J Solution Chem (2011) 40:1819–1834DOI 10.1007/s10953-011-9758-3

Evaluation of the Standard Potential of the Ag/AgClElectrode in a 50 wt-% Water–Ethanol Mixture

Daniela Stoica · Catherine Yardin · Géraldine Ebrard ·Sophie Vaslin-Reimann · Paola Fisicaro

Received: 7 December 2010 / Accepted: 2 March 2011 / Published online: 10 November 2011© Springer Science+Business Media, LLC 2011

Abstract This paper deals with the evaluation of the standard potential of the Ag/AgClelectrode in a water–ethanol mixture (50 wt-%). A potentiometric method was applied us-ing a cell without liquid junction. Mean activity coefficients of HCl in the same mixturehave been also determined. The measurements were performed in the HCl molality rangefrom 0.005 to 0.1 mol·kg−1. The Debye–Hückel theory and Pitzer’s model, based on theinteractions present in the solution, have been applied. Good agreement was found betweenthe results obtained with the two approaches. Uncertainties of the Pitzer parameters andinterionic forces are discussed based on the values found. The variation of the standard po-tential as a function of the temperature was used to calculate the transfer thermodynamicfunctions. The effects of the solvent composition on the thermodynamic properties of HClallow to highlight structural changes in water–ethanol mixtures.

Keywords Standard potential · Activity coefficient · Water–ethanol mixture · Pitzermodel · Uncertainty evaluation · pH measurements

1 Introduction

pH is one of the most often measured parameters for the physico-chemical characterizationof solutions. Although the determination of pH is often regarded as a trivial measurement,in mixed solvent systems the chemical processes are more complicated compared to thoseoccurring in single solvents, since the solvent properties are different. Moreover, the multi-tude of available solvents and resulting large number of mixed solvents adds an additionaldifficulty to this task.

This study can be seen as a contribution to IUPAC (International Union of Pure andApplied Chemistry) efforts in the standardization of pH measurements in aqueous–organicsolvent mixtures [1–4]. Our work is focused on water–ethanol mixtures, within the frame-work of a project aiming to respond to some requirements for quality specifications for thecharacterization of bioethanol fuel [5].

D. Stoica (�) · C. Yardin · G. Ebrard · S. Vaslin-Reimann · P. FisicaroLaboratoire national de métrologie et d’essais, 1 Rue Gaston Boissier, 75015 Paris, Francee-mail: daniela.stoica@lne.fr

1820 J Solution Chem (2011) 40:1819–1834

The lack of standard calibration solutions containing ethanol leads users to calibrate theirpH-electrode by means of aqueous buffer solutions, creating serious problems for the relia-bility of the measured results. Moreover, in order to ensure a global comparability of data,an uncertainty that takes into account all of the measurement steps has to be associated withthe results.

No primary pH methods, able to provide data traceable to SI units, have so far beenidentified for aqueous–alcoholic solvent mixtures. One possible way to develop primary pHmethods for aqueous–alcoholic media is start by application of the existing protocol forprimary pH measurements in aqueous buffers.

An overview of the uncertainties associated with aqueous pH primary measurementsshows that the determination of the standard potential of a reference electrode is the stepmaking the largest contribution to the overall uncertainty of pH [6]. Therefore, the mainobjective of this study is to determine the standard potential (E0) for the Ag/AgCl referenceelectrode in the water–ethanol mixture (50 wt-%), by (i) adopting the same experimentalprotocol normally used for aqueous solutions, and (ii) by applying the existing theories usedfor aqueous media, considering the mixture to be a continuous solvent environment.

The measurements have been carried out by a potentiometric method using HCl as theelectrolyte. The standard potential was determined by measuring the electromotive force(emf) of the following cell without transference (Harned cell): H2(g), Pt | HCl + solvent| AgCl | Ag.

As for aqueous solutions, the value of E0 depends on the calculation of the mean activitycoefficient. Several semi-empirical relations allow the determination of mean activity coef-ficients. These methods are based on estimation models that take into account the solutionstructure as well as interactions occurring in the mixtures. The Debye–Hückel theory andPitzer’s model are the most commonly applied. Thus, for this study, the two theories wereapplied and the results obtained for water–ethanol mixture (50 wt-%) are compared.

2 Theoretical Background

For potentiometric pH measurements, the E0 value directly affects the final pH value, as thefollowing relation shows:

pH = − log10

(as

H

) = limmCl→0

{(Emeasured − E0

RTF

ln(10)

)+ log10(mCl)

}+ log10

(γ 0

Cl

)(1)

where asH is the activity of hydrogen ion in the solvent; R, T , F have their usual meanings,

γ 0Cl is the activity coefficient of the chloride ion, and mCl is the total molality of chloride.

