Section: 5 | pH Scale for Aqueous Solutions |
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John R. Rumble, ed., CRC Handbook of Chemistry and Physics, 102nd Edition (Internet Version 2021), CRC Press/Taylor & Francis, Boca Raton, FL.
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pH SCALE FOR AQUEOUS SOLUTIONS

A. K. Covington

A Working Party of IUPAC, after extensive considerations over five years, has produced a report (1) which sets pH firmly within the International System of Units (SI). A summary of these important developments is given below.

The concept of pH is unique among the commonly encountered physicochemical quantities in that, in terms of its definition,

pH = − lg aH     (1)

it involves a single ion quantity, the activity of the hydrogen ion, which is immeasurable by any thermodynamically valid method and requires a convention for its evaluation.

pH was originally defined by Sørensen (Ref. 2) in terms of the concentration of hydrogen ions (in modern nomenclature) as pH = − lg (cH/co) where cH is the hydrogen ion concentration in mol dm–3, and co = 1 mol dm–3 is the standard amount concentration. Subsequently (Ref. 3), it was accepted as more satisfactory to define pH in terms of the relative activity of hydrogen ions in solution

pH = − lg aH = − lg (mHγH/mo)      (2)

where aH is the relative (molality basis) activity and γH is the molal activity coefficient of the hydrogen ion H+ at the molality mH, and mo the standard molality. The quantity pH is intended to be a measure of the activity of hydrogen ions in solution. However, since it is defined in terms of a quantity that cannot be measured by a thermodynamically valid method, eqn. (2) can only be considered a notional definition of pH.

pH being a single ion quantity, it is not determinable in terms of a fundamental (or base) unit of any measurement system, and there is difficulty providing a proper basis for the traceability of pH measurements. A satisfactory approach is now available in that pH determinations can be incorporated into the International System (SI) if they can be traced to measurements made using a method that fulfils the definition of a ‘primary method of measurement’ (Ref. 4).

The essential feature of a primary method is that it must operate according to a well-defined measurement equation in which all of the variables can be determined experimentally in terms of SI units. Any limitation in the determination of the experimental variables, or in the theory, must be included within the estimated uncertainty of the method if traceability to the SI is to be established. If a convention were used without an estimate of its uncertainty, true traceability to SI would not be established. The electrochemical cell without liquid junction, known as the Harned cell (Ref. 5), fulfills the definition of a primary method for the measurement of the acidity function, p(aHγCl), and subsequently of the pH of buffer solutions.

The Harned cell is written as

Pt | H2 | buffer S, Cl | AgCl | Ag (Cell I)

and contains a standard buffer, S, with chloride ions, as potassium or sodium chloride, added in order to use the silver–silver chloride electrode as reference electrode. The application of the Nernst equation to the spontaneous cell reaction of Cell I:

½ H2 + AgCl → Ag(s) + H+ + Cl

yields the potential difference EI of the cell (corrected to 1 atm [101.325 kPa], the partial pressure of hydrogen gas used in electrochemistry in preference to 100 kPa) as

EI = Eo − (RT/F)In 10 lg [(mHγH/mo)(mClγCl/mo)]       (3)

which can be rearranged, since aH = mHγH/mo, to give the acidity function

p(aHγCl) = − lg(aHγCl) = (EIEo)/[(RT/F)In 10] + lg(mCl/mo)      (4)

where Eo is the standard potential difference of the cell, and hence of the silver–silver chloride electrode, and γCl is the activity coefficient of the chloride ion.

The standard potential difference of the silver–silver chloride electrode, Eo, is determined from a Harned cell in which only HCl is present at a fixed molality (e.g., m = 0.01 mol kg–1)

Pt | H2 | HCl (m) | AgCl | Ag (Cell Ia)

The application of the Nernst equation to the HCl cell (Ia) gives

Ela = Eo – (2RT/F)In 10 lg[(mHCl/mo)(γ±HCl)]      (5)

where EIa has been corrected to 1 atmosphere partial pressure of hydrogen gas (101.325 kPa) and γ±HCl is the mean ionic activity coefficient of HCl.

