Calculation and Use 

of Ion Activity

GEOLOGICAL SURVEY WATER-SUPPLY PAPER 1535-C
Calculation and Use 

of Ion Activity

By JOHN D. HEM

GEOCHEMISTRY OF WATER

GEOLOGICAL SURVEY WATER-SUPPLY PAPER 1S35-C

Processes by which natural water 

acquires or relinquishes solutes, 

in relation to ionic activity

UNITED STATES GOVERNMENT PRINTING OFFICE, WASHINGTON : 1961
UNITED STATES DEPARTMENT OF THE INTERIOR 

STEWART L. UDALL, Secretary

GEOLOGICAL SURVEY 

Thomas B. Nolan, Director

For sale by the Superintendent of Documents, U.S. Government Printing Office 

Washington 25, D.C.
CONTENTS

	Page 

Abstract__ _-___--_--_____________________-..-_______-___-___--__- C-1

Chemical composition of natural water_______________________________ 1

Computation of ion-activity coefficients._____________________________ 2

Mass-action law_--__________________________________________ 2

Graphical aids in computation________________________________ 4

Ionic strength___________________________________________ 4

Activity coefficients______________________________________ 5

Dissociation of carbonic acid. ______________________________________ 6

Solution and deposition of calcite____________________________________ 7

Redox equilibria_________________________________________________ 15

Literature cited.__________________________________________________ 17

ILLUSTRATIONS

Page

PLATE 1. Nomograph for computing ionic strength of natural water.. In pocket 

FIGUBE 1. Relation of activity coefficients of ions to ionic strength of

solutions _____________________________________________ C-5

2. Equilibrium pH for calcium and bicarbonate activities-______ 15

TABLES

Page 

TABLE 1. Chemical analyses of ground water._______________________ C-8

2. Computation of undissociated carbon dioxide-______________ 10

3. Computation of equilibrium pH___._______________________ 12

in

GEOCHEMISTRY OF WATER

CALCULATION AND USE OF ION ACTIVITY

By JOHN D. HEM

ABSTKACT

A nomograph and a set of curves can be used to simplify the computation of 

activity from the measured concentration for constituents of natural water containing as much as 5,800 parts per million of dissolved solids. These devices 

are applied in mass-action computations of undissociated dissolved carbon 

dioxide, and computations of saturation status with respect to calcite in 16 samples 

of natural water whose dissolved solids content ranges from 91 to 4,170 ppm. 

When activity values are used to compute undissociated dissolved carbon dioxide, 

the value obtained is 5 to 25 percent less than when the computation is made 

without activity corrections. Computed saturation pH with respect to calcite is 

from 0.1 to 0.6 of a pH unit greater when activity corrections are made than when 

they are not.

Activity corrections are also used in computing redox potentials from dissolved 

iron and pH in natural water, but corrections are no more than 0.02 or 0.03 volt 

where ionic strength is 0.10 or less.

CHEMICAL COMPOSITION OF NATURAL WATER

Many of the processes by which natural water acquires dissolved 

impurities are reversible chemical reactions. The solutes come from 

the air, from minerals in the soil and rocks, and from vegetation or 

organic debris. The resulting solutions are normally dilute in the 

ordinary chemical sense. Opportunity for chemical equilibrium to be 

reached among solutes and minerals may occur, especially in ground 

water. Some reactions affecting natural-water composition are irreversible, and never reach equilibrium. If a reversible reaction proceeds slowly a water may retain metastable concentrations of some 

ions for a long time.

The processes by which natural water acquires or relinquishes solutes 

are complicated by many factors that seldom can be adequately 

evaluated. Thus, although techniques have been available for many 

years for the exact evaluation of the status of reversible reactions under 

laboratory conditions, the unknown, and perhaps unknowable factors 

in natural-water systems have discouraged the use of these evaluation

techniques in natural-water chemistry.

C-l
C-2 GEOCHEMISTRY OF WATER

In this report several methods are given for simplifying the calculation of ion activity from standard water analyses. The calculation 

of activity is a first step in the quantitative application of the massaction law to natural-water chemistry. Examples of applications of 

the ion-activity data are cited, to show the magnitude of corrections 

introduced by the activity calculation, and to test, in a preliminary 

way, chemical equilibria in simple systems that can occur in ground 

water. These applications are made by using available generalpurpose chemical analyses.

The systems dealt with in natural-water chemistry are complex and 

the ordinary water analysis may not be an accurate representation of 

all the properties of the solution that should be known to apply the 

mass-action law and related chemical calculations. Discussions presented in this report on the equilibria among solids and solutes in 

ground-water systems do not imply ignorance of the complex factors, 

nor do they minimize such factors as unimportant. However, progress 

in the understanding of natural-water chemistry will be served by 

applying mass-law and related computations where some possibility of 

approximate equilibrium exists. The results may help evaluate the 

limitations that such computations have, as well as their potential 

usefulness, and show how that usefulness can be extended and improved.

