6.1 Lime-Soda Ash Softening Chemistry & Stoichiometry

6.1 Lime-Soda Ash Softening Chemistry & Stoichiometry

Hardness in public water supplies originates from the dissolution of subterranean sedimentary mineral deposits—predominantly limestone ($CaCO_3$), dolomite ($CaCO_3\cdot MgCO_3$), and gypsum ($CaSO_4\cdot 2H_2O$)—by groundwater containing dissolved carbon dioxide. While hardness does not present an adverse human health risk, elevated concentrations cause severe economic and aesthetic liabilities: scale buildup in domestic hot water heaters, high friction head loss and restricted flow in distribution mains, premature boiler tube failure, and wasted soap and synthetic detergents. Lime-soda ash softening is the primary municipal precipitation technology used to reduce divalent cation concentrations to acceptable levels.

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Water Hardness Classifications and Mineral Constituents

Water hardness is defined chemically as the total concentration of polyvalent metallic cations dissolved in water, expressed in milligrams per liter as calcium carbonate equivalent ($\text{mg/L as }CaCO_3$). In virtually all natural drinking water sources, divalent calcium ($Ca^{2+}$) and magnesium ($Mg^{2+}$) account for more than 98% of total hardness. Other polyvalent cations, including ferrous iron ($Fe^{2+}$), manganous manganese ($Mn^{2+}$), strontium ($Sr^{2+}$), barium ($Ba^{2+}$), and aluminum ($Al^{3+}$), contribute slightly to total hardness but are typically classified as trace metals or aesthetic contaminants.

Hardness Classification Scale

The United States Geological Survey (USGS) and the American Water Works Association (AWWA) classify drinking water hardness into four operational categories based on total concentration:

Note: To convert hardness between units, $1.0\text{ grain per gallon (gpg)} = 17.12\text{ mg/L as }CaCO_3$.

Total Hardness vs. Calcium and Magnesium Hardness

Laboratory titrations (Standard Method 2340C, EDTA Titrimetric Method) distinguish between total mineral hardness and specific cation fractions:

$\text{Total Hardness (TH)} = \text{Calcium Hardness (CH)} + \text{Magnesium Hardness (MH)}$

Where all parameters are expressed in $\text{mg/L as }CaCO_3$. Determining the exact ratio of calcium to magnesium is critical for chemical feed design because magnesium removal requires higher pH setpoints and twice the stoichiometric lime dosage of calcium removal.

Carbonate vs. Noncarbonate Hardness

Hardness is further categorized by the associated anions balancing the divalent metal cations:

                             [ Total Hardness ]
                              /              \
                             /                \
    [ Carbonate Hardness (CH) ]              [ Noncarbonate Hardness (NCH) ]
     - Associated with HCO3- and CO3(2-)      - Associated with SO4(2-), Cl-, NO3-
     - "Temporary Hardness"                   - "Permanent Hardness"
     - Removable by boiling or lime alone     - Requires soda ash (Na2CO3) for removal

1. Carbonate Hardness (CH): The portion of total hardness chemically balanced by bicarbonate ($HCO_3^-$) and carbonate ($CO_3^{2-}$) alkalinity anions. Historically termed temporary hardness because it precipitates out of solution as calcium carbonate scale when heated or boiled. It can be completely removed using hydrated lime alone.
2. Noncarbonate Hardness (NCH): The portion of total hardness chemically balanced by non-alkalinity mineral anions—principally sulfate ($SO_4^{2-}$), chloride ($Cl^-$), and nitrate ($NO_3^-$). Historically termed permanent hardness because thermal heating does not cause precipitation. Noncarbonate hardness requires the addition of soda ash ($Na_2CO_3$) for precipitation.

To establish the distribution between carbonate and noncarbonate fractions from standard water quality laboratory data:

• If Total Alkalinity (TA) $\ge$ Total Hardness (TH): $\text{Carbonate Hardness} = \text{Total Hardness}$ $\text{Noncarbonate Hardness} = 0$
• If Total Alkalinity (TA) < Total Hardness (TH): $\text{Carbonate Hardness} = \text{Total Alkalinity}$ $\text{Noncarbonate Hardness} = \text{Total Hardness} - \text{Total Alkalinity}$

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Chemical Precipitation Softening Reactions and pH Thresholds

Precipitation softening alters chemical equilibria to exceed the solubility limits of calcium and magnesium compounds, causing them to precipitate as crystalline calcium carbonate ($CaCO_3$) and amorphous magnesium hydroxide ($Mg(OH)_2$). The chemicals added are commercial hydrated lime (calcium hydroxide, $Ca(OH)_2$), quicklime (calcium oxide, $CaO$), and soda ash (sodium carbonate, $Na_2CO_3$).