The determination of the standard potential is based on the Nernst equation, Eq. 2, whichinvolves the mean ionic activity coefficient (γ±):

E = E0 − 2RT

Fln(mγ±) (2)

Estimating the activity coefficient of dissolved species is based on thermodynamic theoriesand can help to understand the particular nature of the aqueous–organic solvent systems;therefore it is discussed here.

The Debye–Hückel theory is based on long-range interactions, i.e., electrostatic interac-tions, and predicts rather well the behavior of dilute electrolyte solutions. There are a fewhypothesis associated to this theory, such as [7, 8]: (i) the relative permittivity of the elec-trolyte solution is the same as that of the pure solvents and its variation with electrolyte

J Solution Chem (2011) 40:1819–1834 1821

Table 1 Properties of the water–ethanol mixture (50 wt-%) and of the corresponding two pure solvents from15 to 35 °C

t

(°C)Component Density

(g·mL−1)

[24]

Saturatedvaporpressure(Pa)[15, 25–28]

Dielectricconstant[29, 30]

A (kg1/2

·mol−1/2)

a0B Aφ (kg1/2

·mol−1/2)

15 Water 0.99910 1703.75 81.95 – – –

Ethanol 0.79351 4258.85 26.08 – – –

Water–ethanol(50 wt-%)

0.91990 3559.48 51.80 0.9602 1.8104 0.7374

25 Water 0.99704 3167.24 78.30 – – –

Ethanol 0.78495 7827.10 24.85 – – –

Water–ethanol(50 wt-%)

0.91200 6579.82 49.02 0.9868 1.8131 0.7577

30 Water 0.99565 4243.24 76.55 – – –

Ethanol 0.78065 10420.82 24.28 – – –

Water–ethanol(50 wt-%)

0.90796 8793.03 47.69 1.0079 1.8148 0.7685

35 Water 0.99403 5623.53 74.83 – – –

Ethanol 0.77631 13720.64 23.74 – – –

Water–ethanol(50 wt-%)

0.90388 11629.50 46.39 1.0156 1.8167 0.7798

concentration is neglected, and (ii) the electrolyte is assumed to be completely dissociated,which requires an electrostatic interaction energy level much lower than the kinetic energyof thermal agitation. These conditions restrict its use to dilute solutions, typically below0.1 mol·kg−1.

Electrolyte dissociation depends on the solvent’s dielectric constant, which depends inturn on the solvent properties. This dissociation affects the relative freedom of ions found inthe solution. When the electrolyte is dissociated, ions are surrounded by an ionic atmospherehaving an equal but opposite charge. The Coulombic attraction among ions is inverselyproportional to the dielectric constant, as shown in the equation below:

�i = zie

4πε

(1

r

)(3)

where zie is the charge of ion i, ε is the solution’s dielectric constant, and r represents thedistance from ion i.

Below a certain value of the dielectric constant, the Coulombic attraction becomes pre-ponderant with respect to thermal agitation. Charged species then are able to approach oneanother more closely. Therefore, ion association plays a major role in solvents with lowdielectric constants. In order to avoid these phenomena and to apply the Debye–Hückel the-ory, the use of solvents with a dielectric constant above 35 is needed [5]. Pure ethanol has alow dielectric constant (see Table 1), but mixing it with water in different proportions leadsto modifications of the physical properties of the mixed solvent system, such as solvationqualities. Thus, the solvent mixture with 50 wt-% of ethanol presents a suitable dielectricconstant for all the temperatures studied in this work, as can be seen in Table 1.

1822 J Solution Chem (2011) 40:1819–1834

IUPAC recommends that the practical determination of the standard potential of Ag/AgClelectrodes in water should be done through measurements of the emf of the cell with HCl at0.01 mol·kg−1 [1, 9]. This approach is based on the Debye–Hückel equation [1, 10]:

log10 γi = −Az2i

( √I

1 + a0B√

I

)(4)

where A and B are the temperature-dependent Debye–Hückel constants, a0 is the meandistance of closest approach of the ions, and I the ionic strength of the solution. I is definedas I = 1

2

∑N

i=1 miz2i , where N is the number of ions, and m and z are the molal concentration

and the charge of ion i, respectively.

A = F 3

4πNA ln(10)

√ρmixture

2(ε0εrRT )3(5)

where F , NA, ρmixture, ε0, εr, R, T are the Faraday constant, the Avogadro constant, thedensity of the mixture, the vacuum permittivity, the dielectric constant of the mixture, thegas constant and the temperature, respectively.

