# Traceability of pH to the Mole

^{*}

## Abstract

**:**

_{H+}(c/mol·dm

^{−3}or m/mol·kg

^{−1}of the free hydrogen ions in solution, H

^{+}) soon (1910) was changed to pH = pa

_{H+}= −lga

_{H+}, integrating the new concepts of activity, a

_{i}and activity coefficient γ

_{i}, for the ionic species i under concern, H

^{+}in this case; it is a

_{i}= −lg(m

_{i}γ

_{i}). Since individual ions do not exist alone in solution, primary pH values cannot be assigned solely by experimental measurements, requiring extra thermodynamic model assumptions for the activity coefficient, γ

_{H+}, which has put pH in a unique situation of not being fully traceable to the International System of Units (SI). Also the concept of activity is often not felt to be as perceptible as that of concentration which may present difficulties, namely with the interpretation of data. pH measurements on unknown samples rely on calibration of the measuring setup with adequate reference pH buffers. In this work, the assignment of pH values to buffers closely matching the samples, e.g., seawater, is revisited. An approach is presented to assess the quantity pm

_{H+}= −lgm

_{H+}profiting from the fact that, contrary to single ion activity coefficients, mean activity coefficients, ${\gamma}_{\pm}=\sqrt{\left({\gamma}_{{H}^{+}}{\gamma}_{C{l}^{-}}\right)}$ can be assessed based on experimentally assessed quantities alone, ${\text{\gamma}}_{\pm}^{Exp}$, thus ensuring traceability to the mole, the SI base unit for amount of substance. Compatibility between ${\text{\gamma}}_{\pm}^{Exp}$ and mean activity coefficient calculated by means of Pitzer model equations, ${\text{\gamma}}_{\pm}^{Ptz}$, validates the model for its intended use.

## 1. Introduction

_{2}O, undergoes self-ionization giving origin to the hydronium ion, H

_{3}O

^{+}(or H

^{+}for simplicity of writing) and hydroxide ion, OH

^{−}, in equal concentrations of the order of 10

^{−7}mol·dm

^{−3}, depending on temperature, T, pressure, p, and ionic strength, I; H

^{+}confers free acidity to the system, commonly expressed in terms of pH = −lga

_{H+}, where a

_{H+}= m

_{H+}γ

_{H+}stands for activity of the indicated chemical species, H

^{+}, m

_{H+}and γ

_{H+}being its molality and activity coefficient respectively. For simplicity of writing, when calculating logarithms, a

_{i}, and m

_{i}, are used instead of the mathematically correct forms a

_{i}/${a}_{i}^{0}$ and m

_{i}/${m}_{i}^{0}$, where the quantities with superscript “

^{0}” represent the standard state for the chemical species i under concern and equal the value 1 mol·kg

^{−1}. Since, in general terms, water is likely to come in contact with almost every other substance and materials, chemical reactions tend to take place in aqueous solutions, therefore affecting and being affected by H

^{+}concentration, e.g., m

_{H+}/mol·kg

^{−1}, hence its activity, a

_{H+}. Activity, a

_{H+}, and concentration, m

_{H+}, of chemical species are equivalent quantities, i.e., the activity coefficient, γ

_{H+}= 1 in the limiting condition of ideal dilute solutions, as it is the case of pure water [1]. Aqueous systems range from simple dilute solutions, e.g., distilled water, to complex mixtures of high ionic strength, e.g., seawater [2]. In the very dilute solutions, the concentration of chemical species, molecules or ions, is very small and the approximation γ

_{H+}= 1 may be acceptable for some intended uses. Solutions of high complexity, both in terms of species concentration and diversity, introduce interactions that reflect upon deviations from ideality, with activity coefficients becoming significantly different from 1.