Reaction 1: Neutralization of Free Carbon Dioxide

Dissolved carbon dioxide ($CO_2$) in raw groundwater reacts instantly with added lime before any hardness can be removed. Although dissolved $CO_2$ is an uncharged dissolved gas and does not contribute to mineral hardness, it imposes an immediate stoichiometric chemical demand on lime:

$CO_2 + Ca(OH)_2 \rightarrow CaCO_3\downarrow + H_2O$

Stoichiometric Rule: Every mole of free $CO_2$ neutralizes one mole of hydrated lime, generating an insoluble calcium carbonate precipitate. If raw groundwater contains high concentrations of free carbon dioxide (>20–30 mg/L), plants often install raw water cascade or tray aerators upstream of softening to strip $CO_2$ gas to the atmosphere, slashing chemical lime costs.

Reaction 2: Removal of Calcium Carbonate Hardness

Calcium associated with bicarbonate alkalinity precipitates when hydrated lime converts bicarbonate ions into carbonate ions at pH 9.0 to 9.6:

$Ca(HCO_3)_2 + Ca(OH)_2 \rightarrow 2CaCO_3\downarrow + 2H_2O$

Stoichiometric Rule: Each mole of calcium bicarbonate hardness consumes one mole of hydrated lime, precipitating two moles of solid calcium carbonate—one mole derived from the raw water calcium and one mole derived from the added lime.

Reaction 3: Removal of Magnesium Carbonate Hardness

Magnesium carbonate is relatively soluble at pH 9.5 and cannot be removed as a carbonate precipitate. Instead, the pH must be elevated to 10.6 to 11.0 by adding excess hydrated lime to force magnesium to precipitate as insoluble magnesium hydroxide ($Mg(OH)_2$). The reaction proceeds in two distinct steps:

$\text{Step 1: } Mg(HCO_3)_2 + Ca(OH)_2 \rightarrow MgCO_3 + CaCO_3\downarrow + 2H_2O$ $\text{Step 2: } MgCO_3 + Ca(OH)_2 \rightarrow Mg(OH)_2\downarrow + CaCO_3\downarrow$ $\text{Overall: } Mg(HCO_3)_2 + 2Ca(OH)_2 \rightarrow Mg(OH)_2\downarrow + 2CaCO_3\downarrow + 2H_2O$

Stoichiometric Rule: Removing one mole of magnesium carbonate hardness requires two moles of hydrated lime—twice the chemical requirement of calcium carbonate hardness—and produces two moles of calcium carbonate plus one mole of gelatinous magnesium hydroxide.

Reaction 4: Removal of Calcium Noncarbonate Hardness

Calcium noncarbonate hardness (such as calcium sulfate, $CaSO_4$) lacks the carbonate ion necessary for precipitation. Soda ash ($Na_2CO_3$) must be dosed at pH 9.0 to 9.6 to supply the requisite carbonate anion ($CO_3^{2-}$):

$CaSO_4 + Na_2CO_3 \rightarrow CaCO_3\downarrow + Na_2SO_4$

Stoichiometric Rule: Each mole of calcium noncarbonate hardness consumes one mole of soda ash, precipitating one mole of calcium carbonate while leaving soluble, non-hardness sodium sulfate ($Na_2SO_4$) in the finished water.

Reaction 5: Removal of Magnesium Noncarbonate Hardness

Removing magnesium noncarbonate hardness (such as magnesium sulfate, $MgSO_4$) is the most chemical-intensive softening process, requiring both hydrated lime and soda ash in two linked stages:

$\text{Stage A (Lime addition at pH 10.6–11.0): } MgSO_4 + Ca(OH)_2 \rightarrow Mg(OH)_2\downarrow + CaSO_4$ $\text{Stage B (Soda ash addition at pH 9.0–9.6): } CaSO_4 + Na_2CO_3 \rightarrow CaCO_3\downarrow + Na_2SO_4$ $\text{Combined: } MgSO_4 + Ca(OH)_2 + Na_2CO_3 \rightarrow Mg(OH)_2\downarrow + CaCO_3\downarrow + Na_2SO_4$

Stoichiometric Rule: Hydrated lime precipitates the magnesium ion as insoluble $Mg(OH)_2$, but in doing so, converts magnesium noncarbonate hardness into an equivalent concentration of calcium noncarbonate hardness ($CaSO_4$). Soda ash must then be added to precipitate the resulting calcium ion as $CaCO_3$. Thus, one mole of magnesium noncarbonate hardness consumes one mole of lime plus one mole of soda ash.