COMPUTATION OF ION-ACTIVITY COEFFICIENTS

MASS-ACTION LAW

The mass-action law states that the rate of a chemical reaction is 

proportional to the active masses of the reacting substances. A 

reversible reaction ultimately reaches a state of equilibrium at which 

the rates of the forward and reverse reactions are equal. When the 

reaction

(where the reactants and products are represented by capital letters 

and the number of moles of each by the lowercase letters) is at 

equilibrium, the law can be stated

where quantities in brackets represent active concentration (activity) 

of the reactants and products and K is the equilibrium constant.

The activity of the dissolved ions in solutions of moderate dilution, 

such as potable natural water, is somewhat less than the concentration 

of the ions measured and reported in water analyses. To apply the 

mass-law computation accurately to such solutions, a factor should
CALCULATION AND USE OF ION ACTIVITY C~3

first be computed by which the reported concentration may be multiplied to yield active concentration or activity. This factor is the 

activity coefficient.

In most of the literature where mass-action calculations are applied 

to water chemistry, an activity coefficient of unity has been assumed. 

This assumption introduces no significant error at very low concentrations. The concentration range of potable water, however, may be 

above the level where unit activity coefficients can be accurately 

assumed. A limit of 500 or 800 ppm is sometimes given as the 

maximum for reasonably accurate application of mass-action calculations without activity corrections.

Actually, for the concentration range of water that is dilute enough 

to be of practical value as a water supply, activity coefficients can be 

simply and accurately computed using an empirical statement of the 

Debye-Hiickel law of interionic attraction which has been widely used 

in studying solutions of low to moderate concentrations. The activity 

coefficient T< for an ion in such a solution is given by the expression

 Azt2 VM

where

A is a constant relating to the solvent whose value for water at 25 °C

is 0.5085

Zi is the charge of the ion 

B is a constant relating to the solvent, which for water at 25°C

is 0.3281 X108 

a* is a constant relating to the effective diameter of the ion in solution 

ju is the ionic strength of the solution

Values for A, B, and a< for different temperatures and common ions 

are given by Klotz (1950, p. 330-331) who selected them from published literature.

The ionic strength of a sample of natural water is computed from 

the expression

2

This equation may also be expressed

mjSi2 M= s r

where /i is the sum of a series of terms each of which is divided by 2. 

In both equations, m-=molar concentration of a kind of ion in the
C-4 GEOCHEMISTRY OF WATER

solution, and z is the charge on that ion. Most of the ionic strength 

of natural water will generally come from 2 to 6 ions.

GRAPHICAL AIDS IN COMPUTATION

IONIC STRENGTH

The computation of ionic strength, ju, and the values of the activity 

coefficient, 7, for the ions reported in a large number of water analyses 

can be tedious. Plate 1 is a nomograph that shortens the process of 

calculating the ionic strength from water analyses in parts per million.

The right- and left-hand scales in plate 1 are identical and represent

rtyj ^

the value of ~- increments to the total ionic strength corresponding

to the concentration of individual ions given on other scales in parts 

per million. To use the nomograph, a transparent straight edge is 

laid on it horizontally, intersecting the concentration value given in 

the analysis for a particular ion. That ion's contribution to the total 

ionic strength is read on either the right- or left-hand scales and 

recorded. The process is repeated for other ions present in amounts 

sufficient to affect the total ionic strength significantly, and the total 

for ju is then determined by adding the increments together. The 

nomograph gives results accurate to about 2 significant figures and 

covers the concentration ranges of the substances commonly found in 

natural water up to an ionic strength of about 0.10.

A part per million of magnesium contributes more to the total ionic 

strength of the solution than an equal amount of any other cation 

given in plate 1. Some cations commonly present in minor amounts 

such as Al+3 or H+, however, have greater effects on the ionic strength. 

Of the anions given in plate 1, carbonate and fluoride are the most 

effective in increasing the ionic strength, but these anions rarely are 

present in major amounts. Sulfate is the most effective of the common 

anions.

The expression for computing activity coefficients is applicable up to 

an ionic strength of 0.1. A solution of magnesium sulfate with 3,100 

ppm dissolved solids is near this value. However, when other ions 

predominate, a much greater amount of dissolved solids can give the 

same ionic strength. For water containing dissolved sodium chloride, 

for example, about 5,800 ppm total dissolved solids gives a value of 

ju as 0.10; for water containing sodium bicarbonate, the total concentration would be 8,400 ppm. In computing or determining the 

total dissolved solids for such water, however, by conventional procedure, the bicarbonate would be reported in terms of equivalent 

amounts of carbonate, so the dissolved-solids. value for this water 

would be given as 5,300 ppm.
CALCULATION AND USE OF ION ACTIVITY C-5

ACT COEFFICIENTS

After the ionic strength, M, has been computed, figulre 1 can be used 

to give the activity coefficient for each ion in the solution. Figure 1 

was prepared from the mathematical expression for 7 (s|ee p. C-3) using 

values for the constants given by Klotz. Similar gitaphs have been 

published by many authors.

