The mean pH of mixed fresh waters

War. Res. Vol. 23, No. 10, pp. 1331-1334, 1989 Printed in Great Britain

0043-1354/89 $3.00+0.00 Pergamon Press plc

TECHNICAL NOTE THE MEAN pH OF MIXED FRESH WATERS C. JORDAN Freshwater Biological Investigation Unit, Department of Agriculture (NI), Greenmount Road, Antrim, Northern Ireland

(First received September 1988; accepted in revised form April 1989) Abstract--In the calculation of the mean pH of mixtures of fresh waters it is necessary to consider not only the relative volumes and pH values of the waters being mixed but also their alkalinity. For waters of very low alkalinity mixed in equal proportions, the error involved in assuming the arithmetic mean pH value instead of the arithmetic mean H + ion concentration may be quite significant and it depends only on the difference in pH values between the waters being mixed.

Key words--pH, alkalinity, mixture, fresh waters

NOMENCLATURE

n = number of water samples in a mixture ~ = sum of samples 1-n i=1

Vi = volume of water sample i in a mixture (1.) pH~ = pH value of water sample i in a mixture p H = m e a n pH value of a mixture of water samples [H +] = concentration of hydrogen ions in solution (mol I-~ or M) [HCO~-] = concentration of bicarbonate ions in solution (moll -I or M) [CO~-] = concentration of carbonate ions in solution (moll -~ or M) [OH-] = concentration of hydroxide ions in solution (tool I-L or M) • ~=fraction of dissolved carbonic species present as HCO~ a2=fraction of dissolved carbonic species present as CO~K~ = first acidity constant of the carbonate buffer system (tool I-~ or M) K2 = second acidity constant ot tl~e carbonate buffer system (tool 1-~ or M) CT = total concentration of dissolved carbonic species (tool 1-~ or M) CT = mean concentration of dissolved carbonic species (tool 1-~ or M) alk = mean alkalinity of a mixture of water samples (molH + 1-h or equiv 1-~); I tool H + 1-I = 1 equiv 1-~ = 50mgl -I CaCO3 K s = equilibrium constant (Henry's Law) for the dissolution of CO 2 gas in water (tool 1-t atm -I) Pco: = partial pressure of CO2 in the atmosphere (atm). INTRODUCTION

The calculation of mean p H is a c o m m o n data reduction step in studies of acidification of fresh waters. M a n y authors seldom state the procedure used to calculate such means (Smith et al., 1987;

Kelso et al., 1986) and, for example, often cite a drop in mean pH with time as evidence of acidification of fresh waters. Often the arithmetic mean appears to have been used to calculate mean p H and. other statistics even though this can lead to substantial" underestimates of the true H + ion concentration. With a mixture o f equal volumes of poorly buffered waters with p H 3 and 5, for example, the mean pH is 3.3 not 4 and the arithmetic mean underestimates the true H + ion concentration by 80%. This note shows that for such mixtures, the arithmetic error term depends only on the difference in the pH values being averaged. In the calculation of the mean p H of a number of water samples taken from one or more locations, or the mean p H of mixtures of fresh waters when one or more of the samples or waters has a significant buffering capacity, both the p H and the alkalinity of each of the waters involved must be known. Since alkalinity is determined by titrating a water sample with acid to pH 4.5 ( H M S O , 1982), the units of alkalinity are moles of monovalent acid (or H + ion) per litre. These units are c o m m o n l y referred to as equivalents per litre (equiv 1-~) and will be used throughout the remainder of this paper. CALCULATION OF MEAN pH Fresh waters may be roughly classified into two categories--those with and those without a significant buffering capacity. There are, therefore, three cases to be considered when determining the pH value of a mixture of two or more of these waters whose individual pH values are known. The first case involves the mixing of waters which have a negligible buffering capacity. In such a case, acid inputs meet little or no buffering resistance and the pH of the mixture will fall rapidly. This situation

1331

1332

Technical Note

is often important when acid rain falls on upland lakes or on the headwaters of small streams in upland areas where the soils are strongly acid. In the second case to be considered, each of the waters to be mixed has a significant buffering capacity. In this case, the buffer components in the waters being mixed will resist a change in pH by neutralizing any H ÷ ion inputs. The final pH of such a mixture depends on the acid neutralizing capacity (alkalinity) of the individual waters comprising the mixture and whether or not the mixture is allowed to equilibrate with CO 2 in the atmosphere. This behaviour is typically observed when water samples from lowland streams or lakes are mixed. The third and final case involves those mixtures in which one or more waters with a high buffering capacity are mixed with waters having a negligible or even a negative alkalinity. In the latter case the water is described as possessing acidity or base neutralizing capacity. Initially, H + ions from such water(s) will be neutralized by the buffer components of the high alkalinity water and the pH of the mixture will change slowly. Eventually the situation will be reached in which all the acid neutralizing capacity of the high alkalinity water has been consumed. At this point further addition of the acid water will cause the pH of the mixture to fall rapidly. This case is equivalent to the titration of a weak base with a strong acid (Vogel, 1962). A typical example would be the mixing of a well buffered lowland stream with a poorly buffered upland stream. The following derives equations describing all three cases and illustrates their use by means of numerical examples.

