results
Electroneutrality. The principle of electroneutrality requires that the sum of the positive ions (cations) must equal the sum of negative ions (anions) in solution, thus
© cations 5© anions (2–3)
where cations 5 positively charged species in solution expressed in terms of equivalent weight per liter, eq/L or milliequvalent weight per liter, meq/L
anions 5 negatively charged species in solution, eq/L or meq/L The equivalent weight of a compound is defined as:
Equivalent weight, g/eq5 molecular weight, g
Z (2–4) where Z 5 (1) the absolute value of the ion charge, (2) the number of H1 or OH2 ions a
species can react with or yield in an acid-base reaction, or (3) the absolute value of the change in valence occurring in an oxidation reduction reaction (Sawyer et al., 2003).
Equation (2–3) can be used to check the accuracy of chemical analyses by taking into account the percent difference defined as follows (Standard Methods, 2012):
Percent difference51003 aS cations2S anions
S cations1S anionsb (2–5) The application of Eq. (2–2) is illustrated subsequently in Example 2–1.
EXAMPLE 2–1 Determination of Mole Fraction Determine the mole fraction of oxygen in water if the concentration of dissolved oxygen is 10.0 mg/L.
Solution
1. Determine the mole fraction of oxygen using Eq. (2–2) written as follows:
xO25 nO2
nO21nw
a. Determine the moles of oxygen.
nO25 (10 mg/L)
(323103 mg/mole O2)53.12531024 mole/L b. Determine the moles of water.
nw5 (1000 g/L)
(18 g/mole of water) 555.556 mole/L c. The mole fraction of oxygen is:
xO25 3.12531024
3.12531024155.556 55.6231026
2–2 Sampling and Analytical Procedures
69
The acceptance criteria are as given below.
S anions, meq/L Acceptable difference
0–3.0 60.2 meq/L
3.0–10.0 62%
10–800 5%
From Standard Methods (2012).
The application of Eqs. (2–3) and (2–5) is illustrated in Example 2–2.
EXAMPLE 2–2 Checking the Accuracy of Analytical Measurements The following analy- sis has been completed on a filtered effluent, from an extended aeration wastewater treat- ment plant, that is to be used for landscape watering. Check the accuracy of the analysis to determine if the analysis is sufficiently accurate, based on the criteria given above.
Cation
Conc.,
mg/L Anion
Conc., mg/L
Ca21 82.2 HCO32 220.0
Mg21 17.9 SO422 98.3
Na1 46.4 Cl2 78.0
K1 15.5 NO32 25.6
Solution
1. Prepare a cation-anion balance.
Cation
Conc.,
mg/L mg/meqa meq/L Anion
Conc.,
mg/L mg/meqa meq/L
Ca21 82.2 20.04b 4.10 HCO32 220.0 61.02 3.61
Mg21 17.9 12.15 1.47 SO422 98.3 48.03 2.05
Na1 46.4 23.00 2.02 Cl2 78.0 35.45 2.20
K1 15.5 39.10 0.40 NO32 25.6 62.01 0.41
π cations 7.99 π anions 8.27
a Molecular weight in grams/Z
b For calcium, eq wt 5 40.08/2 5 20.04 g/eq or 20.04 mg/meq
2. Check the accuracy of the cation-anion balance using Eq. (2–5).
Percent difference51003 aS cations2S anions S cations1S anionsb Percent difference51003 a7.9928.27
7.9918.27b 5 21.72%
For a total anion concentration between 3 and 10 meq/L, the acceptable difference must be equal to or less than 2 percent (see table given above), thus, the analysis is of sufficient accuracy.
Comment If the cation-anion balance is not of sufficient accuracy, the problem may be analytical or a constituent of significant concentration may be missing.
Chemical Equilibrium. A reversible chemical reaction in which reactants A and B combine to yield products C and D may be written as
aA1bB dS cC1dD (2–6) Where the stoichiometry coefficients a, b, c, and d correspond to the number of moles of constituents A, B, C, and D, respectively. The stoichiometry of a reaction refers to the definition of the quantities of chemical compounds involved in a reaction (e.g., a of A, b of B, etc.). When the chemical species come to a state of equilibrium, as governed by the law of mass action, the numerical value of the ratio of the products over the reactants is known as the equilibrium constant K and is written as
[C]c[D]d
[A]a[B]b 5K (2–7) For a given reaction, the value of the equilibrium constant will change with temperature and the ionic strength of the solution. It should also be noted that in Eq. (2–7) it is assumed that that activity of the individual ions is equal to one.
Brackets are used in Eq. (2–7) to denote molar concentrations. The use of molal concentrations (see Table 2–3) is more correct theoretically, but for dilute solutions encountered in wastewater applications, molar concentrations are used. Molal concentra- tions must be used for brine solutions and sea water. To account for non-ideal conditions encountered due to ion-ion interactions, a new concentration term called activity is used.
