Water Hardness

There are different types of water hardness:

•  TH Total Hardness  
•  CH Carbonate Hardness (temporary hardness)
•  NCH Non-carbonate Hardness (permanent hardness)
•  PsH Pseudo Hardness  

Total Hardness TH

Total hardness is defined as the molar concentrations of all multivalent cations in water (i.e. all but the mono-valent cations):

(1) Total Hardness TH   =   Σ multivalent cations

In practice, it is often identified by the sum of the two major fresh-water cations, calcium and magnesium,

(1a) TH   ≈   [Ca] + [Mg]

But this is only an approximation, albeit a good one.1 The correct formula is, according its definition in Eq.(1),

(1b) TH   =   [Ca] + [Mg] + [Sr] + [Ba] + [Fe] + [Mn] + [Al] + …

There are several water hardness scales in use. For example, it can be expressed in meq/L (SI units), or in ppm CaCO3, or in German hardness degrees:

(2a)   TH   in meq/L = 2 × ( [Ca in mM] + [Mg in mM] )
(2b)   TH   in ppm = 100.1 × ( [Ca in mM] + [Mg in mM] )
(2c)   TH   in °d = 0.1339 × [Ca in mg/L] + 0.2307 × [Mg in mg/L]

Conversation rules to other units are given in the table below. The total hardness TH is used to classify aqueous solutions into four water hardness categories.

Note: When the concentrations of Ca and Mg ions are expressed as “ppm CaCO3”, Eq.(1a) is also presented in the form:

(3) Total Hardness TH  =  Calcium Hardness  +  Magnesium Hardness

Carbonate Hardness CH

While total hardness is usually introduced by the cationic content in Eq.(1), there is the opposite viewpoint taken from the anionic content of water, which divides total hardness into carbonate hardness CH (temporary hardness) and non-carbonate hardness NCH (permanent hardness):

(4) Total Hardness TH   =   CH  +  NCH

Carbonate hardness is directly related to alkalinity:

(5a)   CH   in meq/L = [Alk in meq/L]
(5b)   CH   in °dH = 2.809 × [Alk in meq/L]

This means that, in water chemistry the terms carbonate hardness and alkalinity (i.e. M alkalinity or total alkalinity) are synonyms:

(6) Carbonate Hardness CH   =   Alkalinity

The program calculates TH and CH, and displays both quantities in the output tables – as shown here.

Non-Carbonate Hardness NCH

Both total hardness and alkalinity (i.e. carbonate hardness) are common water analysis parameters. The non-carbonate hardness is then – according to Eq.(4) – the difference of both quantities:

(7) NCH  =  TH  –  CH

Carbonate hardness is named temporary hardness because Ca and Mg carbonates precipitate as minerals upon heating, whereas Ca and Mg associated with sulfates, chlorides, or nitrates do not precipitate upon heating (therefore NCH is called permanent hardness).

Alternative Interpretation. The non-carbonate hardness is a measure of the excess of strong acids over strong bases:

(8) NCH   =   strong acids  –  strong bases

Given a water analysis, strong acids manifest themselves in the presence of non-carbonate anions such like sulfate, chloride, nitrate. On the other hand, strong bases are indicated by mono-valent cations (e.g. Na, K, ammonium). In this way, Eq.(8) converts to

(9) NCH [meq/L]  ≈  (2×sulfate + chloride + nitrate)  –  (Na + K + ammonium)

where the measured concentrations should be entered in units of mmol/L.

Pseudo Hardness PsH

The three quantities, TH, CH, and NCH, form the standard vocabulary of water hardness in almost all textbooks. However, there remains one problem when the measured alkalinity (i.e. CH) exceeds the total hardness: CH > TH. In this special case Eq.(4) becomes senseless.

To overcome the problem the hardness vocabulary should be extended by an additional quantity – the pseudo hardness PsH (or “apparent hardness”). It is defined as the “negative NCH” and expressed by the reverse of Eq.(8):

(10) PsH   =   strong bases  –  strong acids   =   - NCH
(11) PsH [meq/L]  ≈  (Na + K + Ammonium)  –  (2×sulfate + chloride + nitrate)

In other words, PsH is the portion of the carbonate hardness that belongs to the mono-valent cations (Na, K, ammonium).

Extension of the Common Hardness Relation

Due to the additional quantity PsH we are able to extend the “standard hardness equation” (4):

(12) Water Hardness:   TH + PsH  =  CH + NCH

It is important to note that in the above formula PsH and NCH are mutually exclusive: Either the strong bases outweight the strong acids (then NCH=0) or vice versa (then PsH=0). Thus, Eq.(12) becomes:

(13a) TH ≥ CH: TH  =  CH + NCH PsH = 0
(13b) TH < CH: TH + PsH  =  CH NCH = 0

where Eq.(13a) represents the “standard” hardness relation of common textbooks.