Values of the activity coefficient (γ±HCl) at molality 0.01 mol kg–1 and various temperatures were given by Bates and Robinson (Ref. 6). The standard potential difference depends on the method of preparation of the electrodes, but individual determinations of the activity coefficient of HCl at 0.01 mol kg–1 are more uniform than values of Eo. Hence the practical determination of the potential difference of the cell with HCl at 0.01 mol kg–1 is recommended at 298.15 K at which the mean ionic activity coefficient is 0.904. (It is unnecessary to repeat the measurement of Eo at other temperatures but simply to correct published smoothed values by the observed difference in Eo at 298.15 K.)

In national metrology institutes (NMIs), measurements of Cells I and Ia are often done simultaneously in a thermostat bath. Subtracting eqn. (5) from eqn. (3) gives

ΔE = ElEla = – (RT/F)In 10{lg[(mHγH/mo)(mClγCl/mo)] – lg[(mHCl/mo)2γ2±HCl]}

which is independent of the standard potential difference. Therefore, the subsequently calculated pH does not depend on the standard potential difference and hence does not depend on the assumption that the standard potential of the hydrogen electrode is zero at all temperatures. Therefore, the Harned cell gives an exact comparison between hydrogen ion activities at different temperatures.

The quantity p(aHγCl) = − lg (aHγCl), on the left-hand side of (4), is called the acidity function (5). To obtain the quantity pH according to eqn. (2) from the acidity function, it is necessary to evaluate lg γCl independently. This is done in two steps: (i) the value of lg (aHγCl) at zero chloride molality, lg (aHγCl)o, is evaluated and (ii) a value for the activity of the chloride ion γoCl , at zero chloride molality (sometimes referred to as the limiting or ‘trace’ activity coefficient) is calculated using the Bates-Guggenheim convention (Ref. 7). The value of lg (aHγCl)o corresponding to zero chloride molality is determined by linear extrapolation of measurements using Harned cells with at least three added molalities of sodium or potassium chloride (I < 0.1 mol kg–1).

The value of lg (aH γCl)o corresponding to zero chloride molality is determined by linear extrapolation of measurements using Harned cells with at least three added molalities of sodium or potassium chloride (I < 0.1 mol kg–1) in accord with eqn. (7):

− lg (aHγCl) = − lg (aHγCl)o + SmCl        (7)

where S is an empirical, temperature dependent, constant.

The Bates-Guggenheim convention (Ref. 7) assumes that the trace activity coefficient of the chloride ion γoCl is given by

lg γoCl = − A I1/2/(1 + Ba I1/2)         (8)

where A is the Debye-Hückel temperature dependent constant (limiting slope), a is the mean distance of closest approach of the ions (ion size parameter), Ba is set equal to 1.5 (mol kg–1)–1/2 at all temperatures in the range 5–50 °C, and I is the ionic strength of the buffer (which for its evaluation requires knowledge of appropriate acid dissociation constants).

The various stages in the assignment of primary standard pH values are combined in eqn. (9), which is derived from eqns. (4), (5), and (8)

pH(PS) = lim mCl→o {(EIEo)/[(RT/F)In 10] + lg (mCl/mo)} − A I1/2/[1 + 1.5 (I/mo)1/2]          (9)

In order for a particular buffer solution to be considered a primary buffer solution, it must be of the “highest metrological” quality (Ref. 4) in accordance with the definition of a primary standard. It is recommended that it have the following attributes (Ref. 9):

  1. High buffer value in the range 0.016–0.07 (mol OH)/pH.
  2. Small dilution value at half concentration (change in pH with change in buffer concentration) in the range 0.01–0.20.
  3. Small dependence of pH on temperature less than ±0.01 K–1.
  4. Low residual liquid junction potential <0.01 in pH.
  5. Ionic strength ≤0.1 mol kg–1 to permit applicability of Bates-Guggenheim convention.
  6. NMI certificate for specific batch.
  7. Reproducible purity of preparation (lot to lot differences of |ΔpH(PS)| < 0.003).
  8. Long-term stability of stored solid material.

Values for the above and other important parameters for the primary and secondary buffer materials are given in Table 1. Column definitions are as follows.