For all aqueous–organic solvent media, the product a0B in Eq. 4 is “normalized” bytaking into account the properties of the mixtures [11]. This implies that

a0B = 1.5

(εwaterρmixture

εmixtureρwater

)1/2

T

(6)

which is evaluated at each measurement temperature T . ε and ρ are the dielectric constantand the density, respectively. The superscripts represent the solvent to which the quantitypertains. This approach is known as the modified Bates–Guggenheim convention. In thisway, continuity of properties from pure water to the desired solvent mixture is ensured.

For aqueous solutions, Fisicaro et al. [12, 13] proposed the use of the Davies equation todefine the activity coefficient, which yields more reliable and accurate results.

Davies’ equation is derived from the previous approach and takes the form expressed bythe Eq. 7:

log10 γi = −Az2i

( √I

1 + √I

− CI

). (7)

This equation better characterizes the chemical conditions used for primary buffers that havean ionic strength close to 0.1 mol·kg−1. However, for water–alcohol mixtures the correctiveterm C, associated with ionic entropies [14], is unknown. For solutions of a uni-univalentelectrolyte, like HCl, the ionic strength I is equal to the molality m.

Taking into account Eq. 7, the Nernst equation can be rearranged to put all experimentallyaccessible quantities on the left-hand-side (E′) and the unknown parameters on the right-hand-side:

E′ = E0 − 2RT ln(10)

FACm (8)

where

E′ = Ecorrected + 2RT ln 10

Flog10 m − 2RT ln 10

F

(A

√m

1 + √m

)(9)

Ecorrected denotes the experimental emf value corrected to 1 atm partial pressure of hydrogengas (see the experimental section). For each molality m, a corresponding E′ value is obtained

J Solution Chem (2011) 40:1819–1834 1823

and E0 can be determined by using a statistical least-squares method that allows a linearextrapolation to zero concentration [15].

Studies performed in different aqueous–organic media have shown that the presence ofthe organic solvent influence the nature of the interactions occurring in the solution, andcause an increase of the uncertainty associated to the mean activity coefficient [16]. Withthe aim of overcoming limitations due to the use of the Bates–Guggenheim convention,the 2002 IUPAC recommendation [1] recommends the use of Pitzer’s model. This modelhas been successfully applied to concentrated aqueous solutions up to ionic strengths of6 mol·kg−1 [17], and indicates that the properties of electrolyte solutions can be expressedby an electrostatic term plus a virial expansion that accounts for the influence of short-rangeforces [18, 19].

According to the Pitzer formalism, the mean activity coefficient of a strong electrolyte ofthe 1:1 charge type, such as HCl, is given by the above equation:

lnγ± = f γ + mBγ + m2Cγ (10)

where

f γ = −Aφ

[ √I

1 + b√

I+ 2

bln(1 + b

√I )

](11)

The term f γ accounts for the long-range forces (electrostatic term). In Eq. 11, Aφ is theDebye–Hückel slope for the osmotic coefficient function and is defined as:

Aφ = 1

3

(2πNAρmixture

)1/2(

e2

4πε0εrkBT

)3/2

(12)

where NA, ρmixture, e, ε0, εr, and kB are Avogadro’s constant, the density of mixture, theelectronic charge, the vacuum permittivity, the dielectric constant of the mixture, and Boltz-mann’s constant, respectively. The value of Aφ depends on the temperature and solventproperties.

Pitzer’s virial equation, Eq. 13, is an empirical equation based on statistical thermody-namics [20]:

Bγ = 2β(0) + 2β(1)

α2I

[1 − e−α

√I

(1 + α

√I − 1

2α2I

)](13)

The function Bγ , which describes the interactions of pairs of oppositely charged ions,represents a measurable combination of the second virial coefficients, β(0) and β(1) [21].

The values b = 1.2 kg1/2·mol−1/2 and α = 2 kg1/2·mol−1/2 were used for this study,according to Koh et al. [22] who showed that these values, originally defined for aqueoussystems, are also appropriate water–alcohol solvent mixtures.

For dilute solutions the third virial coefficient, Cγ , corresponding to the short-range in-teractions of ion triplets, can be neglected. Consequently, the Pitzer model then becomeslinear with respect to the binary interaction parameters [23].