A value for 7 greater than about 0.95 is close t|o unity, and it 

normally is not worth while to change measured concentration to 

activity where the correction is so small. For wjater containing 

sodium chloride of ionic strength 0.002 (about 11^ ppm dissolved 

solids) or less, the 7 value for sodium and chloride is more than 0.95. 

For water containing magnesium sulfate however, ithe 7 value for 

magnesium and sulfate will only be 0.95 if the ionic strength is below 

0.0001 (dissolved-solids concentration less than 31 pppm).

For many purposes, an uncertainty in the final result greater than 5 

percent could be tolerated. However, the activity coefficient for

o.io

0.01

o.ooi

0.0001

0.00001 ^

0.8 0.7 0.6 0.5 0.4 0.3 

ACTIVITY COEFFICIENT ( y )

0.2 0.1

FIGURE 1. Relation of activity coefficients for important ions in natural 4-ater to ionic strength of

solutions.
C-6 GEOCHEMISTRY OF WATER

divalent ions in water of 0.01 ionic strength is shown by figure 1 to 

be from 0.65 to 0.70. An ionic strength of 0.01 is equivalent to 300 to 

500 ppm dissolved solids, and is a common range for natural-water 

concentrations.

The following computations represent simple applications of the 

mass-action law to water analyses. The effect on final results of 

using or ignoring activity corrections will be indicated. In all the 

applications made in this report, the formation of complex ions in 

solution is not considered.

DISSOCIATION OF CARBONIC ACID

The mass-action law provides a convenient approach to a study of 

the behavior of dissolved carbon dioxide present in natural water. 

Many reports have been published on this topic. Dissociation constants for carbonic acid, pH values, and titrated "alkalinity" (as this 

term is used in water analysis) provide a basis for computing the 

amount of carbon dioxide that may be in solution in an undissociated 

form (Moore, 1939). Activity corrections make this computation 

valid for higher dissolved-solids concentrations than the 500 to 800 

ppm dissolved solids sometimes given as the top limit by some authors.

For convenience dissolved undissociated carbon dioxide is often 

written as H2CO3. Actually, only a small part of the dissolved carbon 

dioxide is hydrated. However, published dissociation constants for 

carbonic acid are actually to be applied to total dissolved carbon 

dioxide unless the value is specifically stated to apply only to the 

hydrated part. For the dissociations

and

the values for equilibrium constants K\ and Kz, given by Latimer 

(1952, p. 135) are 4.16 X10~ 7 and 4.84 XIO" 11, respectively. Therefore

[H2C03] 

and

[HCO,-]

In the second equilibrium, where the pH is less than about 8.3, the 

value of the ratio [CO3]:[HCO3] is less than 0.01. Consequently, 

below pH 8.3, the amount of depletion of the dissolved carbon dioxide
CALCULATION AND USE OF ION ACTIVITY C~7

resulting from the second dissociation is a relatively minor factor. 

For the water analyses in table 1, the equation

will yield satisfactory values for dissolved undissociated carbon 

dioxide. Two sets of computed values for dissolved undissociated 

carbon dioxide are given in table 2. One set was obtained by substituting measured concentrations of HCO3~ from the analysis in the 

equation. The other set was obtained by using determined HCO3~ 

and a computed activity coefficient. In computing the second value 

for dissolved carbon dioxide, the activity coeficient enters the numerator of the right-hand side of the equation, and results in a lower 

computed value for dissolved undissociated carbon dioxide. The 

maximum correction amounts to 25 percent at ionic strength 0.11, 

and is relatively minor for the more dilute water samples of table 1. 

The computed carbon dioxide value is the activity of dissolved undissociated carbon dioxide in the solution; but for undissociated 

species, an activity coefficient of unity can be assumed here. Hence, 

the computed value can be taken as the molal concentration.

If the pH of the solution is above about 8.3, the second dissociation 

must also be considered in computing the activity of undissociated 

dissolved carbon dioxide.

Calculation of the amount of undissociated dissolved carbon dioxide 

in laboratory samples is readily possible when the temperature is near 

25 °C, and pH and bicarbonate ion concentration are measured at the 

same time, provided enough information is available to calculate 

ionic strength. Application of laboratory data to field conditions, 

however, may not be valid because of changes that can occur in the 

chemical system during transfer from field to laboratory.

SOLUTION AND DEPOSITION OP CALCITE

Another computation of great interest in natural-water chemistry 

relates to the equilibrium between carbonate minerals in contact with 

water containing H+, and the dissociated ions in solution that result. 