Column heading Definition
Name Name of salt or solid substance
Formula Molecular formula of solid or substance
Molality Molality of standard buffer solution, in mol kg-1
Molar mass Molar mass of salt or solid substance, g mol-1
Density Density of standard buffer solution, in g mL-1
Amount conc. at 20 °C Concentration amount at 20 °C, in mol dm-3
Mass in g to make 1 dm3 Mass of salt or solid substance needed to make 1 dm3; = molar mass × Amount conc. at 20 °C; in g
Dilution value ΔpH1/2 Change of pH value at half concentration, in pH units
Buffer value (β) Strength of pH buffering, in units mol (OH-) dm-3
pH Temperature coefficient Change of pH with temperature, in pH units K-1

Primary Standard Buffers

As there can be significant variations in the purity of samples of a buffer of the same nominal chemical composition, it is essential that the primary buffer material used has been certified with values that have been measured with Cell I. The Harned cell is used by many national metrological institutes for accurate measurements of pH of buffer solutions.

Typical values of the pH(PS) of the seven solutions from the six accepted primary standard reference buffers, which meet the conditions stated above, are listed in Table 2. Batch-to-batch variations in purity can result in changes in the pH value of samples of at most 0.003. The typical values in Table 2 should not be used in place of the certified value (from a Harned cell measurement) for a specific batch of buffer material.

The required attributes listed above effectively limit the range of primary buffers available to between pH 3 and 10 (at 25 °C). Calcium hydroxide and potassium tetraoxalate are excluded because the contribution of hydroxide or hydrogen ions to the ionic strength is significant. Also excluded are the nitrogen bases of the type BH+ (such as tris(hydroxymethyl)aminomethane and piperazine phosphate) and the zwitterionic buffers (e.g., HEPES and MOPS [Ref. 10]). These do not comply because either the Bates-Guggenheim convention is not applicable, or the liquid junction potentials are high. This means the choice of primary standards is restricted to buffers derived from oxy-carbon, -phosphorus, -boron and mono, di- and tri-protic carboxylic acids. The uncertainties (Ref. 11) associated with Harned cell measurements are calculated (Ref. 1) to be 0.004 in pH at NMIs, with typical variation between batches of primary standard buffers of 0.003.

Secondary Standards

Substances that do not fulfill all the criteria for primary standards, but to which pH values can be assigned using Cell I are considered to be secondary standards (Table 3). Reasons for their exclusion as primary standards include difficulties in achieving consistent and suitable chemical quality (e.g. acetic acid is a liquid), suspected high liquid junction potential, or inappropriateness of the Bates-Guggenheim convention (e.g., other charge-type buffers). The uncertainty is higher (e.g., 0.01) for biological buffers. Certain other substances, which cannot be used in cells containing hydrogen gas electrodes, are also classed as secondary standards.

Calibration Procedures

  1. One-point calibration
    A single-point calibration is insufficient to determine both slope and one-point parameters. The theoretical value for the slope can be assumed but the practical slope may be up to 5% lower. Alternatively, a value for the practical slope can be assumed from the manufacturer’s prior calibration. The one-point calibration therefore yields only an estimate of pH(X). Since both parameters may change with age of the electrodes, this is not a reliable procedure.
  2. Two-point calibration [target uncertainty: 0.02–0.03 at 25 °C]
    In the majority of practical applications, glass electrodes cells are calibrated by a two-point calibration, or bracketing, procedure using two standard buffer solutions, with pH values, pH(S1) and pH(S2), bracketing the unknown pH(X). Bracketing is often taken to mean that the pH(S1) and pH(S2) buffers selected from Table 2 should be those that are immediately above and below pH(X). This may not be appropriate in all situations and choice of a wider range may be better.
  3. Multi-point calibration [target uncertainty: 0.01–0.03 at 25 °C].
    Multi-point calibration is carried out using up to five standard buffers. The use of more than five points yields no significant improvement in the statistical information obtainable.
    Details of uncertainty computations (Ref. 11) have been given (Ref. 1).

