Calculation for SI

Given the chemical analysis of ground water from Cascade spring in the Southern Black
Hills .It is required to calculate the saturation Index (SI) w.r.t calcite, dolomite and
gypsum at pH and temperature of 6.89 and 25oC respectively.

 IAP 
SI is given by the relationship SI = log  .Therefore, it is important to calculate IAP
 Ksp 
and Ksp values for calcite, dolomite and gypsum.

Note:-Thermodynamic data used in this calculation is taken from Krauskopf,1979.

a) Calcium:-As the environment is acidic ,the dominant reaction involving calcite is

given by:

−1128.8 − 623.1 −558.6 −2*586.8

CaCO3 + H 2 CO3 ⇔ Ca + 2 HCO3−

2+

∆G Ro = [(−553.6 − 2 * 586.8) − (−1128.8 − 623.1)] = 24.7 KJ / mole

 ∆G Ro 
− 
 5.709 
0  
@\ 25 C , K sp = 10 = 10 − 4.33

IAP =
(M Ca 2 +
(
* γ Ca 2 + ) M HCO − * γ HCO −
3 3
)
[H 2 CO3 ]
As the value of carbonic acid is not provided, it is demanding to back calculate it
from the dissociation of it including the activity for carbonate ion so that the SI value
will include the contribution of carbonate ion although its concentration is
significantly small under the given pH value.

H 2 CO3 ⇔ H + + HCO3−

[H CO ] =
[H ][HCO ] = 10 + −
3
− 6.89
* 0.004
= 0.0012M
2 3 − 6.36
K 1 10

Similarly
HCO3− ⇔ H + + CO32−

[CO ]2−
=
[ =
]
K 2 * HCO3− 10 −10.33 * 0.004
= 0.000001452M ≈ 0.0 M
3
H+ [ ] 10 −6.89

1
As the environment is slightly acidic, the concentration of carbonate is sufficiently
small so that its contribution to the ionic strength (I) of the water is insignificant.
However, it is included in the calculation of I for completeness.

b) Dolomite:- As the environment is slightly acidic, the dominant reaction for

dolomite is given by

−2163.4 2*623.1 −553.6 −454.80 4*586.8

CaMg (CO3 ) 2 + 2 H 2 CO3 ⇔ Ca 2+ + Mg 2+ + 4 HCO3−

∆G Ro = [(− 4 * 586.8 − 454.8 − 553.6 ) − (− 2163.4 − 2 * 623.1)]

=54 KJ/mole

− ∆G R0
0
∴ @ 25 C , log( K sp ) = ⇒ K sp = 10 −9.46
5.709

Finally Ionic strength, I is calculated from the following relationship taking into
account the contribution of all the given ions including carbonate ion except for
chargless species like silica and carbonic acid.

10
I = 0.5 * ∑ M i * Z i2
1

[ ( 3
) (
= 0.5 * 1* M Na + + M K + + M Cl − + M F − + M HCO− + 4 * M Ca2+ + M Mg 2+ + M Sr 2+ + M CO2−
2
)]
= 0.07
(The converted concentration in M used in this calculation is provided in table 2).

2
Using both Huckel-Buy and I − γ plot, the values of ion activity, γ is calculated and/or
interpolated .The following table gives the values for the required species.

Species Mi γ
(M) Debye-Huckel Values interpolated
from I − γ plot
Ca 2+ 0.013348
0.31 0.46

Mg 2+ 0.003414 0.31 0.5

SO42− 0.015095 0.3 0.37

HCO3−
0.003999 0.74 0.83

0.000001 0.3 0.42

CO32−

a) SI-calcite , using gamma from the above table:

IAP ( Debye − Huckel ) =

(M Ca 2 +
(
* γ Ca 2 + ) M HCO − * γ HCO −
3 3
)
[H 2 CO3 ]
(0.013348 * 0.31)(0.003999 * 0.74)
= = 3.01 * 10 −5
(0.001180)

and IAP(calcite)= 5.73 * 10 −5 using γ value estimated from I- γ plot.

Therefore two values of SI will be determined based on Debye-Huckel approach and

IAP interpolated from I- γ plot.

IAP  3.01 * 10 −5 
SI(Debye-Huckel)= log( ) = log − 4.33
 = −0.20 >0(unsaturated).
K sp  10 
SI(I- γ plot)=0.09 >0 ⇒ saturated w.r.t calcite

3
 b) SI-dolomite,
( Mca 2+ * γ Ca 2 + ) * ( M Mg 2 + * γ Mg 2 + ) * ( M HCO − * γ HCO − ) 4
3 3
IAP ( Debye − Huckel ) = 2
[ H 2 CO3 ]

= 2.33 * 10 −10
and IAP= 9.14 * 10 −10 based on γ value calculated from I- γ .