A water analysis provides information relating to the equilibrium

Langelier (1936) suggested a computation for evaluating the stability of water that had been softened in water-treatment plants that 

used this equilibrium. The equilibrium is also of interest in studies 

of the chemistry of natural water which may be associated with 

carbonate minerals. From free-energy data, a value of 0.97 X102 is
TABLE 1. Chemical analyses of ground wa

[Analyses, in parts per million, by U.S. Geological Surv

Constituent or property

Silica (Si02).~. . 

Bicarbonate (HCOj).... .

Sulfate (S04).. - 

Chloride (Cl)_ .. . 

Fluoride (F)..... ............

Nitrate (NOs) . - -

Specific conductance (mipH. ...-..- .-.-..... ...

Analyses of samples, at indicated dat

Apr. 14, 

1955

1

6.7 

.0

.29 

.15

26 

1.1 

.8 

.2 

76 

0 

4.3 

2.0 

.1 

4.0 

.0

91 

79

7

135

7.5 

59

May 15, 

1946

2

5.6

.04

38 

2.1 

1.7 

.4 

118 

0 

.9 

2.5 

.0 

5.7

115 

103 

6

205 

7.5 

70

Mav 24, 

1954

3

8.6

.05

48 

3.6

}"

152 

0 

3.2 

8.0 

.0

.0

135 

10

269 

7.5 

62

Mar. 18, 

1955

4

62 

.0

.39 

.02

74 

17 

9.1 

2.8 

327 

0 

.1 

5.4 

.6 

1.5 

.0 

347 

255 

0

518 

7.5 

62

Sept. 2, 

1952

5

11

.81

100 

19 

20 

1.9 

281 

0 

69 

31 

.0 

17

425 

328 

98

633 

7.2 

68

May 23, 

1949

6

8.0

.65

74 

28

} 37 

338

90 

5.0 

.3 

1.2

440 

300 

23

8.2 

52

Feb. 21, 

1957

7

20 

.2 

.25 

.00 

79 

42 

19 

3.0 

418 

0 

35 

9.0 

.3 

22 

.00 

412 

370 

27

724 

7.5 

55

May 23, 

1955

8

16 

.2 

5.2 

.14 

101 

46 

10 

1.4 

524 

0 

26 

7.0 

1.2 

.6 

.0 

465 

443 

13

791 

7.3 

59

Oct. 23, 

1950

9

14

146 

16

}»

262 

0 

197 

37

.0

588 

430 

216

871 

7.6

Jan. 16

1952

10

12

1.

112 

63 

47 

21 

590 

0 

65 

66 

.1 

14

689 

540 

55

1,150 

7.5
1 Source of data: Samples 1, 3, 4, 7, 8, 11-14, 16, unpublished data, U.S. Geological 

Survey flies; sample 2, Ferguson and others (1947); Sample 5, Robinson and others 

(1953); sample 6, Mozola (1951); sample 9, Petitt and George (1956); sample 10, Drescher 

and others (1953); sample 15, Stramel and others (1954).

2 Calculated value.

SAMPLE 1. Spring, NWM sec. 23, T. 9 S., R. 7 E., Collinsville, DeKalb County, Ala. 

Water-bearing formation: Bangor limestone; discharge: 3.77 cfs.

2. Blue Springs about 5 miles northeast of Marianna, Jackson County, Fla. 

Water-bearing formation: limestone; discharge: average 158 cfs.

3. Huntsville Big Springs, SWH see. 36, T. 3 S., R. 1 W., Huntsville, Madison 

County, Ala. Water-bearing formation: Tuseumbia limestone.

4. Drilled well, B-84, Bath, Beaufort County, N.O. Water-bearing formation: Castle Hayne limestone from 140 to 145 feet.

5. Drilled well, Birmingham, Jefferson County, Ala. Water-bearing formation: Conasauga limestone.

6. Drilled well, Se 175, Seneca County, N.Y. Water-bearing formation: 

Onondaga limestone; discharge: 10 gpm.

7. Well, SEM sec. 17, T. 7 S., R. 4 E., Green, Clay County, Kans. Water 

bearing formation: limestone in Barneston formation from 140 to 170 

feet; discharge: 16 gpm pump estimated.

8. Drilled well, SWJ^sec. 10, T. 3 N., R. 6 E., O.D.W.-M.T.-122, at Vandalia, 

Montgomery County, Ohio. Water-bearing formation: Niagara limestone; discharge: 100 gpm pump estimated.

9. Well, H-5 162

limestone.

10. Drilled well 

Waukesha 

11. Drilled indus

cia County, 

San Andres 

gpm pump 

12. Well, NEM s

N. Mex. W

gpm estima

13. Well, SEMN

Water-bear

charge: 1 gp

14. Blue Spring, 

County, A

sippian); di

15. Well 4, Grand 

Sandstone 

16. Drilled well, 

County, In

feet; dischar
C-10 GEOCHEMISTRY OF WATER

TABLE 2. Computation of undissociated carbon dioxide

Sample

1. ... .... .... .

