Understanding the Calculation of Water Hardness: A Technical Overview

Water hardness calculation quantifies dissolved minerals, primarily calcium and magnesium ions. This process is essential for water quality assessment and treatment design.

In this article, you will find detailed formulas, common values, and real-world applications for precise water hardness determination.

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Calculate total hardness in mg/L CaCO3 given Ca2+ = 80 mg/L and Mg2+ = 12 mg/L.
Determine hardness classification for water with 150 mg/L CaCO3 equivalent.
Convert hardness from ppm CaCO3 to grains per gallon (gpg).
Calculate temporary hardness from bicarbonate concentration of 200 mg/L.

Comprehensive Tables of Common Water Hardness Values

Parameter	Typical Concentration Range	Units	Notes
Calcium (Ca²⁺)	10 – 200	mg/L	Primary contributor to hardness
Magnesium (Mg²⁺)	5 – 50	mg/L	Secondary contributor to hardness
Total Hardness (as CaCO₃)	0 – 500	mg/L	Sum of Ca and Mg expressed as CaCO₃
Temporary Hardness (Carbonate Hardness)	0 – 300	mg/L CaCO₃	Due to bicarbonates of Ca and Mg
Permanent Hardness (Non-Carbonate Hardness)	0 – 200	mg/L CaCO₃	Due to sulfates, chlorides of Ca and Mg
Hardness Classification			Soft: 0-60 mg/L Moderately Hard: 61-120 mg/L Hard: 121-180 mg/L Very Hard: >180 mg/L
Grains per Gallon (gpg)	0 – 30	gpg	1 gpg = 17.1 mg/L as CaCO₃

Fundamental Formulas for Calculating Water Hardness

Water hardness is primarily expressed as equivalent calcium carbonate (CaCO₃) concentration. The calculation involves converting concentrations of calcium and magnesium ions into CaCO₃ equivalents using their molar masses and valences.

1. Total Hardness Calculation

The total hardness (TH) in mg/L as CaCO₃ is calculated by summing the hardness contributions from calcium and magnesium:

TH = (Ca2+ concentration × 50 / 20.04) + (Mg2+ concentration × 50 / 12.15)

TH: Total hardness (mg/L as CaCO₃)
Ca²⁺ concentration: Calcium ion concentration (mg/L)
Mg²⁺ concentration: Magnesium ion concentration (mg/L)
50: Equivalent weight of CaCO₃ (g/mol)
20.04: Atomic weight of Ca (g/mol)
12.15: Atomic weight of Mg (g/mol)

This formula converts the mass of Ca and Mg ions to their equivalent mass as CaCO₃, considering their molar masses and valences.

2. Temporary Hardness (TH_temp)

Temporary hardness is caused by bicarbonates of calcium and magnesium and can be removed by boiling. It is calculated as:

THtemp = (HCO3– concentration × 50 / 61) – (Ca2+ + Mg2+ concentrations as CaCO3)

HCO₃^– concentration: Bicarbonate ion concentration (mg/L)
61: Molecular weight of HCO₃^– (g/mol)

Note: The subtraction accounts for the portion of hardness not due to bicarbonates.

3. Permanent Hardness (TH_perm)

Permanent hardness is the portion of hardness that remains after boiling, caused by sulfates and chlorides:

THperm = TH – THtemp

4. Conversion Between mg/L and Grains per Gallon (gpg)

Water hardness is often expressed in grains per gallon (gpg), where 1 gpg = 17.1 mg/L as CaCO₃. Conversion formulas:

Hardness (gpg) = Hardness (mg/L) / 17.1

Hardness (mg/L) = Hardness (gpg) × 17.1

5. Hardness Classification Based on CaCO₃ Concentration

Water hardness is classified as follows:

Soft: 0 – 60 mg/L
Moderately Hard: 61 – 120 mg/L
Hard: 121 – 180 mg/L
Very Hard: >180 mg/L

This classification guides treatment decisions and industrial applications.