As with the Davies approach, we can rewrite the Nernst equation, Eq. 2, taking intoaccount the definition of activity coefficient according to the Pitzer model:

E′′ = E0 − 4RT

Fβ(0)m (14)

where

E′′ = Ecorrected + 2RT

Flnm − 2RT

FAφ

{[ √m

1 + 1.2√

m

]+

[2

1.2ln(1 + 1.2

√m)

]}

+ 2RTβ(1)

F

{1 − exp

[−2√

m][

1 + 2√

m − 2m]}

(15)

1824 J Solution Chem (2011) 40:1819–1834

The value of E0 and the Pitzer coefficients β(0) and β(1) may be obtained by applying themethod described by Marshal et al. [23], which involves the equation:

α0 = E0 − α1β(1) − 4RT

Fmβ(0) + ε̃ (16)

α0 = Ecorrected + 2RT

Flnm − 2RT

FAφ

{[ √m

1 + 1.2√

m

]

+[

2

1.2ln(1 + 1.2

√m)

]}(17)

α1 = 2RT

F

{1 − exp

[−2√

m][

1 + 2√

m − 2m]}

(18)

where ε̃ denotes the error of the model.Since the measurements have been done at several molalities, Eq. 16 can be written in

the following matrix form:⎛

⎜⎝

α10...

αN0

⎞

⎟⎠ =

⎛

⎜⎝

1 −α11 − 4RT

Fm1

......

...

1 −αN1 − 4RT

FmN

⎞

⎟⎠

(E0

β(1)

β(0)

)

+⎛

⎝ε1...

εN

⎞

⎠ (19)

where N is the number of measurements. If (MT M) is invertible, then the solution of Eq. 19,using the least-squares method, is given by X = (MT M)−1MT Y with MT being the trans-pose of the matrix M .

The properties of the water–ethanol mixture (50 wt-%) and those of the two pure solventsare reported in Table 1.

3 Experimental

A 0.1 mol·kg−1 hydrochloric acid stock solution in the water–ethanol mixture (50 wt-%) wasprepared from HCl suprapur (30%). Its chloride content was determined as silver chlorideby a titration method with a maximum accuracy of ±0.2%. Anhydrous absolute ethanol(purity >99.5%) and HCl (suprapur) were purchased from Merck and were used withoutfurther purification. For the preparation of solutions, the quantity of water initially presentin the HCl solution was taken into account.

All solutions were freshly prepared before measurements. The emf measurements werecarried out over the molality range from 0.005 to 0.1 mol·kg−1. Eight HCl concentrationswere tested within this range. The dilutions were made with the HCl stock solution and awater–ethanol solvent mixture (50 wt-%). All solutions were prepared gravimetrically. Allof the concentrations are expressed on the molality scale (mol·kg−1).

The measurements system is composed by a potentiometric cell, commonly referred to asa Harned cell, a platinum hydrogen gas electrode an Ag/AgCl reference electrode. The sys-tem is designed to avoid any liquid junction, that why it is the only one accepted by IUPACfor primary pH measurements. The electrodes were prepared in our laboratory according tothe methods described by Bates [31].

The primary procedure has been thoroughly described elsewhere [31].The measurements were carried out from 15 and 35 °C. The cells were placed in a Tam-

son thermostatic bath that was regulated to ±0.01 °C. The temperature of this thermostaticbath was measured with a Telna 8 M thermometer equipped with a platinum resistancePt100. A Keithley multimeter with a resolution of 1 µV and measurement uncertainty of

J Solution Chem (2011) 40:1819–1834 1825

Table 2 Mean emf values, corrected to 1 atm partial pressure of hydrogen, as a function of the molality andtemperature

m

(mol·kg−1)Mean Ecorrected (V)

15 °C 25 °C 30 °C 35 °C

0.050 0.463657 0.464890 0.465062 0.465013

0.010 0.431596 0.431811 0.431525 0.430995

0.015 0.413161 0.412837 0.412219 0.411477

0.020 0.400184 0.399396 0.398521 0.397548

0.025 0.390278 0.389188 0.388237 0.387128

0.030 0.381955 0.380656 0.379587 0.378362

0.050 0.35936 0.357360 0.355996 0.354390

0.100 0.328770 0.325984 0.324195 0.322204

Table 3 Standard potentials of the Ag/AgCl electrode in the water–ethanol mixture (50 wt-%) as a func-tion of the temperature, and the approaches used for obtaining the results. The values are given with theirassociated expanded uncertainty, U (k = 2)

t

(°C)E0 (V)

Debye–Hückel equation Davies equation Pitzer equation

Value U (k = 2) Value U (k = 2) Value U (k = 2)

15 0.193588 9.7e−5 0.193443 8.4e−5 0.19363 1.2e−4

25 0.185280 9.2e−5 0.185083 8.6e−5 0.18525 1.4e−4

30 0.180646 9.2e−5 0.180417 9.2e−5 0.18065 1.5e−4

35 0.17589 1.1e−4 0.17563 1.0e−4 0.17578 2.0e−4

20 µV was employed for electromotive force measurements (emf). Measurements of the at-mospheric pressure were carried out with a Druck sensor having a resolution of 1 Pa and anuncertainty of 100 Pa.