Example: TH ≥ CH

The case TH ≥ CH is typical for almost all natural waters. For illustration we take the example water C1.sol (Button Open) with the following parameters:

example: total hardness, carbonate hardness, noncarbonate hardness

T 10 °C
pH 7.34
Alk 2.50 mM
Ca 1.40 mM
Mg 0.23 mM
Na 0.30 mM
K 0.05 mM
Cl 0.25 mM
SO4 0.38 mM
NO3 0.15 mM

Run the water sample by click on button Start. The program complains about a small CBE, and we establish exact charge balance by adjustment of DIC (which decreases alkality from 2.50 to 2.45 meq/L). The obtained hardness values are (shown in table column ‘Output 1’):

TH = 2.45 meq/L
CH = 3.26 meq/L

From the difference we get: NCH = 0.81 meq/L. There is no pseudo hardness (PsH=0). The total hardness is determined by the sum of Ca and Mg: TH = (2×1.40 + 2×0.23) meq/L = 3.26 meq/L – as presented in the diagram above.

example: total hardness, carbonate hardness, noncarbonate hardness, pseudo hardness

Example: TH < CH

We use the same example water C1.sol and add 2 mM NaOH (with button Reac)2.

Due to the addition of 2 meq/L of a strong base the alkalinity and, hence, CH increase by 2 meq/L while the total hardness remains unchanged:

CH = 4.45 meq/L
TH = 3.26 meq/L

Here the difference is given by the pseudo hardness: PsH = 1.19 meq/L. There is no NCH.

The value of PsH can also be calculated by inserting the molar concentrations of the input water into Eq.(11):

(14) PsH  =  (2.30 + 0.05)  –  (2×0.38 + 0.25 + 0.15)  =  1.19 meq/L

Note that Na also incorporates the addition of 2 mM NaOH: Na = (2.0 + 0.3) meq/L. The calculated value of Eq.(14) is explicitly shown as ‘Diff’ in the above diagram.

Calcite Precipitation. Due to the addition of 2 mM NaOH the water becomes super-saturated with respect to calcite. The bottom diagram shows the results when 1.37 mM calcite precipitates (column table ‘Output 2’):

CH = (4.45 – 2×1.37) meq/L = 1.71 meq/L
TH = (3.26 – 2×1.37) meq/L = 0.52 meq/L
PsH = CH – TH = 1.19 meq/L   (unchanged)

More Examples

In the following we add one or two reactants with an amount of 1 mM to pure water (H2O).3 The result of about 20 example calculations are presented in the table below where each row refers to one calculation. The values of TH, CH, NCH, and PsH are given in the last columns.

table of example calculations for water hardness

Note: In those cases when CaCO3 (calcite) becomes super-saturated (SI > 0) an extra line was added in order to show the results after calcite precipitation, too.4

Conversion Chart

Water hardness is expressed in different units:

(15a)   German Degrees: 1 °d = 10 mg/L CaO
(15b)   French Degrees: 1 °f = 10 mg/L CaCO3
(15c)   English Degrees: 1 °e = 10 mg CaCO3 in 0.7 L
(15d)   1 ppm CaCO3 (US norm): 1 °a = 1 mg/L CaCO3 5

These units are interrelated by simple conversion rules:


  ppm CaCO3 °d °f °e meq/L mmol/L
1 ppm CaCO3 1 0.06 0.10 0.07 0.02 0.01
1 °d 17.8 1 1.78 1.25 0.357 0.178
1 °f 10.0 0.56 1 0.70 0.2 0.1
1 °e 14.3 0.80 1.43 1 0.285 0.142
1 meq/L 50.04 2.8 5 3.51 1 0.5
1 mmol/L 100.09 5.6 10 7.02 2 1

The conversion factors in this table follow directly from the molar weights of calcium carbonate and calcium oxide:

             
(16a)   CaCO3: 100.09 g/mol   or   1 mg/L CaCO3 = 0.010 mM
(16b)   CaO: 56.08 g/mol   or   10 mg/L CaO = 0.178 mM

Remarks & Footnotes

  1. Other alkaline-earth metals like strontium (Sr) and barium (Ba) have no significant contribution to TH because they are found in trace amounts in natural waters. The same applies also for all other metals (such like Fe, Mn, Al) in pH neutral waters.

  2. To establish exact charge balance click on Setup in the reaction module.

  3. To add reactants to a given water click on Reac.

  4. The reaction 1 mM CaCO3 + 1 mM MgCO3 generates a water which is super-saturated with respect to both calcite and brucite . In the present example, we consider the precipitation of calcite, but we ignore the precipitation of brucite, i.e. Mg(OH)2.

  5. ppm (parts per milion) is defined as 1 ppm = 1 mg / 1 kg = 10-6. For diluted solutions with density ≈1 kg/L the numerical value of both concentration units, ppm and mg/L, are practically equal.

[last modified: 2015-06-23]