Detailed Explanation of Variables and Their Typical Values

Calcium (Ca²⁺): Typically ranges from 10 to 200 mg/L in natural waters. It is the dominant ion contributing to hardness due to its prevalence in minerals like limestone and gypsum.
Magnesium (Mg²⁺): Usually found between 5 and 50 mg/L. Though less abundant than calcium, magnesium significantly affects hardness and water taste.
Bicarbonate (HCO₃^–): Concentrations vary widely, often between 50 and 300 mg/L. It determines temporary hardness and influences alkalinity.
Total Hardness (TH): Sum of calcium and magnesium hardness expressed as CaCO₃. Values above 180 mg/L indicate very hard water, which can cause scaling and reduce soap efficiency.
Grains per Gallon (gpg): A traditional unit in the US, where 1 gpg equals 17.1 mg/L as CaCO₃. Useful for water softener sizing and industrial standards.

Real-World Applications and Case Studies

Case Study 1: Municipal Water Supply Hardness Assessment

A municipal water treatment plant receives raw water with the following ion concentrations:

Calcium (Ca²⁺): 85 mg/L
Magnesium (Mg²⁺): 15 mg/L
Bicarbonate (HCO₃^–): 220 mg/L

The goal is to calculate total hardness, temporary hardness, permanent hardness, and classify the water hardness.

Step 1: Calculate Total Hardness (TH)

TH = (85 × 50 / 20.04) + (15 × 50 / 12.15) = (85 × 2.495) + (15 × 4.115) = 212.1 + 61.7 = 273.8 mg/L as CaCO3

Step 2: Calculate Temporary Hardness (TH_temp)

HCO3– equivalent as CaCO3 = 220 × 50 / 61 = 180.3 mg/L

Temporary hardness is the lesser of the bicarbonate equivalent or total hardness:

THtemp = 180.3 mg/L

Step 3: Calculate Permanent Hardness (TH_perm)

THperm = TH – THtemp = 273.8 – 180.3 = 93.5 mg/L

Step 4: Hardness Classification

With total hardness of 273.8 mg/L, the water is classified as very hard.

Case Study 2: Industrial Boiler Feed Water Hardness Control

An industrial facility must maintain boiler feed water hardness below 100 mg/L as CaCO₃ to prevent scaling. The raw water analysis shows:

Calcium (Ca²⁺): 40 mg/L
Magnesium (Mg²⁺): 10 mg/L
Bicarbonate (HCO₃^–): 150 mg/L

The plant uses ion exchange softening and needs to calculate the hardness before and after treatment.

Step 1: Calculate Raw Water Total Hardness

TH = (40 × 50 / 20.04) + (10 × 50 / 12.15) = (40 × 2.495) + (10 × 4.115) = 99.8 + 41.15 = 140.95 mg/L

Step 2: Calculate Temporary Hardness

HCO3– equivalent = 150 × 50 / 61 = 122.95 mg/L

Temporary hardness is 122.95 mg/L.

Step 3: Calculate Permanent Hardness

THperm = 140.95 – 122.95 = 18 mg/L

Step 4: Post-Treatment Hardness

The ion exchange system removes all calcium and magnesium ions, reducing hardness to near zero. However, residual hardness may remain due to incomplete removal or other ions.

Regular monitoring using the formulas above ensures compliance with boiler feed water standards.

Additional Considerations in Water Hardness Calculation

Ion Interference: Other ions such as iron (Fe²⁺), manganese (Mn²⁺), and strontium (Sr²⁺) can contribute to hardness but are typically minor.
Analytical Methods: Hardness is commonly measured by EDTA titration, atomic absorption spectroscopy, or ion chromatography, each with specific accuracy and detection limits.
Regulatory Standards: Agencies like the EPA and WHO provide guidelines on acceptable hardness levels for drinking water and industrial use.
Water Softening Technologies: Understanding hardness calculation is critical for designing ion exchange, lime softening, and membrane treatment systems.