http://www.e-journals.net
ISSN: 0973-4945; CODEN ECJHAO
E-Journal of Chemistry
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
∑ Pk , j Ak = Y j
j = 1,2, L , n
(1)
k =1
When reactions (1) reached equilibrium, the equilibrium constants can be expressed as:
K j = {c j } /
∏{ak }P
k, j
j = 1,2,L, n
(2)
In equation (2), { } denotes activities of species. The relations between activity and
concentration are as follows:
{a k } = α k a k
(3)
{cj} = βj cj
(4)
th
th
Where αk and βj represent the activity coefficients of k component and j aqueous
complex respectively, which can be calculated by extended Debye-Huckel equation.
Substituting equations (3) and (4) into equation (2), we have:
m
c j = K j F j−1
Pk , j
∏ ak
j = 1,2, L , n
(5)
k =1
Where,
Fj = β j / ∏ α k
Pk , j
j = 1,2,..., n
(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
Tk = ak +
∑ Pk , j c j
j =1
k = 1,2L , m
(7)
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
∑
n
ak zk +
k =1
∑c j z j = 0
(8)
j =1
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:
u=
m
n
k =1
j =1
∑ ak z k + ∑ c j z j − u 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:
Subject to
m
n
k =1
j =1
(10)
∑ ak z k + ∑ c j z j − u0
min u =
n
ak +
∑ Pk , j c j = Tk
k = 1,2, L , m
(11)
j =1
m
c j = K j F j−1
Pk , j
∏ ak
(12)
j = 1,2, L , n
k =1
6
4
2
u
0
-2
-4
-6
5
6
7
8
9
10
11
pH
Figure 1. Calculated relation between u and pH values for three water samples
660
QIAN HUI et al.
6
5
abs, u
4
3
2
1
0
5
6
7
8
9
10
11
pH
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:
a1 = b + 0.618(a − b)
(13)
b1 = a + 0.618(b − a)
(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.
Calculation of pH Values for Mixed Waters
661
Table 1. Chemical analysis results of the water samples (unit: mg/L)
No.
1
2
3
No.
1
2
3
pH
9.62
6.84
8.02
Na+
280
100
6.6
T oC
25
25
25
K+
1.5
2
1.5
Cl78.7
175.12
10.64
Ca2+
29.08
53.66
16.79
SO42122.87
76.88
30.9
Mg2+
36
58.55
5.09
CO32246.28
0
12.31
Al3+
0
0
0.1
HCO3197.84
378.14
58.85
Fe2+
0.03
0.01
0.8
CO2
0
47.54
0
Fe3+
0.05
0.06
0
SiO2 F25
0.4
20
0.8
5
0.2
u0 mmol/L
2.12
-0.271
-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
ClSO42CO32H2SiO42FNa+
K+
Ca2+
Mg2+
Al3+
Fe2+
Fe3+
H+
OH-
Aqueous complexes
NaCl0
KCl0
FeCl2+
FeCl2+
FeCl30
FeCl4FeCl+
FeCl20
H2SO40
HSO4NaSO4KSO4CaSO40
MgSO40
FeSO40
FeSO4+
AlSO4+
Al(SO4)2H2CO30
HCO3CaCO30
MgCO30
CaHCO3+
MgHCO3+
H4SiO40
H3SiO4NaH3SiO40
Chemical reaction
Na+ + Cl- = NaCl0
K+ + Cl- = KCl0
Fe3+ + Cl- = FeCl2+
Fe3+ + 2Cl- = FeCl2+
Fe3+ + 3Cl- = FeCl30
Fe3+ + 4Cl- = FeCl4Fe2+ + Cl- = FeCl+
Fe2+ + 2Cl- = FeCl20
2H+ + SO42- = H2SO40
H+ + SO42- = HSO4Na+ + SO42- = NaSO4K+ + SO42- = KSO4Ca2+ + SO42- = CaSO40
Mg2+ + SO42- = MgSO40
Fe2+ + SO42- = FeSO40
Fe3+ + SO42- = FeSO4+
Al3+ + SO42- = ALSO4+
Al3+ + 2SO42- = AL(SO4)22H+ + CO32- = H2CO30
H+ + CO32- = HCO3Ca2+ + CO32- = CaCO30
Mg2+ + CO32- = MgCO30
2+
Ca + H+ + CO32- = CaHCO3+
Mg2+ + H+ + CO32- = MgHCO3+
2H+ + H2SiO42- = H4SiO40
H+ + H2SiO42- = H3SiO4+
Na + H+ + H2SiO42- = NaH3SiO40
Contd…
662
QIAN HUI et al.