_{H+}and m

_{Cl−}, are known. Nevertheless owing to the inevitable presence of both anions and cations in solution, experimental assessment of the activity coefficients of individual ions, γ

_{+}or γ

_{−}cannot be done. It can only be achieved through model approaches [3]. Experimentally assessed mean values, ${\text{\gamma}}_{\pm}^{Exp}$, or modeled mean, γ

_{±}, or single values γ

_{H+}or γ

_{Cl−}introduce uncertainty to pH values assigned to the reference buffer solutions subject to measurement. Particularly complex mixtures of high ionic strength, as is the case of seawater, I ≈ 0.7 mol·kg

^{−1}, present major deviations from ideality as well as problems of chemical interferences, e.g., due to the presence of sulphate.

^{−1}[4], which are used by the great majority of practitioners who are not fully aware of this potential source of uncertainty. This may eventually bring the result to an unacceptably low level of quality, not fit for its intended purpose. The use of other more appropriate pH buffer solutions, when available, is advisable and is becoming more common practice among experts.

^{−1}. Above this metrological level an additional source of uncertainty, 0.01, is introduced by the purely electrostatic Debye-Hückel model with the Bates-Guggenheim convention for the distance of closest approach of chloride ions, adopted for the assessment of the individual activity coefficient, γ

_{Cl−}[2].

**Figure 1.**Metrological traceability scheme for pH. U

_{c}denotes the expanded measurement uncertainty (coverage factor k = 2; the value of the measurand lies with a probability of approximately 95% within the interval of values) assigned to the pH values obtained for aqueous solutions up to an ionic strength of 0.1 mol·kg

^{−1}[7].

^{−1}, a more elaborate model, as it is the Pitzer model, accounting for electrostatic and specific interactions [8], is bound to match reality more closely.

**i**on, consistently approaching a seawater matrix, ASW, for the assessment of experimental values of the acidity function, p(a

_{H+}γ

_{Cl−}) and mean activity coefficients, ${\gamma}_{\pm}=\sqrt{\left({\gamma}_{{H}^{+}}{\gamma}_{{Cl}^{-}}\right)}$. This allows evaluation of pm

_{H+}, with the corresponding uncertainty, without extra-thermodynamic assumptions, thus establishing full traceability to the mole. Mean activity coefficients can also be calculated, ${\text{\gamma}}_{\pm}^{Calc}$, upon taking into account a value of m

_{H+}derived from the literature value for the acidity constant of the hydrogen sulphate ion, Ka (HSO

_{4}

^{−}) [9]. Compatibility with mean activity coefficients calculated from the semi empirical Pitzer model equations ${\text{\gamma}}_{\pm}^{Ptz}$, validates the use of the model [10], also available for single ion activity coefficients.

## 2. Method

_{H+}values, are conventionally assigned [2] to primary standard pH buffer solutions (S) of ionic strength, I ≤ 0.1 mol·kg

^{−1}, through a primary method, pH (S). This includes measurement of Harned cell (H+ sensitive hydrogen gas electrode and Cl

^{−}sensitive silver, silver chloride electrode, without transference) potentials, E, calculations with the Nernst equation (Equation (1)) and adoption of extra-thermodynamic assumptions concerning models of electrolyte solutions, for the assignment of mean or single activity coefficients:

^{0}is the standard cell potential, assessed from measurement in 0.01 mol·kg

^{−1}HCl, R is the gas constant, T is the absolute temperature and F is the Faraday constant.

_{H+}γ

_{Cl−}), (Equation (2)):

_{H+}𝛾

_{Cl−}) = p(𝑎

_{H+}𝛾

_{Cl−})

_{Cl−}.

_{H+}= − lg(𝑎

_{H+}𝛾

_{Cl−}) +lg𝛾

_{Cl−}

_{Cl−}, valid to ionic strengths, I, below 0.1 mol·kg

^{−1}, and assuming the value 0.01 in pH for the respective uncertainty, pH becomes traceable to the internationally accepted SI [2]. Above this value, the model is no longer valid and the use of the Pitzer model [8] has been debated.