The solution density was measured at 20 °C with a DMA 45 densimeter from Anton Paar.Using the information from Ref. [24], it was possible to express the densities at the studiedtemperatures. The calibration of the densimeter was carried out using certified ultrapurewater and n-heptane as liquid density standards.

All experimental emf values (Eexp) were corrected to 1 atm partial pressure of hydrogen(Ecorrected), by making use of the calculated vapor pressure of the mixture (p

vapmixture) and

atmospheric pressure (p), according to the following relations:

Ecorrected = Eexp − E (20)

E = RT ln 10

2Flog10

(p − p

vapmixture + 0.4ρgh

patm

)(21)

The term 0.4ρgh is an empirical bubbler-depth correction, and patm represents the standardcondition of atmospheric pressure.

The mean values of the corrected potentials for each molality and investigated tempera-ture are given in Table 2.

The hydrogen gas used for experiments has a reported purity of 99.9999% and it waspurchased from Air Liquid.

1826 J Solution Chem (2011) 40:1819–1834

Fig. 1 Standard potential of the Ag/AgCl electrode obtained in the water–ethanol mixture (50 wt-%) at25 °C from this study compared with values from the literature [3, 15, 32, 33]. The uncertainties (k = 2) forthe values obtained in the present study are indicated by the bars. The standard deviation of the interceptassociated to the value from Ref. [15] is 0.02 mV and to the value from Ref. [33] is 0.06 mV

4 Results and Discussion

4.1 Standard Potential in the Water–Ethanol Mixture (50 wt-%)

Table 3 summarizes all of the E0 values obtained by applying the Debye–Hückel equation,the Davies equation and the Pitzer model over the temperature range from 15 °C to 35 °C at50% ethanol mass fraction.

Figure 1 shows the values of E0 at 25 °C, comparing the values obtained in the presentstudy with those found in the literature at this same temperature. All reported values inthe literature were obtained according to the extrapolation approach explained in the sec-tion Theoretical Background, and very little information was found about their uncertaintyevaluations.

As can be seen in Fig. 1, there is good agreement among the three estimation approachesused to evaluate E0. These values are slightly lower than those reported in the literature.Differences in the parameters used for the calculations of the literature data could explaintheir bias with respect to our values.

The E0 values in the water–ethanol solvent mixture (50 wt-%), E0mixture, are lower than

those obtained in pure water, E0water. Typically, the value for E0

water at 25 °C is 0.22229 V.The difference between E0

water and E0mixture is a direct measure of the medium effect for the

transfer process from water to water–ethanol (50 wt-%) solvent. The primary medium effectis related to the transfer Gibbs energy (G0

tr) of the charged species H+ and Cl−, from waterto the mixed solvent, at infinite dilution, as shown in the relation:

G0tr = zF

(E0

water − E0mixture

). (22)

Table 4 reports these standard thermodynamic functions. The E0 values used for calculationswere those obtained by applying the Debye–Hückel equation.

J Solution Chem (2011) 40:1819–1834 1827

Table 4 Standard thermodynamic functions for the transfer process of HCl from water to the water–ethanolmixture (50 wt-%), along with their associated expanded uncertainty, U (k = 2)

t

(°C)G0

tr (J·mol−1) S0tr (J·mol−1) H 0

tr (J·mol−1)

Value U (k = 2) Value U (k = 2) Value U (k = 2)

15 3352 16 −19 18 3065 281

25 3571 16 −23 23 2999 568

30 3671 17 −25 26 2928 754

35 3814 18 −7 28 2881 969

The variation of the Gibbs energy of transfer with temperature was used to calculate thestandard enthalpy of transfer (H 0

tr) and the standard entropy of transfer (S0tr). Values of

these functions can provide precious information about ion–solvent interactions as well ason structural changes occurring in the solution [34].

