_T. While higher molalities enhance buffering capacity, the relationship is not strictly linear. At elevated concentrations, the ionic strength of the solution may influence TRIS dissociation, introducing further complexity.

By addressing these variables together, the study offers a comprehensive framework for understanding TRIS buffer behavior in ASW. The fitting function, Equation (11), with a correlation coefficient (r²) of 0.99998, demonstrates the strong dependency of salinity, temperature, and molality in determining pH_T. This model has important implications for accurate pH measurements in marine systems, particularly in a scenario of climate change and ocean acidification.

Despite the robust metrological characterization presented here, the broader implications of the findings—particularly the deviations observed between absolute salinity (S_A) and practical salinity (S) in TRIS-buffered solutions—warrant further research. While these differences are systematically quantified, their practical significance for Certified Reference Material (CRM) production or for the interpretation of pH_T in real seawater samples remains to be fully established. Specifically, it is not yet clear whether the S_A deviations introduced by TRIS components meaningfully affect the traceability or comparability of pH_T measurements between laboratories or under field conditions.

The insights gained from this research pave the way for further studies in TRIS buffer behavior under more extreme salinity conditions, such as those found in equatorial waters and closed seas. Investigating the impact of natural seawater constituents, such as organic matter and trace metals, could provide a clearer picture of TRIS buffers’ performance in complex marine settings. Testing in real-world samples would further validate the buffer’s reliability.

Additionally, the empirical regression model developed in this study based on the functional form originally proposed by DelValls and Dickson [14] and incorporating the molality term as proposed by Müller et al. [17] demonstrated high quality of fit (r² = 0.99998) within the studied matrix (20 ≤ S ≤ 50 and 270.36 K ≤ T ≤ 303.15 K), suggesting that the main thermodynamic effects could be captured.

This assumption may not hold outside the calibration range, particularly at salinity or temperature extremes where ionic activity coefficients, speciation equilibria, and buffer dissociation constants can exhibit non-linear behaviors. For example, at S > 50, the thermodynamic interactions governing TRIS protonation may no longer be well-approximated by this model [20,24]. Consequently, while the model performs well within the experimental conditions, its predictive reliability under extrapolated conditions—such as sea ice brines or hypersaline lagoons—has not been independently validated. In this context, it is important to recognize that the practical application of TRIS buffers, particularly in the form of CRMs, depends on their physical stability, temperature dependence, and compatibility with in situ sensors. These factors critically influence the traceability and comparability of pH_T measurements in field work and interlaboratory comparability [9,14,24].

Future work should assess the robustness of the current regression model by systematically evaluating residuals across extended S and T ranges. Validation against independent experimental datasets, particularly those employing natural seawater matrices or extreme boundary conditions, will be essential for confirming the model’s generalizability.

Finally, integrating machine learning into pH_T and E⁰* predictions could improve accuracy and applicability by using larger datasets to train algorithms that would allow for reliable forecasts under conditions that have not yet been tested.

5. Conclusions

This study provides a comprehensive metrological characterization of TRIS buffers in artificial seawater (ASW), emphasizing the combined effects of salinity, temperature, and buffer molality on accurate pH_T determination by Harned cell. The results demonstrate high accuracy and metrological traceability to the primary potentiometric method, which is aligned with the GOA-ON climate quality standards.

The incorporation of molality as a variable in the model equation provides a good fit (r² = 0.99998) for pH_T calculations, extending reliable predictions to more extreme conditions (T ≤ 30 °C, S ≤ 50) previously outside the validated range of existing pH_T models. However, extending this model to more extreme conditions requires additional experimental data at higher salinities (S > 50) and temperatures (T > 30 °C), particularly considering discrepancies between practical salinity (S) and absolute salinity (S_A), which could affect interlaboratory comparability and traceability of pH_T measurements. In this context, the practical application of TRIS buffers as Certified Reference Materials relies on their chemical stability and on compatibility with field sensors, thereby becoming indispensable to extend testing in marine environments and under diverse ecological conditions.

For future work, the adoption of machine learning approaches is proposed to enhance the predictive power of pH_T and E⁰* models, expanding their usability under conditions that extend beyond current experimental boundaries. In addition, extending the dataset towards higher temperatures and performing source-robustness checks will be pursued to reinforce the validity domain and robustness of the fitted function.

Finally, this work presents a metrological characterization for pH_T measurements in chemical oceanography, with direct implications for monitoring ocean acidification and ensuring global comparability of environmental data.