_{Cl−}and m

_{H+}, independent of model assumptions:

_{H+}) = − lg𝑚

_{H+}= p(𝑎

_{H+}𝛾

_{Cl−}) + 2lg𝛾

_{±}= p𝑚H (S)

_{H+}γ

_{Cl−})], can be calculated [11] by Equation (10).

_{2}is the partial pressure of hydrogen in the cell and ∆E is the bias potential of the Ag/AgCl electrodes.

_{H}(S) is assessed through Equations (11)–(13).

_{Cl−}, and valid γ

_{±}values have been assessed, it is possible to obtain pm

_{H}(S). Making the reference buffers further available for the calibration of measuring devices, allows evaluation of pm

_{H+}values for unknown aqueous samples, based on experimental values, hence fully traceable to the mole. This is of particular relevance for speciation studies, in environmental sciences and toxicological studies.

_{H+}and pm

_{H+}, are related through the activity coefficient of the single hydrogen ion, γ

_{H+}.

_{H+}= p𝑚

_{H+}− lg𝛾

_{H+}

_{±}, γ

_{H+}and γ

_{Cl−}can also be calculated by means of Pitzer electrolyte solutions model equations [8,12,13,14], thus enabling assessment of the consistency between experimental and model approaches.

_{H+}values will always be affected by an uncertainty equal or bigger than the one respectively associated to the reference pH (S), or pm

_{H}(S) values assigned to the calibrating solutions, S.

_{H+}, hence on the properties of the aquatic systems under study. This is illustrated by a double traceability chain, represented schematically by Figure 2, which enables assessment of compatibility of measured results [18].

**Figure 2.**Schematic representation of a double traceability chain, showing traceability of two different types of analytical signals pertaining to hydrogen ion, H

^{+}, to common references of pH or pm

_{H+}, and higher up to the conceptually defined pH value.

## 3. Results and Discussion

^{−1}HCl + (NaCl + KCl + Na

_{2}SO

_{4}+ CaCl

_{2}+ MgCl

_{2}), at I = 0.67 mol·kg

^{−1}[19]. All measurements were performed at 25 °C [10].

^{®}30% HCl solution and were reassessed by titration against tris(hydroxymethyl) methylamine (Sigma-Aldrich ≥99.8% purity, supplied by Química, S.L., Sintra, Portugal) according to reference procedures [20]. All solutions were prepared using high purity Millipore

^{®}Milli-Q Advantage water, with resistivity higher than 18.2 MΩ·cm.

_{H+}γ

_{Cl−}), mean activity coefficients, ${\text{\gamma}}_{\pm}^{Exp}$, hence also p(m

_{H+}). Mean activity coefficients calculated from experimental data, ${\text{\gamma}}_{\pm}^{Exp}$, for sulphate-free solutions were co-opted for solutions of the same ionic strength where sodium sulphate substituted an equivalent amount of sodium chloride. Mean activity coefficients can also be calculated, ${\text{\gamma}}_{\pm}^{Calc}$, upon taking into account a value of m

_{H+}based on the literature value for the acidity constant of the hydrogen sulphate ion, Ka (HSO

_{4}

^{−}).

- Solution A—0.01 mol·kg
^{−1}HCl + NaCl - Solution B—0.01 mol·kg
^{−1}HCl + (NaCl + KCl + CaCl_{2}+ MgCl_{2}) - Solution C—0.01 mol·kg
^{−1}HCl + (NaCl + KCl + Na_{2}SO_{4}+ CaCl_{2}+ MgCl_{2})

_{H+}. In parallel, as an alternative procedure for this later, Ka = 1.2 × 10

^{−2}[8], was used to calculate m

_{H+}, which on its turn led to ${\text{\gamma}}_{\pm}^{Calc}$.

_{H}(S), (Equation (12)), for HCl in saline background, are also presented in Table 1. These values lead to estimation of the combined uncertainty u(pm

_{H+}).