The Gibbs energy of transfer G0tr is positive at all of the studied temperatures. At 25 °C,

for the water–methanol mixture (50 wt-%), the value of G0tr reported by Feakins et al. and

cited by Bates [31] is 3490 J·mol−1. This value is slightly lower than 3571 J·mol−1 obtainedin the present study for the water–ethanol mixture (50 wt-%) at the same temperature andmass fraction of alcohol. The difference between the two values indicates that HCl is moresolvated in the water/methanol mixtures.

This thermodynamically unfavorable transfer could be the result of an endothermic pro-cess, H 0

tr being positive, or could be due to a more orderly system since S0tr is negative.

Water/isopropanol mixtures analyzed by Roy et al. [35] show the same characteristics athigh alcohol concentrations, e.g. 46 wt-%. The authors concluded that hydrogen bonds areresponsible for this behavior because they highly reorganize the molecules in the mixtures.This conclusion was confirmed by Wakisaka et al. [36], who showed that water–ethanolmixtures are characterized by the presence of microheterogeneity. Their interpretation isthat the alcohol’s self-association, appearing around 40–55 wt-% mass fraction of alcohol,highly organizes the water structure: the hydrogen bonds of water are broken and watermolecules hydrate the alcohol molecules. Thus, the clusters formed by the self-associationof ethanol molecules affect ion solvation in the mixture and lead to deviations from idealbehavior. However, the greater contribution of enthalpy suggests that structural changes arenot very important. Indeed, taking into account the 95% confidence intervals, the entropiesvalues are slightly negative or fall around zero, which suggests that the unfavorable transferprocess is not dominated by an increase of order in the solvent system.

4.2 Activity Coefficient for the Water–Ethanol Mixture (50 wt-%)

Water–organic mixtures can be characterized by two pH scales, depending on the standardstate chosen for the proton at infinite dilution. The first one is based on infinite dilution ofthe hydrogen ion in water, whereas in the second case it is infinite dilution for the hydro-gen ion in the aqueous–organic solvent [37, 38]. The standard state for aqueous solutionsis chosen in such a way that the activity of HCl approaches its concentration at infinite di-lution. For dilute electrolytes, activity coefficient values reflect solute–solvent interactions.If water is chosen as the standard state for water–organic mixtures, the activity coefficientis not expected to equal unity at infinite dilution, mainly due to the effects induced by thepresence of the organic solvent in the mixture. Moreover, since the standard potential mayvary by up to 0.2 mV as a result of variations in preparative techniques of the electrodes,

1828 J Solution Chem (2011) 40:1819–1834

Fig. 2 Mean activity coefficientsof HCl versus molality in the50 wt-% water–ethanol solventmixture at 25 °C: (!) Debye–Hückel equation, (1) Daviesequation, (Q) Pitzer equation.The values obtained by differentapproaches are shown along withtheir associated expandeduncertainty U (k = 2)

Table 5 Mean activity coefficients of HCl at molality 0.01 mol·kg−1 as a function of the temperature andthe approach used for obtaining the parameters. The associated expanded uncertainties, U(k = 2), are alsoindicated

t

(°C)γ±Debye–Hückel equation Davies equation Pitzer equation

Value U (k = 2) Value U (k = 2) Value U (k = 2)

15 0.8293 1e−4 0.8270 3e−4 0.830 6e−3

25 0.8250 1e−4 0.8223 3e−4 0.825 7e−3

30 0.8217 1e−4 0.8188 3e−4 0.822 7e−3

35 0.8205 1e−4 0.8172 4e−4 0.819 9e−3

as recognized by IUPAC [1, 9], it is preferable to characterize the system in terms of meanactivity coefficients.

Figure 2 shows the variation of the mean activity coefficient for HCl in the water–ethanolmixture (50 wt-%) as a function of the molality at 25 °C: its values decrease with increaseof molality and the decrease is more pronounced in the mixed solvent than in water. The ob-served deviations from ideal behavior confirms the particular characteristics of the aqueous–alcoholic solvent mixture.

For example, at the HCl molality of 0.01 mol·kg−1 at 25 °C, application of the Debye–Hückel equation in the water–ethanol mixture (50 wt-%) leads to a mean activity coefficientof 0.825, which is smaller than the one obtained in pure water by Bates et al. [39], namely0.904.

Table 5 shows the mean activity coefficient of 0.01 mol·kg−1 HCl, along with their as-sociated uncertainties in water–ethanol solvent mixture (50 wt-%), obtained in the presentstudy at all investigated temperatures. Generally, an increase of the temperature leads to adecrease of the activity coefficient.

