# GEOL Lecture

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```					THE GEOCHEMISTRY OF
NATURAL WATERS

CHEMICAL EQUILIBRIUM
CHAPTER 2c - Kehew (2001)

1
LEARNING OBJECTIVES
 Learn to use some tools of thermodynamics.
 Become acquainted with equilibrium and the equilibrium
constant.
 Become acquainted with activity and activity coefficients.
 Learn to calculate and use IAP and SI.
 Learn to calculate variation of the equilibrium constant with
temperature.
 See and be able to calculate the effect of activity
coefficients on mineral solubility.
 See and be able to calculate the effect of complexation on
mineral solubility.

2
IONIC STRENGTH - I

 Recall that activity and concentration are related
through the activity coefficient according to:
ai = i·Mi
 Activity coefficients different from unity arise
because of the interaction of ions as
concentration rises.
 The degree of ion interaction depends on ionic
charge as well as concentration.

3
IONIC STRENGTH - II
 Ionic strength (I ) is a quantity that is required
to estimate activity coefficients. It takes into
account both concentration and charge:
I    1
2   M z    2
i i

 The calculation of ionic strength must take into
account all major ions:
I  1 [ M Na   4 M Ca 2  4 M Mg 2  M HCO  M Cl   4 M SO2 ]
2                                         3                  4

4
DEBYE-HÜCKEL EQUATION
 Az I
2
log  i      i

1  Ba o I
 Used to calculate activity coefficients for ions at
ionic strengths < 0.1 mol L-1.
 A, B are functions of temperature and pressure
and are given in Table 2-1 of the text.
 ao is the distance of closest approach and it is a
property of the specific ion.

5
DEBYE-HÜCKEL
PARAMETERS
T(C)     A      B(108)
0      0.4883   0.3241
5      0.4921   0.3249
10     0.4960   0.3258
15     0.5000   0.3262
20     0.5042   0.3273
25     0.5085   0.3281
30     0.5130   0.3290
40     0.5221   0.3305
50     0.5319   0.3321
60     0.5425   0.3338
6
DISTANCES OF CLOSEST
APPROACH FOR SELECTED IONS

Ion      a0 (10-8)        Ion          a0 (10-8)
Ca2+        5.0        HCO3-, CO32-        5.4
Mg2+         5.5           NH4+             2.5
Na+         4.0         Sr2+, Ba2+         5.0
K+, Cl-      3.5       Fe2+, Mn2+, Li+      6.0
SO42-        5.0       H+, Al3+, Fe3+       9.0

7
THE DAVIES EQUATION
  I          
log  i   Az 
2
i  1 I  0.2 I 

             
 Used to calculate activity coefficients for ions at
ionic strengths < 0.5 mol L-1.
 The value of A is the same as the one employed
in the Debye-Hückel equation.
 The advantage over the D-H equation is that
the only ion-dependent parameter is the ionic
charge, zi.
8
ACTIVITY COEFFICIENTS
1.2                                                      +
H

HCO3-
1.0                                                          +
Na
Activity coefficient, 

K+, Cl-

0.8

0.6

0.4
2+      2-
Mg , CO3
Ca2+, SO42-
0.2
0.0   0.2   0.4          0.6           0.8   1.0           1.2
Ionic strength (I)
9
NEUTRAL SPECIES

 Deviations from ideal behavior are least
important for neutral solutions, e.g., H2CO30.
 In other words, their activity coefficients will not
be greatly different from unity.
 A useful approximation for neutral species is:

  10       0.1I

10
EFFECT OF ACTIVITY COEFFICIENTS
ON GYPSUM SOLUBILITY
Question: what is the solubility of gypsum in water at 25°C
and 1 bar? For the equilibrium:
CaSO4·2H2O(s)  Ca2+ + SO42- + 2H2O(l)
we can calculate log KSP = -4.41. We can also see that, if
gypsum dissolves in pure water, then the stoichiometry
of the reaction is such that the molarity of calcium
should equal the molarity of sulfate.
Thus, solubility of gypsum = MCa2+.
So what’s the problem? Catch 22! We don’t know the
concentrations, so we can’t calculate the ionic strength,
so we can’t calculate the activity coefficients, so we
can’t calculate the concentrations!
11
WHAT TO DO?
We start by making an initial assumption that the activity
coefficients are equal to 1 and solve the problem by
iteration.

               
We write:
K  aCa 2 aSO2   Ca 2 M Ca 2   SO2 M SO2  10   4.41
4                        4      4
but because we assume activity coefficients are equal to 1,
we write: K  M 2  M 2   M 2 2   104.41
Ca   SO4        Ca
or
M Ca 2  102.205  6.24  103
This is what the solubility would be if we ignored activity
coefficients altogether.
12
THE NEXT STEP
Now, having the concentration of Ca2+ and SO42-, we can
calculate the ionic strength according to:

I   1
2   46.24 10   46.24 10   2.5 10
3                3        2
mol L1

Applying the Debye-Hückel formula we get:
 Ca   SO  0.548
2     2
4
which are about half the originally assumed values. We
calculate a new estimate for the molality of Ca2+:
4.41
10                      3
M Ca 2                   11.40  10
0.548  0.548
This is used to calculate a new ionic strength and the
whole process is repeated until convergence.              13
RESULTS OF ITERATIONS

Iteration            MCa2+          I
1         1     6.2410-3    2.5010-2
2       0.548   11.4010-3   4.5610-2
3       0.476   13.1110-3   5.2410-2
4       0.460   13.5610-3   5.4210-2
5       0.454   13.7510-3   5.5010-2
6       0.452   13.7810-3   5.5110-2
7       0.453   13.7810-3   5.5110-2

14
 The final calculated molarity of Ca2+ is
13.7810-3.
 This is 2.21 times the calculated molarity of
Ca2+ assuming activity coefficients are unity.
 We see that activity coefficient corrections are
very important for this solution.
 It is customary to express solubility in g L-1 of
gypsum:
(13.7810-3 mol L-1)(172.1 g mol-1) = 2.37 g L-1

15

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