2 .... .....

3- 

4. ... ... ..... .

5 

6 

7- 

8. ..... .

9-. ... ... ... ..

10 

11. 

12- 

13. 

14. _ 

15 

16. 

Ionic 

strength,

M

0.0022 

.0030 

.0040 

.0078 

.011 

.011 

.012 

.014 

.016 

.019 

.025 

.029 

.040 

.052 

.052 

.11

HCO3

Equivalents per 

million

1.246 

1.934 

2.491 

5.36 

4.61 

5.54 

6.85 

8.59 

4.29 

9.67 

5.92 

4.26 

6.16 

15.80 

5.87 

2.95

(Moles per 

liter)

1.2X10-3 

1.9X10-3 

2.5X10-3 

5.4X10-8 

4.6X10-3 

5.5X10-3 

6.8X10-8 

8.6X10-3 

4.3X10-3 

9. 7X10-3 

5. 9X10-3 

4.3X10-8 

6.2X10-3 

1.6X10-2 

5.9X10-3 

3.0X10-3

Activity coefficient, t

0.69 

.95 

.94 

.92 

.90 

.90 

.89 

.88 

.87 

.87 

.86 

.85 

.84 

.82 

.82 

.77

Activity 

(moles per 

liter)

1.2X10-3 

1.8X10-3 

2.4X10-3 

5.0X10-3 

4. 1X10-3 

5.0X10-3 

6.0X10-3 

7. 6X10-3 

3.7X10-3 

8.4X10-3 

5. 1X10-' 

3. 7X10-3 

5. 2X10-3 

1. 3X10-2 

4.8X10-' 

2.3X10-3

CO2 (ppm)

From 

concentration

4.2 

6.6 

8.5 

18 

31 

3.7 

23 

46 

11 

33 

6.3 

9.0 

26 

536 

62 

16

Prom 

activity

4.0 

6.3 

7.9 

17 

28 

3.3 

21 

40 

9.9 

29 

5.4 

7.7 

22 

440 

51 

12

pH

7.5 

7.5 

7.5 

7.5 

7.2 

8.2 

7.5 

7.3 

7.6 

7.5 

8.0 

7.7 

7.4 

6.5 

7.0 

7.3

obtained for Kes for the calcite equilibrium. Aragonite is somewhat 

more soluble, and the equilibrium constant for the equilibrium involving this mineral is 1.48X102 . At normal temperature and pressure in 

ground-water environments, aragonite would therefore be preferentially dissolved if the water were saturated with respect to calcite and 

calcite would be preferentially precipitated if the water became 

saturated with respect to aragonite. Accordingly, calcite is likely to 

be much more common in aquifers than aragonite.

If calcite is present in an aquifer, and if the calcite is relatively pure, 

unit activity may be assigned to the mineral and the following relation 

assumed

"log [Ca+2] [HCO3 -

-=pH (calculated)

If the ground water is in equilibrium with calcite, the calculated pH 

value should closely agree with the actual value in the water.

For conducting an equilibrium experiment in the laboratory under 

controlled conditions, methods could be devised to determine pH and 

calcium and bicarbonate contents of solutions with assurance that the 

measurements represented actual conditions in the system. Obtaining 

measurements of equal dependability on ground water in its native 

state, however, is a more difficult problem. The calcium and bicarbonate contents of most ground water are relatively stable, and 

samples of the water taken and handled with ordinary care will give 

values on analysis that can be safely assumed to represent the composition of water in the aquifer. The same cannot always be said for the
CALCULATION AND USE OF ION ACTIVITY C~ll

measured pH. When the pH measurement is made carefully in the 

field at the time samples are obtained, it should represent reasonably 

well the pH of the water in the aquifer. When the pH measurement 

is made on the sample several days or weeks after collection, however, 

the value obtained may differ from that of the water in its natural 

surroundings.

The hydrogen-ion activity of most natural water is a small quantity 

in comparison with the dissolved constituents determined in the normal water analysis. At pH 7, for example, the hydrogen-ion activity 

is equivalent to only 0.0001 ppm H+. The degree to which this 

quantity can be held constant until a measurement can be made is 

uncertain. In some water, the pH is stable, in other water it is 

unstable. During research by the Geological Survey from 1955 to 

1957, a large number of field pH determinations were made by 

experienced personnel with accurate equipment. Laboratory pH 

measurements were made later on the samples collected at the same 

time as the field measurements were made.

A comparison of 165 sets of field and laboratory pH data shows 

that the average difference is about ±0.3 pH unit. Some of the 

water was unstable, and field and laboratory pH values for these 

samples differed by 1.0 pH unit or more. However, the unstable 

water was from geologic sources in which calcium carbonate was in 

very short supply; hence, the water did not resemble samples whose 

analyses are given in this report.

