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ISSN: 0973-4945; CODEN ECJHAO

E-Journal of Chemistry

http://www.e-journals.net 2011, 8(2), 657-664

Calculation of pH Values for Mixed Waters

QIAN HUI*, SONG XIULING, ZHANG XUEDI, YANG CHAO and LI PEIYUE

School of Environmental Science and Engineering

Chang’an University, Xi’an, 710054, China

[email protected]

Received 7 September 2010; Accepted 6 November 2010

Abstract: Mixing of waters with different compositions is a common

phenomenon. The pH of mixed water can be calculated by introducing charge

neutrality equation into the equations for equilibrium distribution calculation

of species in water. In this paper, the equations thus obtained were solved by

golden section method. To verify the calculation method, laboratory

experiments were done for three sets of mixing water. The results showed the

calculated pH values are in good agreement with measured ones.

Keywords: Mixing of different waters, pH calculation, Experimental verification

Introduction

The pH is an important property of natural water. It has important effects on the

concentration of elements and distribution of species in water as well as the water-rock

interaction1,2

. In hydrogeological studies, for purpose of deeply understanding various

geochemical processes, the saturation indices of minerals in water under different conditions

need to be accurately calculated. This in turn requires the accurate calculation of pH. When

two waters of different composition are mixed, how will the pH of mixed water change?

This is the key question in understanding hydrogeochemical processes caused by mixing.

Many researchers have studied the calculating methods of pH values. Jordan3 presented

the calculating methods of pH for the mixtures of fresh waters under three cases. Hunter4

gave a method to calculate the temperature dependence of pH in surface seawater as a

function of salinity and CO2 composition. Partanen and Minkkinen5 discussed the methods

for calculation of the pH of buffer solutions containing sodium or potassium dihydrogen

phosphate, sodium hydrogen phosphate and sodium chloride. Bi Shuping6 proposed a simple

computer model for predicting the pH values of acidic natural waters. Piedrahita7 developed

a procedure for the calculation of pH in fresh and salt waters. For more general cases, pH

values of water under different conditions can be calculated by introducing a new equation

into the mass action and mass balance equations. In this regard, Plummer et al.8 introduced

658 QIAN HUI et al.

charge neutrality equation. Arnorsson et al.9 introduced the mass balance equation of ionic

hydrogen, while Reed and Spycher10

introduced mass balance equation of all hydrogen ions.

No matter what equation is introduced, the main purpose is to make the number of equations

equal to the number of unknowns. Although each of the methods used by the above

researchers can be employed to calculate pH values of mixed water, all those methods are

rarely verified by experimental data. In this paper, the charge neutrality equations as used by

Plummer et al.8 are employed and solved by Golden section method instead of quadratic

interpolation method which was used by Plumer8. Actual calculation shows the golden

section method is more stable than quadratic interpolation method. To verify the calculated

results, laboratory mixing experiments were done for three sets of mixed waters and the

results showed the calculated pH values are in good agreement with measured ones.

Method of pH Calculation

Dissolved species in water can be classified as components and aqueous complexes.

Aqueous complexes exist in water as results of combination of opposite charged

components. Let A1, A2, …, Ak, …, Am denote components, a1, a2, …, ak, …, am represent

their concentrations respectively. Let Y1, Y2, … , Yj, … , Yn denote aqueous complexes and

their concentrations are represented by c1, c2, …, cj, …, cn respectively. By introducing the

stoichiometric coefficient Pk, j, the reactions forming aqueous complexes can be written as

follows11

:

∑=

==m

k

jkjk njYAP

1

, ,,2,1 L (1)

When reactions (1) reached equilibrium, the equilibrium constants can be expressed as:

∏ == njacK jkPkjj ,,2,1}{/}{ , L (2)

In equation (2), { } denotes activities of species. The relations between activity and

concentration are as follows:

kkk aa α=}{ (3)

{cj} = βj cj (4)

Where αk and βj represent the activity coefficients of kth

component and jth

aqueous

complex respectively, which can be calculated by extended Debye-Huckel equation.