The values obtained with Davies equation are slightly smaller than the values obtainedby application of the Debye–Hückel equation. The largest differences are observed at highermolalities. It can be concluded that the Debye–Hückel equation can be successfully appliedfor molalities smaller than 0.05 mol·kg−1. However, most of the buffer solutions used for pH

J Solution Chem (2011) 40:1819–1834 1829

calibrations have an ionic strength close to 0.1 mol·kg−1. For aqueous solutions, limitationsand discrepancies from using the Debye–Hückel equation have already been highlighted byFisicaro et al. [12, 13]. As shown in Fig. 2, at higher ionic strengths (i.e. 0.1 mol·kg−1) thevalues obtained by the application of the Davies equation are closer to those obtained withthe Pitzer equation. This behavior is similar to that observed at all of the studied tempera-tures.

Parameters of Davies’s and Pitzer’s equations, obtained in the present study at 25 °C, arecompared to those available in the literature in Table 6.

Unfortunately, few studies are reported in the literature concerning the activity coefficientof HCl in water–ethanol mixtures. Most of them are not recent publications [3, 15, 32, 33]and do not report uncertainties for their reported values. With regards to Pitzer’s approach,the only available results were published by Deyhimi et al. [40]. The Pitzer parameters re-ported in Ref. [40] for water–ethanol mixtures, and those reported in Refs. [23, 41] forpure water, are indicated in Table 6 for the same temperature, i.e. at 25 °C. Comparison ofthe parameters leads to the identification of some difficulties in applying the Pitzer model,especially at low molalities. In particular, the differences among values characterizing thewater–ethanol mixtures can be justified by considering two reasons: (i) use of different val-ues for physico-chemical constant involved in the calculations, such as dielectric constant,density, etc.; and (ii) parameters determined from data extending to high molalities do notalways provide an accurate extrapolation to lower molalities, as pointed out by Marshall etal. [23]. In fact, the parameters reported by Deyhimi et al. [40] fit the data up to 5 mol·kg−1.Consequently, it is difficult to compare parameters that don’t characterize the same molalityregion.

One of the main advantages of Pitzer’s model is that it highlights the importance of thebalance of forces existing in the system. As expected, for very low concentrations, the termwith the largest contribution to the activity coefficient is the term related to electrostaticinteractions (f γ ). This contribution is up to 99% of the total at 0.005 mol·kg−1, but it de-creases to 86% by 0.1 mol·kg−1. The coefficient Bγ arises from short-range forces betweenpairs of oppositely charged ions. The positive Bγ value for HCl in the water–ethanol system(50 wt-%) indicates the predominance of attractive forces [21]. With an increase of molal-ity, the influence of these forces becomes more pronounced, up to 14% for 0.1 mol·kg−1.Under these conditions the effects of binary interactions are not negligible and should beconsidered.

4.3 Uncertainty Estimation

The uncertainty is a parameter characterizing the dispersion of the quantity values beingattributed to a measurand based on the information used [42]. To ensure the metrologicaltraceability of the results is important to consider all the uncertainty sources for the mea-surement process. For example, a variation of 0.1 mV in E0 can result in a variation of about0.002 pH units. Knowing that the overall uncertainty associated with primary pH measure-ments with the Harned cell, for different buffers, can be of 0.003 (k = 2) pH units, it isevident that variations in the electrode’s standard potential can have a major impact on thecalculated pHs.

In order to estimate the overall uncertainty of the standard potentials, the individual con-tributions to the uncertainty were determined according to ISO guidelines [43]. The standarduncertainties were propagated by using a dedicated software that has been developed andvalidated in our laboratory (Wincert, V.3.11, Implex, France). The uncertainties associatedto the three E0 values obtained by the different methods are reported in Fig. 1. The smallestuncertainty was obtained by using the Debye–Hückel equation.

1830 J Solution Chem (2011) 40:1819–1834

Tabl

e6

Para

met

ers

for

Dav

ies

equa

tion

and

ion-

inte

ract

ion

para

met

ers

for

the

wat

er–e

than

olm

ixtu

re(5

0w

t-%

)at

25°C

Dav

ies

coef

ficie

ntβ

(0)

(kg·m

ol−1

)

U (kg·m

ol−1

)(k

=2)

β(1

)

(kg·m

ol−1

)

U (kg·m

ol−1

)(k

=2)

Cγ

(kg2

·mol

−2)

U (kg·m

ol−1

)(k

=2)

mm

ax

(mol

·kg−1

)

Pres

ents

tudy

0.47

0.25

0.23

0.39

0.19

––

0.1

Ref

.[40

],in

wat

er–e

than

olm

ixtu

re(5

0w

t-%

)

–0.