 9.14 * 10 −10 
SI(Debye-Huckel)= log − 9.46
 =-0.20>0(unsaturated)
 10 
SI(I- γ plot)=0.42 ⇒ saturated w.r.t dolomite.

c) SI-gypsum
( M Ca 2 + * γ Ca 2 + )( M SO 2 − * γ SO 2 − − )
IAP( Debye − Huckel ) = 4 4
= 1.82 * 10 −5
1
and IAP(from I- γ plot)= 3.43 * 10 −5

Ksp of gypsum= 10 −4.6 (taken from previous exercise).

 1.82 * 10 −5 
∴ SI ( Debye − Huckel ) = log − 4.6
 = −0.15 <0 (Unsaturated).
 10 
SI(I- γ plot)=0.14>0 ⇒ Saturated w.r.t. gypsum.

Based on activity coefficient calculated from the plot of I vs γ , the water is saturated
with respect to calcite, dolomite and gypsum while the result calculated based on Debye-
Huckel equation predicted the water is unsaturated with respect to calcite, dolomite and
gypsum. SI values calculated based on I- γ plot is more realistic than the Debye-Huckel
equation as it is based on empirical data. Therefore, the potential of this ground water
from Cascade Spring to dissolve the aquifer is minimal(note that this conclusion is
solely based on SI calculated using I- γ plot).

4
Uncertainty Analysis

The validity of the analysis is checked by the equation:

%error =
∑ McZc − ∑ MaZa
∑ McZc + ∑ MaZa

0.03506528 - 0.033800
= 0.033800 + 0.035065
≈ 0.13%

∴ The chemical analysis has no series error.

Verification for the acidity of the water

This is justified based on pressure of CO2 from

H 2 O + CO2 ⇔ H 2 CO3

K CO2 = [H 2 CO2 ] / PCO2

0.0012
10 −1.47 = ⇒ PCO2 = 0.03atm
P
The water is slightly acidic because the dissolved CO2 attributes to the formation of
carbonic acid which eventually acidified the water.

5
Problem Identification and Engineering the problem

F − (Hardening and staining of teeth but the given concentration is acceptable for this
particular spring), Ca + + and Mg + + (cause for hardness of water), Sulfate (problem if the
water is used for drinking) and Na + (problem if the water is used for irrigation purpose)
and generally increased level of TDS are some of the major potential problems associated
with water from this Spring.

Engineering the problem:-

Some complex inorganic crystal such as ziolite( Na 2 Al 2 Si4 O12 ) could be applied as water
softener to overcome the hardness of the water. As second option boiling the water could
also be removed by boiling, which cause the excess carbon dioxide to be driven off , and
calcite to precipitate. However, as this method is energy intensive process, it is not
economical from engineering economy perspective. Instead the water would rather be
softened by the addition of calcium hydroxide, slaked lime.

Ca 2+ + 2 HCO3− + Ca (OH ) 2 → 2CaCO3 + 2 H 2 O .

However, as the water is rich in sulfate ion, the hardness of water can’t be affected by
boiling. This type of permanently hard water could be softened treatment with sodium
carbonate. The sodium ion of the sodium carbonate will be left in the solution in water
together with the sulfate ion that were already present.

6
All the above computations are summarized in the following table2 and 3.

Table (2) Table presenting chemical analysis of the species in M and useful calculations
for determination of SI at 25 O C (The values of A,B and ao are taken from Drever,1997
pp.28).

log( γ ) γ γ
Species con(M) zi Mi*Zi*Zi A B ao
(Debye) Plot
+
Na 0.001174 1 0.001174
K+ 0.000133 1 0.000133
Ca 2+ 0.013348 2 0.053393
5.00E-
Mg 2+ 0.003414 2 0.013657 0.5085 3281 08 -0.53 0.30 0.46
5.50E-
Sr 2+ 0.000071 2 0.000283 0.5085 3281 08 -0.53 0.30 0.5
HCO3− 0.003999 -1 0.003999
5.40E-
SO42− 0.015095 -2 0.060379 0.5085 3281 08 -0.13 0.74 0.83
5.00E-
Cl − 0.000874 -1 0.000874 0.5085 3281 08 -0.53 0.30 0.37
F− 0.000084 -1 0.000084
H+ 0.00000013 1 0.000000
CO32− 0.000001 2 0.000004
5.40E-
H 2 CO3 0.001180 0 0.000000 0.5085 3281 08 -0.53 0.30 0.42
SiO2 0.000250 0 0.000000

Table(3) summary of SI values based on Debye-Huckel equation and I- γ plot

SI
I Species
Debye- status I- γ plot status
Huckel
Calcite -0.2 unsaturated 0.09 Saturated

0.07 Dolomite -0.2 Unsaturated 0.42 Saturated

gypsum -0.15 unsaturated 0.14 saturated