Substituting equations (3) and (4) into equation (2), we have:

njaFKcm

k

P

kjjjjk ,,2,1

1

1 ,L== ∏

=

− (5)

Where, njF jkP

kjj ,...,2,1/ , == ∏αβ (6)

For every aqueous complex, we have an equation like (5). There are totally n equations

for n aqueous complexes. Let Tk represent the total concentration of species which contain

kth

component and then we have:

∑=

=+=n

j

jjkkk mkcPaT

1

, ,2,1 L (7)

pH

u

Calculation of pH Values for Mixed Waters 659

For every component, we have an equation like (7). There are totally m equations for m

components. Since concentration of H+ and OH

- can be easily calculated by the pH values of water, so

equations (7) usually do not include these two components. Given the pH values of water, equations

(5) and equations (7) formed the basic equations for the equilibrium distribution calculation. These

equations can be solved by Newton-Raphson method11,12

. To calculate the pH values, another

equation must be introduced. Here we employ the following charge neutrality equation:

∑ ∑= =

=+m

k

n

j

jjkk zcza

1 1

0 (8)

Where, zk and zj denote the valent of kth

component and jth

aqueous complex

respectively. In practice, error in chemical analysis is inevitable. So when equations (5) and

(7) are solved under laboratory temperature using the measured pH value and the solutions

are substituted into equation (8), the result usually does not equal to zero. If this error is

denoted by u0, then charge neutrality equation employed for the pH calculation should be:

∑ ∑= =

=−+=m

k

n

j

jjkk uzczau

1 1

0 0 (9)

Thus it can be seen that when pH value is calculated on the basis of chemical analysis

results, it should be chosen in such a way that if this pH value is used to solve equations (5)

and (7), their solution should satisfy equation (9). In this way, the calculation of pH value

under given condition is transformed into reasonably choosing of pH value, so that u in

equation (9) equals to zero.

Large numbers of calculated results show u is a monotone function of pH. With the increase

of pH, u decreases monotonously. This is because the increase of pH is equivalent to the increase

of the concentration of negatively charged OH-, which inevitably makes the calculated results of

equation (9) decrease. The calculated pH~u curves for 3 water samples is given in Figure 1.

Figure 2 is the pH~|u| curves for the same water samples. Figure 2 clearly shows pH~|u| curve

has a sole minimum. At this minimum, u equals to zero. Thus the calculation of pH value under

given condition can be transformed into the following optimization problem:

∑ ∑= =

−+=m

k

n

j

jjkk uzczau

1 1

0min (10)

Subject to mkTcPa k

n

j

jjkk ,,2,1

1

, L==+∑=

(11)

∏=

− ==m

k

P

kjjj njaFKc jk

1

1 ,,2,1,L (12)

5 6 7 8 9 10 11

-6

-4

-2

0

2

4

6

Figure 1. Calculated relation between u and pH values for three water samples

abs,

u

pH

660 QIAN HUI et al.

5 6 7 8 9 10 11

0

1

2

3

4

5

6

Figure 2. Calculated relations between |u| and pH values for the three water samples

This problem can be solved by Golden section method as follows:

(1) Give an initial pH, solve equations (11) and (12), calculate u by substitute the solution

into equation (9). There are three cases for the calculated u: u<0, u>0 and u=0. u<0 indicates

the initial pH is greater than the actual pH of the solution. So we need to decrease pH and do

the above calculation until u>0. u>0 indicates the initial pH is lower than the actual pH of

the solution. We need to increase the pH until u<0. For the above two cases, we can find the

bound of actual pH value of the solution [a, b] by decreasing or increasing the pH value on the

basis of the initial pH. u=0 indicates the initial pH happened to be the actual pH of the solution.