Measurement of pH and Choice of pH Standard Solutions

  1. If pH is not required to better than ±0.05 any pH standard solution may be selected.
  2. If pH is required to ±0.002 and interpretation in terms of hydrogen ion concentration or activity is desired, choose a standard solution, pH(PS), to match X as closely as possible in terms of pH, composition and ionic strength.
  3. Alternatively, a bracketing procedure may be adopted whereby two standard solutions are chosen whose pH values, pH(S1), pH(S2) are on either side of pH(X). Then if the corresponding potential difference measurements are E(S1), E(S2), E(X), then pH(X) is obtained from
    pH(X) = pH(S1) + [E(X) – E(S1)]/%k
    where %k = 100[E(S2) – E(S1)]/[pH(S2) – pH(S1)] is the apparent percentage slope. This procedure is very easily done on some pH meters simply by adjusting downwards the slope factor control with the electrodes in S2. The purpose of the bracketing procedure is to compensate for deficiencies in the electrodes and measuring system.

Information to Be Given about the Measurement of pH(X)

The standard solutions selected for calibration of the pH meter system should be reported with the measurement as follows:

System calibrated with pH(S) = .... at ... K

System calibrated with two primary standards, pH(PS1) = .... and pH(PS2) = .... at .... K

System calibrated with n standards, pH(S1) = ...., pH(S2) = .... etc. at .... K

 

Interpretation of pH(X) in Terms of Hydrogen Ion Concentration

The defined pH has no simple interpretation in terms of hydrogen ion concentration but the mean ionic activity coefficient of a typical 1:1 electrolyte can be used to obtain hydrogen ion concentration subject to an uncertainty of 3.9% in concentration, corresponding to 0.02 in pH.

 

References

  1. Buck, R.P., Rondinini, S., Covington, A.K., Baucke, F.G.K., Brett, C.M.A., Camoes, M.F.C., Milton, M.J.T., Mussini, T., Naumann, R., Pratt, K.W., Spitzer, P., and Wilson, G.S. Pure Appl. Chem. 74, 2105, 2002. [https://doi.org/10.1351/pac200274112169]
  2. Sørensen, S.P.L. Comp. Rend. Trav. Lab. Carlsberg 8, 1, 1909.
  3. Sørensen, S.P.L., and Linderstrøm-Lang, K.L., Comp. Rend. Trav. Lab. Carlsberg 15, 1924.
  4. BIPM, Com. Cons. Quantité de Matière 4, 1998. See also M.J.T. Milton and T.J. Quinn, Metrologia 38, 289, 2001. [https://doi.org/10.1088/0026-1394/38/4/1]
  5. Harned H.S., and Owen, B.B., The Physical Chemistry of Electrolytic Solutions, Ch 14, Reinhold, New York, 1958.
  6. Bates R.G., and Robinson, R.A., J. Soln. Chem. 9, 455, 1980.
  7. Bates R.G., and Guggenheim, E.A., Pure Appl. Chem. 1, 163, 1960. [https://doi.org/10.1351/pac196001010163]
  8. International Vocabulary of Metrology – Basic and General Concepts and Associated Terms (VIM), Third Edition, JCGM 200:2012, BIPM, 2012.
  9. Bates, R.G. Determination of pH, Wiley, New York, 1973.
  10. Good, N.E. et al., Biochem. J. 5, 467, 1966. [https://doi.org/10.1021/bi00866a011]
  11. Evaluation of Measurement Data — Guide to the Expression of Uncertainty in Measurement (GUM), JCGM 100:2008, BIPM, IEC, IFCC, ISO, IUPAC, IUPAP, OIML, 2008.

TABLE 1. Properties of Some Primary and Secondary Standard Buffer Substances and Solutions (Primary Standards in Bold Face)



Salt or solid substanceFormulaMolality/mol kg–1Molar mass/g mol–1Density/g mL-1Amount conc. at 20 °C/mol dm–3Mass/g to make 1 dm3Dilution value ΔpH1/2Buffer value (β)/mol OH dm–3pH Temperature coefficient/K–1
Potassium tetroxalate dihydrateKH3C4O8·2H2O0.1254.1911.00910.0987525.101
Potassium tetraoxalate dihydrateKH3C4O8·2H2O0.05254.1911.00320.0496512.6200.1860.0700.001
Potassium hydrogen tartrate (sat at 25 °C)KHC4H4O60.0341188.181.00360.0346.40.0490.027-0.0014
Potassium dihydrogen citrateKH2C6H5O70.05230.221.00290.0495811.410.0240.034-0.022
Potassium hydrogen phthalateKHC8H4O40.05204.441.00170.0495810.120.0520.0160.00012
Disodium hydrogen orthophosphate (0.025 m) + potassium dihydrogen orthophosphate (0.025 m)Na2HPO4
KH2PO4
0.025141.9581.00380.024923.53790.0800.029-0.0028
Disodium hydrogen orthophosphate (0.03043 m) + potassium dihydrogen orthophosphate (0.00869 m)Na2HPO40.03043141.9591.00200.086654.3020.070.016-0.0028
Disodium tetraborate decahydrateNa2B4O7·10H2O0.05381.3671.00750.0498519.012
Disodium tetraborate decahydrateNa2B4O7·10H2O0.01381.3671.00010.009983.8060.010.020-0.0082
Sodium hydrogen carbonate (0.025 m) + sodium carbonate (0.025 m)NaHCO30.02584.011.00130.024922.0920.0790.029-0.0096
Calcium hydroxide (sat. at 25 °C)Ca(OH)20.020374.090.99910.020251.5-0.280.09-0.033