Analyses in table 1 represent water having a wide range of dissolvedsolids concentration and the samples come from many different 

geologic sources in the United States. With one exception, however, 

the samples all come from limestone or dolomite. In table 3, computed pH values for the water are compared with the values determined in the laboratory. One computed pH value was prepared by 

using the reported Ca+2 and HCO3 ~ concentrations from the analysis; 

another computed value was obtained by using concentrations 

corrected by activity coefficients.

The activity coefficient, % increases the computed pH by the amount

ApH= log Yca+2 YHCOS"

In the examples shown in table 3, A pH is from 0.1 to as much as 0.6 

unit. The pH computed with the aid of activity coefficients comes 

nearer agreement with the observed pH than the uncorrected computed 

value in every sample except the 3 most dilute: sample 1 had a low 

bicarbonate content, and possibly an unstable pH; and agreement of 

all 3 values was close for samples 2 and 3.
TABLK 3. Computation of equilibrium

Sample

! . __ ... _ 

2...- 

3 

4...... ... ... ...

5... ... 

6 

7 

8 

9.. ___

10. - 

11 ___ -

12... 

13 

14.... _

15.. 

16 

Ionic 

strength, 

/ »

0. 0022 

.0030 

.0040 

.0078 

.011 

.011 

.012 

.014 

.016 

.019 

.025 

.029 

.040 

.052 

.052 

.11

Oa

Equivar 

lents per 

million

1.297 

1.896 

2.395 

3.69 

4.99 

3.69 

3.94 

5.04 

7.29 

5.59 

7.09 

8.48 

14.57 

13.17 

22.95 

21.46

Concentration (moles 

per liter)

6. 5X10-* 

9. 9X10-4 

1.2X10-3 

1.8X10-3 

2.5X10-3 

1.8X10-3 

2.0X10-3 

2.5X10-3 

3. 6X10-3 

2.8X10-3 

3.5X10-3 

4. 2X10-3 

7. 3X10-3 

6.6X10-3 

1. 2X10-2 

1.07X10-2

7

0.82 

.79 

.77 

.70 

.67 

.67 

.66 

.64 

.62 

.61 

.57 

.56 

.51 

.48 

.48 

.39

Activity 

(moles per 

liter)

5. 3X10-1 

7.8X10-* 

9. 2X10-* 

1. 3X10-3 

1.7X10-3 

1.2X10-3 

1.3X10-3 

1.6X10-3 

2. 2X10-3 

1.7X10-' 

2.0X10-3 

2.4X10-3 

3.7X10-3 

3. 2X10-3 

5.8X10-3 

4.2X10-3

HCOs

Equivalents per 

million

1.246 

1.934 

2.491 

5.36 

4.61 

5.54 

6.85 

8.59 

4.29 

9.67 

5.92 

4.26 

6.16 

15.80 

5.87 

2.95

Concentration (moles 

per liter)

1. 2X10-3 

1.9X10-3 

2.5X10-3 

5.4X10-3 

4.6X10-3 

5. 5X10-3 

6. 8X10-3 

8.6X10-3 

4.3X10-3 

9.7X10-3 

5. 9X10-3 

4. 3X10-3 

6.2X10-3 

1.6X10-2 

5.9X10-3 

3.0X10-3

7

.96 

.95 

.94 

.92 

.90 

.90 

.89 

.88 

.87 

.87 

.86 

.85 

.84 

.82 

.82 

.77

Activity 

(moles pe

per liter)

1.2X10-3 

1. 8X10-3 

2. 4X10-3 

5.0X10-3 

4.1X10-3 

5.0X10-3 

6. 0X10-3 

7. 6X10-3 

3.7X10-3 

8. 4X10-3 

5.1X10-3 

3.7X10-3 

5. 2X10-3 

1.3X10-2 

4. 8X10-3 

2.3X10-3
CALCULATION AM) USE OF ION ACTIVITY C~13

Differences between measured pH and computed pH on the order 

of 0.3 pH unit could be expected to result solely from possible changes 

in pH during the transfer from field to laboratory. Larger differences 

from the same cause could sometimes occur; accordingly, for data of 

the kind given in tables 1 and 3, differences between measured and 

computed pH should be interpreted with caution.

Although the activity coefficient correction may sometimes be 

minor compared to the uncertainty introduced by assumptions 

regarding the effects of possible environmental variables, the activity 

coefficients eliminate questions as to possible error that is introduced 

when their value is assumed to be unity.

The data in table 3 show how the water samples would react at 

the tune of analysis if they were brought in contact with calcite of 

the composition CaCO3 . If the data are assumed to be representative 

of conditions in the aquifer (this may not always be a safe assumption) 

some interesting speculations about the relation between the chemical 

constituents in the aquifer are possible. Calcite very probably is 

present in the aquifers represented in table 1; but this also has been 

assumed without positive proof. In studies where the chemical 

analyses of water were planned to be used for computation of equilibrium status with respect to calcite, field pH data should be obtained 

and samples of aquifer material should be examined for calcite.