(2) On the basis of the actual pH bound [a, b], two points can be obtained by the following

formulae:

)(618.01 baba −+= (13)

)(618.01 abab −+= (14)

Let the pH value of the solution equal to a1 and b1 respectively, solve equations (11) and

(12) using these pH values, calculate the corresponding u (a1) and u (b1) by substituting the

solutions into equation (9). There are three cases for the calculated u (a1) and u (b1):

(a) |u (a1)|<|u (b1)|. According to the mono extremum property of pH~|u| curve, the

actual pH value of the solution must be in [a, b1]. (b) |u (a1)|>|u (b1)|. The actual pH value of

the solution must be in [a1, b]. (c) |u (a1)|=|u (b1)|. The actual pH value of the solution must

be in [a1, b1].

Thus the bounds of the actual pH value of the solution for the above cases can be

reduced to [a, b1], [a1, b] and [a1, b1] respectively. (3) Repeat step (2) in the reduced bound

until |b-a| is less than a prior given error limit. Then the actual pH value of the solution can

be set to be (a+b)/2. In the following example, the calculation is thought to be reached

required accuracy when |b-a|<0.0001.

Chemical analysis results of mixing waters and the chemical model

Chemical analysis results of mixing waters

Table 1 listed the chemical analysis results of three mixing waters. Water samples 1 and 2

are groundwater collected in Xi'an city, China. Before the chemical analysis and mixing

experiment, some NaOH solution is added to water sample 1 to raise its pH and some HCl

solution is added to water sample 2 to reduce its pH. Chemical analysis and mixing

experiment are done after two days of addition. Water sample 3 is tap water of Xi’an

without any treatment.


Table 1. Chemical analysis results of the water samples (unit: mg/L)

No. pH T oC Cl- SO4

2- CO3

2- HCO3

- CO2 SiO2 F

-

1 9.62 25 78.7 122.87 246.28 197.84 0 25 0.4

2 6.84 25 175.12 76.88 0 378.14 47.54 20 0.8

3 8.02 25 10.64 30.9 12.31 58.85 0 5 0.2

No. Na+ K

+ Ca

2+ Mg

2+ Al

3+ Fe

2+ Fe

3+ u0 mmol/L

1 280 1.5 29.08 36 0 0.03 0.05 2.12

2 100 2 53.66 58.55 0 0.01 0.06 -0.271

3 6.6 1.5 16.79 5.09 0.1 0.8 0 -0.5

Chemical model for the mixed water

According to the chemical analysis results of mixing waters listed in Table 1, chemical

model in Table 2 is used in the calculation of pH of mixed waters. There are totally 62

species in the model, of which 14 species are constituents, 48 species are aqueous

complexes. The species are interrelated through chemical reactions in Table 2. The

equilibrium constants for the chemical reactions in Table 2 are calculated by the empirical

equations given by Arnorsson et al.9.

