Numerical Problems on Hardness and Alkalinity
Definition
Hardness is the property of water that prevents it from readily forming lather with soap, mainly due to dissolved calcium and magnesium ions. It is numerically expressed as equivalent concentration of CaCO3 in mg/L (ppm).
Alkalinity is the capacity of water to neutralize acids, mainly caused by the presence of OH⁻, CO3²⁻, and HCO3⁻ ions, and it is also expressed as mg/L as CaCO3.
A numerical problem in this topic typically involves:
- converting ion concentration into CaCO3 equivalents
- calculating total, temporary, and permanent hardness
- determining phenolphthalein and total alkalinity
- interpreting titration data to find water quality parameters
Main Content
1. Hardness Calculations in Water
- Hardness is caused by salts of calcium and magnesium, such as:
- Calcium bicarbonate, Ca(HCO3)2
- Magnesium sulfate, MgSO4
- Calcium chloride, CaCl2
- Magnesium chloride, MgCl2
- Hardness is classified into:
- Temporary hardness: due to bicarbonates of calcium and magnesium
- Permanent hardness: due to chlorides and sulfates of calcium and magnesium
- Total hardness is the sum of temporary and permanent hardness and is usually calculated using:
Hardness as CaCO3 (mg/L) = (mg/L of ion × 50) / equivalent weight of ion
- Equivalent weights commonly used:
- Ca²⁺ = 20
- Mg²⁺ = 12.15
- CaCO3 = 50
- Example: If water contains 40 mg/L of Ca²⁺, hardness due to calcium is:
(40 × 50) / 20 = 100 mg/L as CaCO3
- If water contains 24.3 mg/L of Mg²⁺, hardness due to magnesium is:
(24.3 × 50) / 12.15 = 100 mg/L as CaCO3
- Total hardness is found by adding all calcium and magnesium contributions.
2. Alkalinity Calculations in Water
- Alkalinity is determined by titrating water with a standard acid, usually HCl or H2SO4.
- It is associated with three main species:
- Hydroxide alkalinity due to OH⁻
- Carbonate alkalinity due to CO3²⁻
- Bicarbonate alkalinity due to HCO3⁻
- Alkalinity is measured in two stages:
- Phenolphthalein alkalinity (P-alkalinity): measured up to pH 8.3
- Total alkalinity (T-alkalinity): measured up to pH 4.5
- Numerical problems often use titration values:
- Volume of acid used
- Normality of acid
- Volume of water sample
- Formula:
Alkalinity (mg/L as CaCO3) = (A × N × 50,000) / V
where:
- A = volume of acid used in mL
- N = normality of acid
- V = volume of water sample in mL
- Example: If 20 mL of 0.02 N acid is used for 50 mL water sample:
Alkalinity = (20 × 0.02 × 50,000) / 50 = 400 mg/L as CaCO3
- Depending on titration results, one can determine whether alkalinity is due to hydroxide, carbonate, or bicarbonate.
3. Combined Numerical Problems: Hardness and Alkalinity Relationships
- In many problems, hardness and alkalinity are compared to predict the type of salts present in water.
- Key relation:
- If alkalinity > hardness, some alkalinity is not associated with hardness-producing ions.
- If hardness > alkalinity, some hardness exists as non-carbonate hardness.
- Important classifications:
- Carbonate hardness = hardness associated with alkalinity
- Non-carbonate hardness = total hardness − carbonate hardness
- Rules used in calculations:
- If alkalinity equals total hardness, all hardness is temporary
- If alkalinity is less than hardness, carbonate hardness = alkalinity
- If hardness is less than alkalinity, carbonate hardness = hardness
- Example:
- Total hardness = 250 mg/L as CaCO3
- Total alkalinity = 180 mg/L as CaCO3
- Carbonate hardness = 180 mg/L as CaCO3
- Non-carbonate hardness = 250 − 180 = 70 mg/L as CaCO3
- Such problems are important in:
- boiler water treatment
- scaling prediction
- softening calculation
- water suitability assessment
Working / Process
1. Identify the given data and convert everything to common units
- Check whether the values are given in mg/L, ppm, moles, normality, or titration volume.
- Convert all quantities into a standard basis, usually mg/L as CaCO3.
2. Apply the correct formula based on the parameter required
-
For hardness from ion concentration, use: (mg/L of ion × 50) / equivalent weight
-
For alkalinity from titration: (A × N × 50,000) / V
-
For carbonate and non-carbonate hardness, compare hardness with alkalinity.
3. Interpret the result carefully
- Add hardness contributions from calcium and magnesium to get total hardness.
- Determine whether alkalinity is due to hydroxide, carbonate, or bicarbonate.
- State the final answer with proper units, usually mg/L as CaCO3.
- If required, classify the water as soft, moderately hard, hard, or very hard based on the result.
Advantages / Applications
- Helps in water quality analysis for drinking and industrial purposes.
- Essential for boiler operation, since hardness and alkalinity affect scaling, corrosion, and foaming.
- Useful in designing water softening and treatment processes such as lime-soda, zeolite, and ion exchange methods.
- Supports environmental monitoring of natural water bodies, groundwater, and wastewater.
- Important in chemical and numerical problem-solving for academic exams and laboratory analysis.
- Assists in estimating soap consumption, detergency, and suitability of water for washing and cleaning.
- Helps predict sodium carbonate or lime requirement in treatment plants.
- Enables comparison of water samples on a common basis using CaCO3 equivalents.
Summary
- Hardness and alkalinity are key water-quality parameters expressed mostly as mg/L as CaCO3.
- Hardness calculations involve calcium and magnesium salts, while alkalinity calculations are based on acid titration data.
- The relationship between hardness and alkalinity is used to determine carbonate and non-carbonate hardness.
- Accurate numerical problem-solving is essential for water treatment, boiler safety, and industrial applications.