Because of possible uncertainties in the pH measurements used in 

this discussion, equilibrium is assumed to exist when the measured 

and computed pH values agree to ±0.5 unit. With better data, 

this range could probably be decreased. Analysis 1 seems to represent 

undersaturation, and analyses 6 and 10-13 are supersaturated with 

respect to calcite. The activity coefficient correction brings computed 

pH values for analyses 7-9, and 15-16 into close agreement with the 

measured value. Uncorrected pH values for these five samples also 

would have indicated supersaturation.

Analysis 1 represents water from a humid region and an aquifer 

where water probab y circulates rapidly. In such a system, the 

ground water could be discharged so rapidly that there would not be 

time for equilibrium with respect to calcite to be reached.

Analyses that seem to show a great degree of supersaturation with 

respect to calcite could precipitate calcium carbonate from solution, a 

significant fact in hydrology and geology. The water of analysis 6 has 

an equilibrium pH a full unit below the observed value. To reach 

equilibrium, about half the calcium and an equivalent cmount of 

bicarbonate would have to precipitate as calcite. From 1,000 gallons 

of this water, nearly 1 pound of calcium carbonate precipitate would 

be obtained. A sudden change in pressure head or other alteration 

in the environment of the water could cause at least some calcium
C-14 GEOCHEMISTRY OF WATER

carbonate to precipitate. When water moves into the area of influence of a pumping well, a marked pressure differential can take 

place, and this could explain calcium carbonate deposits on well 

screens or in the water-bearing material outside the well bore. Such 

deposits may seriously interfere with the production of water. Possibly, such deposits also form when unstable water is left standing in 

the well bore after the pump is shut down.

Changes in pH as calcium bicarbonate water saturated with respect 

to calcite moves into a noncalcareous rock formation can explain the 

commonly observed cementing of sandstones with calcite.

If the solutes in water are in equilibrium with a solid phase that is 

not pure calcite, the assumptions made in computing the equilibrium 

pH are not entirely valid. If the solid phase is dolomite, as in 

analysis 10, an equilibrium constant applicable to dolomite or a 

dolomite-calcite mixture should be used. Values for K for such an 

equilibrium, however, are not readily available.

The apparent metastability of surface water that was supersaturated 

with respect to calcite has been noted in the literature of limnology. 

Ohle (1952) and other authors have attributed this to a colloidal or 

undissociated form of calcium carbonate in solution. The threshold 

effect of small amounts of polyphosphates in water treatment can also 

make solutions metastable. The threshold effect is generally attributed to surface activity of the added polyphosphate which prevents 

the calcium carbonate particles from growing large enough to form a 

precipitate.

Alekin and Moricheva (1957) have discussed apparent supersaturation of the water of major rivers of Russia and attribute the apparent 

stability of the water for periods as long as 30 days to threshold effects 

exerted by organic matter.

Calcium carbonate tends to precipitate from the water in some 

reservoirs. The effect is particularly marked in Lake Mead, in which 

water of the Colorado River is held in storage. The capacity of the 

reservoir is so large that the average residence time of the water may be 

several years. Evidently, the precipitation of calcium carbonate can 

be slow under some circumstances. Evidence for calcium carbonate 

(aqueous phase) or for threshold effects being involved in not conclusive.

Figure 2 may be used to estimate equilibrium values for the pH of 

natural water that is in contact with calcite. To use the diagram, 

ionic strength of the water and activity coefficients for calcium and 

bicarbonate ions should be computed first, by using plate 1 and figure 

1. The activity coefficients then should be applied to the reported 

concentrations of calcium and bicarbonate in the water analyses to 

give activities of these ions.
CALCULATION AND USB OF ION ACTIVITY C-15

1000

500

400

300

200

z 

o

i 100

cc

UJ 

OL.

{2 50 

S 40

^

°- 30

z

~- 20

<3

° 10

>

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3

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1.0

1.0

ACTIVITY OF HC03 - , IN PARTS PER MILLION 

2345 10 20 304050 100 200300 500 1000 2000

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y

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 N

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FIGURE 2. Equilibrium pH for calcium and bicarbonate activities from 1.0 to 2,000 ppm in solutions in 

contact with calcite. Pressure=l atmosphere; temperature=25° O.

The foregoing discussion has outlined a method for determining 

whether water in a given environment is undersaturated or supersaturated with respect to calcite. Two kinds of information, which 

have not generally been obtained in water-resources investigations, 

are required to utilize the method satisfactorily. If (a) accurate field 

pH data are obtained, and (b) mineralogic studies of the minerals associated with the water show that calcite reasonably close to CaCo3 in 

composition is present, then the saturation computation can be applied 

geologically and hydrologically. For example, the degree of saturation should be lowest in recharge areas, and should increase as water 

movers through the aquifer.