Table 2. Chemical model of the aquatic solution

Constituents Aqueous complexes Chemical reaction

Cl- NaCl

0 Na

+ + Cl

- = NaCl

0

SO42-

KCl0 K

+ + Cl

- = KCl

0

CO32-

FeCl2+

Fe3+

+ Cl- = FeCl

2+

H2SiO42-

FeCl2+ Fe

3+ + 2Cl

- = FeCl2

+

F- FeCl3

0 Fe

3+ + 3Cl

- = FeCl3

0

Na+ FeCl4

- Fe

3+ + 4Cl

- = FeCl4

-

K+ FeCl

+ Fe

2+ + Cl

- = FeCl

+

Ca2+

FeCl20 Fe

2+ + 2Cl

- = FeCl2

0

Mg2+

H2SO40 2H

+ + SO4

2- = H2SO4

0

Al3+

HSO4- H

+ + SO4

2- = HSO4

-

Fe2+

NaSO4- Na

+ + SO4

2- = NaSO4

-

Fe3+

KSO4- K

+ + SO4

2- = KSO4

-

H+ CaSO4

0 Ca

2+ + SO4

2- = CaSO4

0

OH- MgSO4

0 Mg

2+ + SO4

2- = MgSO4

0

FeSO40 Fe

2+ + SO4

2- = FeSO4

0

FeSO4+ Fe

3+ + SO4

2- = FeSO4

+

AlSO4+ Al

3+ + SO4

2- = ALSO4

+

Al(SO4)2- Al

3+ + 2SO4

2- = AL(SO4)2

-

H2CO30 2H

+ + CO3

2- = H2CO3

0

HCO3- H

+ + CO3

2- = HCO3

-

CaCO30 Ca

2+ + CO3

2- = CaCO3

0

MgCO30 Mg

2+ + CO3

2- = MgCO3

0

CaHCO3+ Ca

2+ + H

+ + CO3

2- = CaHCO3

+

MgHCO3+ Mg

2+ + H

+ + CO3

2- = MgHCO3

+

H4SiO40 2H

+ + H2SiO4

2- = H4SiO4

0

H3SiO4- H

+ + H2SiO4

2- = H3SiO4

-

NaH3SiO40 Na

+ + H

+ + H2SiO4

2- = NaH3SiO4

0

Contd…

662 QIAN HUI et al.

HF0 H

+ + F

- = HF

0

AlF2+

Al3+

+ F- = AlF

2+

AlF2+ Al

3+ + 2F

- = AlF2

+

AlF30 Al

3+ + 3F

- = AlF3

0

AlF4- Al

3+ + 4F

- = AlF4

-

AlF52-

Al3+

+ 5F- = AlF5

2-

AlF63-

Al3+

+ 6F- = AlF6

3-

CaOH+ Ca

2+ + OH

- = CaOH

+

MgOH+ Mg

2+ + OH

- = MgOH

+

AlOH2+

Al3+

+ OH- = AlOH

2+

Al(OH)2+ Al

3+ + 2OH

- = Al(OH)2

+

Al(OH)30 Al

3+ + 3OH

- = Al(OH)3

0

Al(OH)4- Al

3+ + 4OH

- = Al(OH)4

-

FeOH+ Fe

2+ + OH

- = FeOH

+

Fe(OH)20 Fe

2+ + 2OH

- = Fe(OH)2

0

Fe(OH)3- Fe

2+ + 3OH

- = Fe(OH)3

-

Fe(OH)42-

Fe2+

+ 4OH- = Fe(OH)4

2-

FeOH2+

Fe3+

+ OH- = FeOH

2+

Fe(OH)2+ Fe

3+ + 2OH

- = Fe(OH)2

+

Fe(OH)30 Fe

3+ + 3OH

- = Fe(OH)3

0

Fe(OH)4- Fe

3+ + 4OH

- = Fe(OH)4

-

The composition of mixed water is also needed for the calculation of its pH. When

water A with known composition is mixed with another known composition water B by

proportion PR, the composition of mixed waters can be calculated by following

equation:

mkTPRTPRPRT kkk ,,2,1)()1()()( L=Β×−+Α×= (15)

Where, Tk (A) and Tk (B) are the analyzed concentration of kth

constituents for water A

and B respectively. When the composition of mixed water is calculated by equation (15), the

analytical error of mixing waters is introduced into the composition of mixed water. This

error can be calculated by the following equation:

)()1()()( 000 Β×−+Α×= uPRuPRPRu (16)

Comparison of calculated and measured results

The three mixing waters in Table 1 can be combined into three sets. The first set is the

mixing of water samples 1 and 2. The second set is the mixing of water samples 1 and 3.

The third set is the mixing of water samples 2 and 3. These three sets of mixed water are

mixed in proportions (volume) of 1:9, 2:8, 3:7,…, 7:3, 8:2, 9:1 respectively. The pH

values of the mixed waters are measured within 5 minutes after mixing.