**Table 1.**Acidity function p(a

_{H+}γ

_{Cl−}) and pm

_{H+}values with their respective uncertainty, u, budgets assessed in different background solutions: A, B and C.

Solution | γ_{±} | u(γ_{±}) | u(2lgγ_{±}) | p(a_{H+}γ_{Cl−}) | u(pa_{H+}γ_{Cl−}) | pm_{H+} | u(pm_{H+}) |
---|---|---|---|---|---|---|---|

A * | 0.7440 | 0.0085 | 0.010 | 2.2939 | 0.0038 | 2.037 | 0.011 |

B * | 0.7284 | 0.020 | 0.024 | 2.2919 | 0.0038 | 2.017 | 0.024 |

C * | 0.7284 | 0.020 | 0.024 | 2.2628 | 0.0038 | 1.988 | 0.024 |

C ** | 0.7249 | 0.046 | 0.056 | 2.2628 | 0.0038 | 1.983 | 0.056 |

_{±}= ${\text{\gamma}}_{\pm}^{Exp}$; ** γ

_{±}= ${\text{\gamma}}_{\pm}^{Calc}$.

_{±}) for the sulphate containing solution (C**) is due to the high impact of the acidity constant, Ka(HSO

_{4}

^{−}), on the calculation of the uncertainty of ${\text{\gamma}}_{\pm}^{Calc}$.

_{±}to pm

_{H+}has been calculated for the two values of mean activity coefficient, ${\text{\gamma}}_{\pm}^{Exp}$ and ${\text{\gamma}}_{\pm}^{Calc}$, and for the possible scenario of an order of magnitude lower uncertainty level, C′, Table 2.

**Table 2.**Uncertainty introduced by γ

_{±}to the calculation of pm

_{H+}for acidic solutions in seawater background, function of its uncertainty level.

Solution | γ_{±} | uγ_{±} | u(2lgγ_{±}) | u(pm_{H+}) | Upm_{H+} (Expanded, k = 2) |
---|---|---|---|---|---|

C^{Exp} | 0.7284 | 0.020 | 0.024 | 0.024 | 0.048 |

C′^{Exp} | 0.7284 | 0.0020 | 0.0024 | 0.0024 | 0.0048 |

C^{Calc} | 0.7249 | 0.046 | 0.056 | 0.056 | 0.12 |

C′^{Calc} | 0.7249 | 0.0046 | 0.0056 | 0.0056 | 0.012 |

C^{Ptz} | 0.76998 | 0.00518 | 0.0058 | 0.0058 | 0.012 |

_{H+}) comes from the uncertainty of the mean activity coefficient, uγ

_{±}. As shown, its effect, u(2lgγ

_{±}), can be lowered upon improvement of the quality of the raw data .

## 4. Conclusions

_{±}, for reference pH buffer standards, S, allows further calculation of their respective concentrations, m

_{H+}, hence −lgm

_{H+}= pm

_{H+}= pm

_{H}(S).

_{H+}γ

_{Cl−}) the major uncertainty component largely contributing to the combined uncertainty of pm

_{H+}, u(pm

_{H+}), comes from (γ

_{±}).

_{H}(S) values with their respective uncertainties. This enables calibration of measuring devices in terms of pm

_{H+}, thus ensuring fully traceable sample pm

_{H+}values.

## Acknowledgments

## Author Contribution

## Conflicts of Interest

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**MDPI and ACS Style**

Camões, M.F.; Anes, B.
Traceability of pH to the Mole. *Water* **2015**, *7*, 4247-4255.
https://doi.org/10.3390/w7084247

**AMA Style**

Camões MF, Anes B.
Traceability of pH to the Mole. *Water*. 2015; 7(8):4247-4255.
https://doi.org/10.3390/w7084247

**Chicago/Turabian Style**

Camões, Maria Filomena, and Bárbara Anes.
2015. "Traceability of pH to the Mole" *Water* 7, no. 8: 4247-4255.
https://doi.org/10.3390/w7084247