23–

0.72

–−0

.013

6–

5

Ref

.[23

],in

wat

er–

0.20

488

0.00

295

0.07

662

0.06

631

−0.0

0377

0.00

023

16

Ref

.[41

],in

wat

er–

0.17

75–

0.29

45–

0.00

080

–6

J Solution Chem (2011) 40:1819–1834 1831

Fig. 3 Contribution from the predominant factors to uncertainties for E0

The expanded uncertainty at the 95% confidence level (k = 2) for the Debye–Hückelequation was determined as the sum of two contributions: (i) the highest uncertainty ob-tained for individual measurements, as suggested by IUPAC [1], and (ii) the intermediateprecision, defined as the results obtained with “same procedure, in same location, and repli-cate measurements over an extended period of time”, according to the International Vocabu-lary of Metrology [42]. The last term was considered in order to include contributions suchas the electrode quality or the small variations in the ethanol content in the water–ethanolmixture possibly related to an evaporation phenomenon. The expanded uncertainty (k = 2)

for Davies’s and Pitzer’s models also includes two components: (i) the extrapolation itself,and (ii) the contribution related to the lowest ionic strength in order to maximize the uncer-tainty.

Figure 3 shows the contributions of the predominant factors for the three standard poten-tial values at 25 °C. It can be seen that the main contribution for the Debye–Hückel equa-tion comes from the intermediate precision (almost 85%). This large contribution can beattributed to a poor reproducibility that generates results with high dispersion. This factormay also be evoked when the extrapolation methods are applied, since the extrapolationitself is based on the dispersed experimental data.

The poor measurement precision can be explained by the fact that the E0 values arevery sensitive to the electrode microstructure and solvent composition. Any deviation inpreparation of the solvent mixture will have a consequential effect on the standard potential.A detailed analysis of the factors influencing the reproducibility of standard potential inwater–ethanol media is still under study and the assessment of their influence will not bean easy task. At present, all of them are covered by the terms related to the intermediateprecision and to the extrapolation.

The reliability of E0 can be further improved by minimizing the effects mainly due tothe complexity of the water–ethanol environment (e.g. composition, concentration). Onepossible way to achieve this would be to increase the number of experimental points withinthe range of studied molalities and to reconsider the hypothesis regarding the linearity of thesystem.

The uncertainty obtained by the application of Pitzer’s model is an order-of-magnitudehigher than those obtained with Debye–Huckel’s equation, which can be explained by thefact that the Pitzer approach involves more parameters for its description. It was thereforeexpected to yield the highest uncertainties.

Table 6 shows that the uncertainties associated with the Pitzer’s parameters calculated inthe present work are highly significant, namely up to 85% of their absolute values. Since,

1832 J Solution Chem (2011) 40:1819–1834

these parameters are used to calculate lnγ±, it is useful to evaluate the contribution of asingle parameter on the overall uncertainty of the activity coefficient. Almost 68% of itstotal uncertainty arises from the parameter β(1). This parameter is the most sensitive one, asalready found by Marshall et al. [23]. The same effect was previously found for HCl in pureaqueous solutions.

5 Conclusions

The present work can be seen as a contribution to IUPAC’s efforts in the field of pH mea-surements in aqueous–alcoholic mixed solvents.

The study focused on the determination the standard potential of the Ag/AgCl electrodein water–ethanol (50 wt-%) solvent mixtures. No reference values concerning E0 or activitycoefficients in water–alcohol solvent mixtures are available in the literature. Without assum-ing any existing primary protocols, three different approaches applied previously for aque-ous solutions were used. Thus, the upper molality limit for HCl was fixed at ≤0.1 mol·kg−1,since this concentration domain covers most of the existing aqueous buffers for pH-metercalibrations. The standard potential, the mean activity coefficient of HCl, and the relatedthermodynamic quantities were presented and discussed.

For E0, the Debye–Hückel theory and Pitzer’s model was used and good agreement wasfound between the results obtained by the two approaches.

Estimation of the standard potential of the Ag/AgCl electrode, and of the mean activitycoefficients of HCl in water–ethanol mixed solvent solutions, is an important step in the di-rect determination of pH in these mixtures. For aqueous solutions that are sufficiently dilute,≤0.1 mol·kg−1, the Debye–Hückel theory can be applied. However, this study shows that thepresence of 50% ethanol in the solvent system reduces the validity of Debye–Hückel equa-tion to much lower ionic strengths than for aqueous solutions, i.e., close to 0.05 mol·kg−1.

Acknowledgements The research leading to these results has received funding from the European Unionon the basis of Decision No 912/2009/EC.

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