At the same time, the pH values of the three sets of mixed water are calculated by the

method described above. Table 3 listed the calculated and measured results, where pHm is

measured pH, pHc is calculated pH, PR is mixing proportion. The results in Table 3 are

illustrated in Figure 3. Both Table 3 and Figure 3 show the calculated pH agreed very well

with the measured one for all the three sets of mixed water. For the first set of mixed water,

the maximum error is 0.125 pH unit. The maximum error of the other two sets of mixed

water is less than 0.05 pH unit.


Table 3. Calculated vs. measured pH values for mixed water

First set Second set Third set PR

pHm pHc |pHm-pHc| pHm pHc |pHm-pHc| pHm pHc |pHm-pHc|

0.0 6.84 6.840 0.000 8.02 8.020 0.000 8.02 8.020 0.000

0.1 6.98 6.988 0.008 9.15 9.162 0.012 7.25 7.297 0.047

0.2 7.18 7.177 0.003 9.36 9.374 0.014 7.08 7.116 0.036

0.3 7.37 7.453 0.083 9.48 9.470 0.010 6.99 7.027 0.037

0.4 7.88 7.960 0.080 9.51 9.524 0.014 6.96 6.972 0.012

0.5 8.46 8.585 0.125 9.55 9.558 0.008 6.94 6.934 0.006

0.6 8.82 8.936 0.116 9.57 9.580 0.010 6.91 6.906 0.004

0.7 9.13 9.166 0.036 9.59 9.596 0.006 6.88 6.884 0.004

0.8 9.31 9.343 0.033 9.60 9.607 0.007 6.87 6.867 0.003

0.9 9.49 9.491 0.001 9.61 9.615 0.005 6.85 6.852 0.002

1.0 9.62 9.620 0.000 9.62 9.620 0.000 6.84 6.840 0.000

0.0 0.2 0.4 0.6 0.8 1.0

PR6

7

8

9

10

pH

a

0.0 0.2 0.4 0.6 0.8 1.0

PR6

7

8

9

pH

c

0.0 0.2 0.4 0.6 0.8 1.0

PR8

9

10

pH

b

Figure 3. Comparison of calculated pH values of mixed waters with measured one

a- first set b- second set c- third set

Conclusion

From the above discussion, the following conclusions can be drawn:

• The method described in this paper can be successfully employed to calculate the

pH values of mixed water.

664 QIAN HUI et al.

• Chemical reactions among species in aqueous solution as shown in table 2 can

reach equilibrium state within a short period of time.

• The pH values of mixed water usually do not vary linearly with mixing proportion.

Their specific variation is dependent upon the composition of end member mixing

waters.

Acknowledgment

This research was supported by the projects of National Natural Scientific Foundation of

China (40772160, 40372114). Authors would like to thank the editor and anonymous

reviewers for their valuable comments that have greatly improved the quality of the article.

References

1 Banks D, Markland H and Smith P V, Mendez C, Rodriguez J, Huerta A, Sæther O M, J

Geochemical Exploration, 2004, 84, 141-166.

2 Frengstad B, Bank D and Siewers U, The Science of the Total Environment, 2001, 277,

101-117.

3 Jordan C, Water Res., 1989, 23(10), 1331-1334.

4 Hunter K A, Deep-sea Res I., 1998, 45, 1919-1930.

5 Partanen J I and Minkkinen P O, J Solution Chem., 1997, 26(7), 709-727.

6 Bi S B, Anal Chim Acta, 1995, 314, 111-119.

7 Piedrahita R H, Aquacultural Engg, 1995, 14(4), 331-346.

8 Plummer L N, Parkhurst D L and Kosiur D R, MIX2: A Computer Program for

Modeling Chemical Reaction in Natural Waters, U.S.G.S. Water Resources Division,

1975, 61-75.

9 Arnorsson S, Sigurdsson S and Svavarsson H, Geochim Cosmochim Acta, 1982, 46,

1513-1532.

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11 Crerar D A, Geochim Cosmochim Acta, 1975, 39, 1375-1384.

12 Reed M, Geochim Cosmochim Acta, 1982, 46, 513-528.

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