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Practical Softening Limits and Residual Hardness Equilibrium

Chemical precipitation softening cannot reduce water hardness to zero. Unlike ion exchange, which operates by physical resin adsorption, precipitation is governed by the thermodynamic solubility products ($K_{sp}$) of the precipitating solids:

• Calcium carbonate solubility: $K_{sp} \approx 3.36 \times 10^{-9}$ at 25°C
• Magnesium hydroxide solubility: $K_{sp} \approx 5.61 \times 10^{-12}$ at 25°C

In municipal water treatment plant operations, common-ion effects, organic complexation, temperature depression, and short-circuiting in clarifiers establish practical solubility limits:

• Residual Calcium Hardness: Minimum practical limit of 30 to 40 mg/L as $CaCO_3$.
• Residual Magnesium Hardness: Minimum practical limit of 10 mg/L as $CaCO_3$.
• Total Finished Water Hardness: Under optimum operating conditions, lime-soda softening can reduce total hardness to approximately 40 to 50 mg/L as $CaCO_3$, though municipal plants typically target a finished hardness of 75 to 100 mg/L as $CaCO_3$ (approx. 4.5 to 6.0 gpg) to prevent distribution system pipe corrosion and save chemical operating expenses.

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Stoichiometric Equivalents and Chemical Dosing Math

To calculate required chemical dosages, operators express all water quality constituents and commercial chemical reagents in equivalent weights ($EW = \text{Molecular Weight} / \text{Valence}$):

$\text{Equivalent Weight of } CaCO_3 = \frac{100.09\text{ g/mol}}{2} \approx 50.05\text{ g/eq}$ $\text{Equivalent Weight of Hydrated Lime } [Ca(OH)_2] = \frac{74.09\text{ g/mol}}{2} \approx 37.05\text{ g/eq}$ $\text{Equivalent Weight of Quicklime } [CaO] = \frac{56.08\text{ g/mol}}{2} \approx 28.04\text{ g/eq}$ $\text{Equivalent Weight of Soda Ash } [Na_2CO_3] = \frac{105.99\text{ g/mol}}{2} \approx 53.00\text{ g/eq}$ $\text{Equivalent Weight of Carbon Dioxide } [CO_2] = \frac{44.01\text{ g/mol}}{2} \approx 22.00\text{ g/eq}$

Stoichiometric Chemical Demand Formulas

For pure (100%) chemicals, the required feed concentrations in mg/L are given by:

$\text{Hydrated Lime } [Ca(OH)_2]\text{ Demand (mg/L)} = \left[ CO_2 \times \frac{37.05}{22.00} \right] + \left[ CH \times \frac{37.05}{50.05} \right] + \left[ 2 \times MH \times \frac{37.05}{50.05} \right] + \text{Excess Lime}$

$\text{Soda Ash } [Na_2CO_3]\text{ Demand (mg/L)} = NCH \times \frac{53.00}{50.05} = NCH \times 1.059$

Where:

• $CO_2$ is free carbon dioxide in mg/L as $CO_2$
• $CH$ is calcium carbonate hardness removed in mg/L as $CaCO_3$
• $MH$ is magnesium hardness removed in mg/L as $CaCO_3$
• $NCH$ is noncarbonate hardness removed in mg/L as $CaCO_3$
• Excess Lime is typically 30 to 50 mg/L as $Ca(OH)_2$ required to maintain the pH at 10.8 for complete magnesium hydroxide precipitation.

Accounting for Purity: Commercial hydrated lime is typically 90% to 95% pure $Ca(OH)_2$; quicklime is 85% to 95% pure $CaO$; soda ash is 98% to 100% pure $Na_2CO_3$. Dosing calculations must divide pure chemical demand by the commercial purity decimal fraction ($%\text{ purity} / 100$).

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Reference Summary Tables

Table 1: Lime-Soda Softening Precipitation Chemistry Summary

Table 2: Softening Chemical Equivalents and Operational Factors