This follows from the definition of pH and the importance of calculating the mean H + ion concentration, given by: pH=-log[H

+] or

[ H + ] = 1 0 -pH.

(2)

Thus, for example, the mixing of equal volumes of acid waters of pH 3 and 5 would lead to a mixture having a pH of 3.3, not pH 4. For a mixture made up of equal volumes of two samples with pH, and pH2, respectively, the percentage error (based on H + ion concentrations) involved in taking the arithmetic mean pH of the mixture rather than using equation (1) can be shown to depend only on the value of (pH, - p H 2 ) . The percentage error is given by equation (3) and shown graphically in Fig. 1 for values of ( p H i - pH2) from 0 to 4. % error = 100 (Htrue - Harith)/Htrue = 100 {1 --[(2 × 10°'5(pH'-pH2)) (3)

/(l "b lO (pH' -pH2))]}

where Ht~e = 0.5 (10 -pH' + 10 -pH2) and Harith

~_.

10-0.5 (pHI + pH2)

From equation (3), it may be seen that there will be < 5% error involved in taking an arithmetic mean of pairs of pH values differing by up to 0.3 pH units. This fact, together with the uncertainty associated

100

CASE I--WATERS OF LOW BUFFERING CAPACITY

For waters of low buffering )uffering ccapacity, such as those ~cipitation and in acid surface involved in acid precipitation waters, the mean pHI of n samples sat mixed in equal proportions is given not by th, the arithmetic mean, n ~ (pH,),

eo

~: 60

but by:

40

p-H=--log[!k(lO-pH') ]. L

(la)

3

~:1

Where the waters are not mixed in equal proportions, a weighting factor must be applied to equation (la) for the relative volume (V) of each water in the mixture as follows:

20

0 0

m

pH = - log

k (V i. IO-pHO

i=l

i= 1

I

2

3

4

pHI" pH2

(lb)

Fig. 1. The percentage error in H + ion concentration associated with using the arithmetic mean to calculate the pH value of a mixture composed of equal volumes of two poorly buffered waters having pH~ and pH z, respectively.

Technical Note with the measurement of pH, particularly in poorly buffered waters (Warren et al., 1986), justifies the carrying out of run-mean smoothing and/or seasonal adjustment directly on the pH values concerned during trend analysis only if successive pH values do not differ by more than 0.3 pH units from each other. For larger pH changes, it is necessary to transform the pH values to H ÷ ion concentrations before analysis. CASE 2--WATERS OF H I G H BUFFERING CAPACITY

For natural waters of higher buffering capacity than those considered under Case 1, the situation is more complex as, here, the pH is often controlled by the carbonate buffer system. Typically, the pH of such waters ranges from around 6.0 to 9.0 depending on the concentration of the various components (CO2, HCO~- and CO 2-) of the buffer system. For carbonate buffered waters, the alkalinity which arises from the presence of the buffer components is related to the H ÷ ion concentration by equations (4) and (5), assuming no exchange of carbon dioxide with the atmosphere (Stumm and Morgan, 1981). Alkalinity = [HCO;] + 2[CO:3-] + [OH-] - [H +] = CT(~ + 2~:) + [OH-] -- [H ÷]

(4) (5)

where

1333

expressed in equiv 1-1, equation (7a) reduces to: pH - 11.3 + log (alk).

Equations (5) or (6) permit the evaluation of CT for any sample whose pH and alkalinity is known. When two carbonate buffered waters are mixed in equal proportions, the alkalinity and Cx value of the mixture will be the arithmetic mean of their respective alkalinities and Cr values. Substituting these mean values into equations (5) and (6) permits the calculation of the H + ion concentration, and hence the pH, for the mixture. In the following example, values in parentheses refer to the pH value the water samples would establish if they were allowed to equilibrate with the CO2 in the atmosphere. Thus, mixing equal volumes of a water of pH 8.00 (8.60), alkalinity 2.0 mequiv 1- l and, therefore, CT = 2.04 mM, with a water of pH 6.00 (8.00) and alkalinity 0.5 mequiv I(C T calculated to be 1.50 mM) will result in a mixture with an alkalinity of 1.25 mequiv 1-~, CT = 1.77 mM and a pH of 6.68 (8.40). This value should be compared with the arithmetic mean pH of 7.00 (8.30) and the mean pH of 6.30 (8.20) calculated from equation (!). For waters not mixed in equal proportions, the alkalinity (alk) and C r value (CT) of the mixture will be the volume (V) weighted mean of their respective alkalinities and CT values as follows:

~, Vi" alk~ al----k- i= l

--l-t"÷l+ I-' LK, l+T J

i=1

and

F[H+]2+ [H+] + l]-'. In equation (5), K 1 = 1 0 -6.3 M and K: = 1 0 - 1 0 3 M at 25°C and at ionic strengths up to 10-2 M (Stumm and Morgan, 1981). Over the pH range 5-8, the only significant contribution to the total alkalinity of the water arises from HCO~- ions, and equation (5) reduces to: _

r [H

+]

(7b)

~, Vi" CT~ and

CT = i=l'--2--~

(8)

i=1

These may be substituted into equation (6) as before to calculate the mean pH of the mixture. In order to calculate the pH of mixtures of surface waters of adequate buffering capacity accurately, it is necessary to know the pH and the alkalinity of each of the waters being mixed.

CASE 3 - - M I X T U R E S OF H I G H AND LOW BUFFER CAPACITY WATERS

The third and final mixture of fresh waters to be considered is that of a carbonate buffered water of comparatively high alkalinity ( > 0.5 mequiv 1- t) For alkalinities expressed in equiv i -1, Cr will be which is diluted with an acid water of low buffering expressed in mol 1-1. If the mixture is allowed to capacity and negligible ( < 50/~equiv 1-z) or negative equilibrate with the CO2 in the atmosphere, the final alkalinity. In the former case, the dilution effect of pH of the mixture may be estimated using equation adding the essentially zero alkalinity water will have (Ta) (Morel, 1983): no effect on the pH value of the high alkalinity water. This can be deduced from equation (6). For this case, [H +] -~ gl KH Pc°2__ (7a) the ratio a l k : C r will remain constant on dilution so alk [H ÷] and, therefore, the pH value of the mixture will (KHPco2) is the total analytical concentration of also remain constant. The latter case, in which diludissolved CO 2 in the mixture. tion takes place with a water of negative alkalinity is Typically, K H = 10 -~5 mol ! -I atm -1 Pco2 = 10-35 equivalent to adding strong acid to the carbonate atm (Stumm and Morgan, 1981) and for al----k buffered water. For every mmol of H ÷ added in this alkalinity ",, C'T L1--'0-S~.3+ 1] -1 •

(6)

1334

Technical Note

way, the alkalinity of the buffered water will be reduced by 1 mequiv although the ~"Cr' V value for the mixture will remain constant. The calculation of the mean pH of the mixture remains similar to that outlined for Case 2 above. Thus, for a buffered water of pH 7.08 with an alkalinity of 0.96mequivl -~ (CT = 1.12 mM) mixed in equal proportions with an acid water of pH 3.80 (0.16mmol H + 1-~), the mixture will have an alkalinity of 0.40 mequiv 1-~, a Cr of 0.56mM and a pH of 6.70 (or pH 7.90 if the mixture is allowed to equilibrate with the CO 2 in the atmosphere). The arithmetic mean pH and the mean pH calculated using equation (I) is 5.44 and 4.10, respectively. By mixing these waters in appropriate proportions, and excluding exchange of CO2 with the atmosphere, any desired pH in the range 3.80-7.08 could be achieved.

REFERENCES

HMSO (1982) The determination of alkalinity and acidity in water 1981. Methodsfor the Examination of Waters and Associated Materials. HMSO, London. Kelso J. R. M., Minns C. K., Gray J. E. and Jones M. L. (1986) Acidification of surface waters in eastern Canada and its relationship to aquatic biota. Can. Spec. Publ. Fish. Aquat. Sci. 87, 42p. Morel F. M. M. (1983) Principles of Aquatic Chemistry, pp. 142-t49. Wiley-Interscience, New York. Smith A. S., Alexander R. B. and Wolman M. G. (1987) Water quality trends in the nation's rivers. Science 235, 1607-1615. Stumm W. and Morgan J. J. (1981) Aquatic Chemistry, 2nd edition, pp. 171-222. Wiley-Interscience, New York. Vogel A. I. (1962) A Text-book of Quantitative Inorganic' Analysis, 3rd edition, pp. 68-72. Longmans, London. Warren S. C. et aL (1986) Acidity in United Kingdom fresh waters. U.K. Acid Waters Review Group Interim Report. Department of the Environment, London.