REDOX EQUILIBRIA

Another type of chemical equilibrium in natural water is that involving oxidation and reduction. Theoretical evaluations of such 

equilibria are based on the Nernst equation. The author has used 

this treatment in discussions of the chemistry of iron in natural water
C-16 GEOCHEMISTRY OF WATER

(Hem and Cropper, 1959; Hem, 1960), and details are not repeated 

here.

In computing the redox potential (Eh) of ground water from the pH 

and iron content, the Nernst equation takes the form,

Eh=£»+0.059 log g «£ 3pedesl1 

65 [Fe+2 species]

where E° is the standard potential of the couple considered. 

The value for activity of ferric species is obtained from the pH by 

computation, and no correction by activity coefficient is involved in 

this term. The ferrous concentration, however, is subject to correction because it is obtained from the water analysis results. The 

activity coefficients for ferrous iron are the same as those for calcium 

(fig. 1). The Nernst equation tends to minimize the effect of activity 

corrections, however. In calculating the Eh value based on the ferricferrous equilibrium, the correction amounts to only +0.02 volt for the 

sample in table 1 containing the largest amount of dissolved solids. 

Accordingly, it seems that activity corrections are comparatively 

small in calculations of Eh from iron content and pH according to 

techniques suggested by Hem (1960). Some uncertainty in the calculation of the Eh of natural water may be introducted by ignoring 

possible carbonate or bicarbonate complexes, but no data on such complexes are presently available.

In redox equilibria involving other than ferrous and ferric ions, the 

effects of the activity correction would be different. The correction 

to Eh resulting from the use of activity coefficients (AEh) will be

log ToxMatlon

Treductlon

where n is number of electrons involved in the oxidation. 

If n is greater than 1, the effect of the activity correction becomes less 

important. If Todd^on is greatly different from y^^^ the value of 

AEh will be greater. The maximum value AEh could attain in soluutions up to 0.10 ionic strength, however, is not much over 0.03 volt, 

because when rotation differs greatly from y^^io* the value of n will 

generally be much greater than 1.
CALCULATION AND USE OF ION ACTIVITY C-17

LITERATURE CITED

Alekin, O. A., and Moricheva, N. P., 1957, The problem of the stability of the

carbonate system in natural water: Doklady Akad. Nauk. SSSR [Geochem.

Sec.], v. 117, p. 1030. 

Drescher, W. J., Dreher, F. C., and Brown, P. N., 1953, Water resources of the

Milwaukee area, Wisconsin: U.S. Geol. Survey Circ. 247, p. 32. 

Ferguson, G. E., Lingham, C. W., Love, S. K., and Vernon, R. O., 1947, Springs

of Florida Blue Springs: Dept. Conserv., Florida Geol. Survey, Bull. no. 31,

p. 87. 

Hem, J. D., 1960, Restraints on ferrous iron imposed by bicarbonate, redox

potential, and pH: U.S. Geol. Survey Water-Supply Paper 1459-B, p. 33-56. 

Hem, J. D., and Cropper, W. H., 1959, Survey of ferrous-ferric chemical equilibria

and redox potentials: U.S. Geol. Survey Water-Supply Paper 1459-A

p. 1-31. 

Langelier, W. F., 1936, The analytical control of anticorrosion water treatment:

Am. Water Works Assoc. Jour., v. 28, p. 1500-1521.

Latimer, W. M., 1952, Oxidation potentials: New York, Prentice-Hall, 392 p. 

Klotz, I. M., 1950, Chemical thermodynamics: New York, Prentice-Hall, p.

329-332. 

Moore, E. W., 1939, Graphic determination of carbon dioxide and the three forms

of alkalinity: Am. Water Works Assoc. Jour., v. 31, p. 51-66. 

Mozola, A. J., 1951, The ground water resources of Seneca County, New York:

New York Dept. Conserv., Water Power and Control Comm., p. 31. 

Ohle, W., 1952, Die hypolimnische Kohlendioxyde-Akkumulation als productions

biologischer indikator: Arch. Hydrobiol., v. 46, p. 153-285. 

Petitt, B. M., Jr., and George, W. O., 1956, Ground water resources of the San

Antonio area, Texas: Texas Board Water Engineers, Bull. 5608, v. 2, pt. 3,

p. iv-6. 

Robinson, W. H., Ivey, J. B., and Billingsley, G. A., 1953, Water supply of the

Birmingham area, Alabama: U.S. Geol. Survey Circ. 254, p. 47. 

Stramel, G. J., Wisler, C. O., and Laird, L. B., 1954, Water resources of the

Grand Rapids area, Michigan: U.S. Geol. Survey Circ. 323, p. 26.

O