The activity of an ion is defined as follows:
ai5g[Ci] (2–8) where ai5 activity of ith ion, mole/L
g5 activity coefficient for the ith ion
Ci5 concentration of ith ion in solution, mole/L
If Eq. (2–7) is written in terms of activity and activity coefficients rather than concentra- tions, the resulting expression is:
[aC]c[aD]d
[aA]a[aB]b5 [gcC]C[gDD]d
[gAA]a[gBB]b 5K (2–9) Ionic Strength. The ionic strength of a solution is a measure of the concentration of dissolved chemical constituents. The ionic strength of a solution can be estimated using the following expression:
I51
2S CiZ2i (2–10) where I 5 ionic strength
Ci5 concentration of the ith species, mole/L
Zi5 the valance (or oxidation) number of the ith species [see Eq. (2–4)]
The ionic strength can also be estimated based on the total dissolved solids concentration using the following expression:
I52.5310253TDS (2–11) where TDS 5 total dissolved solids, mg/L or g/m3
2–2 Sampling and Analytical Procedures
71
Equation (2–11) is often used to estimate the ionic strength of treated wastewater in groundwater recharge applications.
Activity Coefficient. The activity coefficient can be estimated using the following expression, derived from the Debye-Huckel theory, as proposed by Davies (1962). Com- putation of the activity coefficient is illustrated in Example 2–3 following the discussion of ionic strength and solubility.
log g5 20.5 (Zi)2a "I
11 "I20.3Ib (2–12)
where Zi5 charge on ith ionic species I 5 ionic strength
The above relationship, without the 20.3 I, term is often used for solutions with an ionic strength that does not exceed 0.1 M. A number of other similar relationships will be found in the literature. Computation of the activity coefficient is illustrated in Example 2–3 following the discussion of ionic strength and solubility.
Solubility Product. The equilibrium constant for a reaction involving a precipitate and its constituent ions is known as the solubility product. For example, the reaction for calcium carbonate (CaCO3) is
CaCO3dSCa211CO223 (2–13) Because the activity of the solid phase is usually taken as 1, the solubility product is written as
[Ca21][CO322]5Ksp (2–14) where Ksp5 solubility product constant.
It is important to note that the value of the equilibrium constant will change with the tem- perature of the solution. Written in terms of activity coefficients, Eq. (2–14) becomes gCa21[Ca21]gCO223 [CO223 ] 5Ksp (2–15) The application of Eq. (2–15) is illustrated in Example 2–3.
EXAMPLE 2–3 Determine the Activity Coefficients and Solubility of Calcium Carbonate Determine the activity coefficients for the mono and divalent ions in the wastewater given in Example 2–2. Using the value of the activity coefficient for a divalent ion, estimate the equilibrium concentration of calcium in solution needed to satisfy the solubility product for calcium carbonate (CaCO3) at 25°C. The value of the solubility product constant Ksp
for CaCO3 at 25°C is 5 3 1029. Solution
1. Determine the ionic strength of the wastewater using Eq. (2–10).
a. Prepare a computation table to determine the summation term in Eq. (2–10) using the data from Example 2–2.
Ion
Conc., C, mg/L
C 3 103,
mole/L z2 cz23 103
Ca21 82.2 2.051 4 8.404
Mg21 17.9 0.736 4 2.944
Na1 46.4 2.017 1 2.017
K1 15.5 0.396 1 0.397
HCO32 220 3.607 1 3.607
SO422 98.3 1.024 4 4.096
Cl2 78.0 2.200 1 2.200
NO32 25.6 0.413 1 0.413
Sum 23.876
b. Determine the ionic strength of the wastewater.
I51
2SCiZ2i 51
2(23.87631023)511.93831023
2. Determine the activity coefficients for Ca21 and CO322. Because both species have a valance (charge) of 2, the activity of each will be the same.
a. For monovalent ions log g5 20.5(Z i)2a "I
1 1 "I 20.3Ib log g 5 20.5(1)2c "11.93831023
1 1 "11.9383102320.3(11.93831023)d5 20.0475 g5 0.896
b. For divalent ions
log g5 20.5(2)2c "11.93831023
1 1 "11.9383102320.3(11.93831023)d5 20.1898 g5 0.646
3. Determine the minimum solubility of calcium using Eq. (2–15).
a. Because the molar concentrations of calcium and carbonate ions are the same, Eq (2–15) can be written as follows:
g2 [C2]5Ksp
b. Solve for the concentration C.
C5 Å
Ksp g2 5
Å
531029
(0.646)2 51.0931024 mole/L
c. Convert the molar concentration of calcium carbonate to mg/L.
Ca51.0931024 mole/L340,000 mg/mole54.36 mg/L
Comment The computed value represents the minimum concentration of calcium that would be required in solution to be in equilibrium with solid calcium carbonate.