TABLE 2. Typical Values of pH(PS) for Primary Standards at 0–50 °C



Primary standards (PS)0 °C5 °C10 °C15 °C20 °C25 °C30 °C35 °C37 °C40 °C50 °C
Sat. potassium hydrogen tartrate (at 25 °C)3.5573.5523.5493.5483.5473.549
0.05 mol kg–1 potassium dihydrogen citrate3.8633.8403.8203.8023.7883.7763.7663.7593.7563.7543.749
0.05 mol kg–1 potassium hydrogen phthalate4.0003.9983.9973.9984.0004.0054.0114.0184.0224.0274.050
0.025 mol kg–1 disodium hydrogen phosphate + 0.025 mol kg–1 potassium dihydrogen phosphate6.9846.9516.9236.9006.8816.8656.8536.8446.8416.8386.833
0.03043 mol kg–1 disodium hydrogen phosphate + 0.008695 mol kg–1 potassium dihydrogen phosphate7.5347.5007.4727.4487.4297.4137.4007.3897.3867.3807.367
0.01 mol kg–1 disodium tetraborate9.4649.3959.3329.2769.2259.1809.1399.1029.0889.0689.011
0.025 mol kg–1 sodium hydrogen carbonate + 0.025 mol kg–1 sodium carbonate10.31710.24510.17910.11810.06210.0129.9669.9269.9109.8899.828


TABLE 3. Values of pH(SS) as a Function of Temperature for Some Secondary Standards from Harned Cell I Measurements



Secondary standards0 °C5 °C10 °C15 °C20 °C25 °C30 °C37 °C40 °C50 °C
0.05 mol kg–1 potassium tetroxalatea1.671.671.671.671.681.681.681.691.691.71
0.05 mol kg–1 sodium hydrogen diglycolateb3.473.473.483.483.493.503.523.533.56
0.1 mol dm–3 acetic acid + 0.1 mol dm–3 sodium acetate4.684.674.674.664.664.654.654.664.664.68
0.01 mol dm–3 acetic acid + 0.1 mol dm–3 sodium acetate4.744.734.734.724.724.724.724.734.734.75
0.02 mol kg–1 piperazine phosphatec6.586.516.456.396.346.296.246.166.146.06
0.05 mol kg–1 tris hydrochloride + 0.01667 mol kg–1 trisc8.478.308.147.997.847.707.567.387.317.07
0.05 mol kg–1 disodium tetraborate9.519.439.369.309.259.199.159.099.079.01
Saturated (at 25 °C) calcium hydroxide13.4213.2113.0012.8112.6312.4512.2912.0711.9811.71

  • aPotassium trihydrogen dioxalate (KH3C4O8).
  • bSodium hydrogen 2,2-oxydiacetate.
  • c2-Amino-2-(hydroxymethyl)-1,3 propanediol or tris(hydroxymethyl)aminomethane.


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Accessibility

The Voluntary Product Accessibility Template (VPAT) is a self-assessment document which discloses how accessible Information and Communication Technology products are in accordance with global standards.

The VPAT disclosure templates do not guarantee product accessibility but provide transparency around the product(s) and enables direction when accessing accessibility requirements.

Taylor & Francis has chosen to complete the International version of VPAT which encompasses Section 508 (US), EN 301 549 (EU) and WCAG2.1 (Web Content Accessibility Guidelines) for its products.

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