Langelier Saturation Index and carbonate equilibrium

The Langelier Saturation Index is based on the study of the carbonate equilibrium in water.

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The elements that are important for the carbonate equilibrium are gaseous carbon dioxide CO₂, aqueous carbon dioxide (CO₂)_aq, carbonic acid H₂CO₃, bicarbonate HCO₃^-, carbonate CO₃^2- and solids containing carbonate like calcium carbonate CaCO₃ or magnesium carbonate MgCO₃. Below you can see all the equations that matters for the carbonate equilibrium. These are all equilibrium reactions symbolized by double arrows.

(A)

(B)

Figure 1

Figure 1 can give you the filling that you have to start with gaseous CO₂ in your water to obtain at the end calcium carbonate CaCO₃. This is not the right interpretation. You can have water that naturally contains bicarbonate ions HCO₃^-. So equilibrium equation (A) may occur without the earlier equilibrium reactions with carbon dioxide.

The formula to calculate the Langelier Saturation Index is:

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where:

	Langelier Saturation Index
	saturation pH
	pH value

The pH of a solution is usually known. Equations (A), (B) and (C) are used to compute the saturation pH (pH_s). Equation (C) is a result of the combination of equations (A) and (B).

								(A)
								(B)
----------------------------------------------------------------------
								(C)

The equilibrium constant K_aof equation (A) can be calculated with the following formula:

where:

	= activity coefficient for hydrogen ion
	= activity coefficient for carbonate
	= activity coefficient for bicarbonate
	= concentration of hydrogen ion
	= concentration of carbonate
	= concentration of bicarbonate

The equilibrium constant K_spof equation (B) can be calculated with the following formula:

where:

= solubility product constant

The equilibrium constant K of equation (C) can be calculated with the following formula:

From this point untill the formula of pH_s you only have to use mathematics.

Simplify

If X=Y then log(X) = log(Y) So:

log (X * Y) = log(X) + log(Y) and log(X/Y) = log (X)-log(Y) so:

The formula for pH_s is then:

If you have a water analysis de calcium and bicarbonate concentration are known so you have to calculate the activity coefficients. These coefficients are approximately equal to one but it is better for the accuracy of pH_s to calculate them.

The activity coefficient can be calculated by the following formula:

where:

	= activity coefficient
	= ionic strengh of the solution here water
	= charge on ith ionic species

The ionic strength can be calculated with the following formula:

where:

= Total Dissolved Solids in mg/L or g/m³

All the parameters are now known and the Langelier Saturation Index can be calculated:

The equilibrium constants Ka and K_sp change with temperature. The following values for the equilibrium constants are given in [1]:

Temperature ^oC
5	2,754	8,128
10	3,236	7,080
15	3,715	6,020
20	4,169	5,248
25	4,477	4,571
40	6,026	3,090

The relation between the constant solubility product K_sp of calcium carbonate and temperature is as follows

This equation is found by adding an exponential trendline in the graph of the solubility product constant as a function of the temperature in degree Celsius.

The relation between the equilibrium constant Ka of carbonate/bicarbonate (CO₃^2-/HCO₃^-) and temperature is as follows:

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References:

[1] :Metcalf & Eddy, Inc. Wastewater Engineering Treatment and Reuse, Fourth Edition