HF0
AlF2+
AlF2+
AlF30
AlF4AlF52AlF63CaOH+
MgOH+
AlOH2+
Al(OH)2+
Al(OH)30
Al(OH)4FeOH+
Fe(OH)20
Fe(OH)3Fe(OH)42FeOH2+
Fe(OH)2+
Fe(OH)30
Fe(OH)4-
H+ + F- = HF0
Al3+ + F- = AlF2+
Al3+ + 2F- = AlF2+
Al3+ + 3F- = AlF30
Al3+ + 4F- = AlF4Al3+ + 5F- = AlF52Al3+ + 6F- = AlF63Ca2+ + OH- = CaOH+
Mg2+ + OH- = MgOH+
Al3+ + OH- = AlOH2+
Al3+ + 2OH- = Al(OH)2+
Al3+ + 3OH- = Al(OH)30
Al3+ + 4OH- = Al(OH)4Fe2+ + OH- = FeOH+
Fe2+ + 2OH- = Fe(OH)20
Fe2+ + 3OH- = Fe(OH)3Fe2+ + 4OH- = Fe(OH)42Fe3+ + OH- = FeOH2+
Fe3+ + 2OH- = Fe(OH)2+
Fe3+ + 3OH- = Fe(OH)30
Fe3+ + 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:
Tk ( PR ) = PR × Tk ( Α) + (1 − PR ) × Tk (Β) k = 1,2, L , m
(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:
u 0 ( PR ) = PR × u 0 ( Α) + (1 − PR ) × u0 (Β)
(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.
Calculation of pH Values for Mixed Waters
663
Table 3. Calculated vs. measured pH values for mixed water
First set
PR
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
pHm
pHc
6.84
6.98
7.18
7.37
7.88
8.46
8.82
9.13
9.31
9.49
9.62
6.840
6.988
7.177
7.453
7.960
8.585
8.936
9.166
9.343
9.491
9.620
Second set
|pHm-pHc| pHm
0.000
0.008
0.003
0.083
0.080
0.125
0.116
0.036
0.033
0.001
0.000
pHc
8.02
9.15
9.36
9.48
9.51
9.55
9.57
9.59
9.60
9.61
9.62
Third set
|pHm-pHc| pHm
8.020
9.162
9.374
9.470
9.524
9.558
9.580
9.596
9.607
9.615
9.620
0.000
0.012
0.014
0.010
0.014
0.008
0.010
0.006
0.007
0.005
0.000
8.02
7.25
7.08
6.99
6.96
6.94
6.91
6.88
6.87
6.85
6.84
pHc
|pHm-pHc|
8.020
7.297
7.116
7.027
6.972
6.934
6.906
6.884
6.867
6.852
6.840
0.000
0.047
0.036
0.037
0.012
0.006
0.004
0.004
0.003
0.002
0.000
pH
a
10
9
8
7
6
PR
0.0
0.2
0.4
0.6
pH
pH
9
0.8
1.0
10
c
b
8
9
7
6
PR
0.0
0.2
0.4
0.6
0.8
1.0
8
PR
0.0
0.2
0.4
0.6
0.8
1.0
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.
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Bi S B, Anal Chim Acta, 1995, 314, 111-119.
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