The calculation of the universal index of water saturation with calcium carbonate

Ivan Tikhonov

The article presents a method for calculating the pH value, which corresponds to the saturation of water by calcium carbonate and compares this pH value with the pH value obtained by calculating with Langelier saturation index and Stiff & Davis index.

Corrosion and scale formation are the main problems in the usage of equipment and pipelines of water supply and water treatment systems. The separation of the solid phase of calcium carbonate from water is a serious problem, but in the absence of conditions for the release of calcium carbonate, conditions for corrosion processes develop. Corrosion in most cases is a more serious problem than scale, because corrosion will eventually require replacement of the equipment and pipelines.

In the physical sense, the following processes occur in water. If the water tends to release the solid phase of calcium carbonate, it means that the concentration of calcium ions [Са2+] and carbonate [СО32- ] in water is higher than the value of the solubility product for calcium carbonate [KspСаСО3]. Obviously, these conditions imply the absence of free carbon dioxide in the water. If free carbonic acid is present, the water will tend to dissolve the solid calcium carbonate rather than release it. Accordingly, when calcium carbonate is released from the water, the condition for corrosion with a hydrogen depolarizer will be absent, since all the dissolved carbon dioxide provides an equilibrium state of existence of dissolved calcium carbonate. All carbon dioxide is “bound”. In this case, corrosion with oxygen depolarization is possible, but the precipitation of calcium carbonate significantly inhibits the corrosion process on the anode section of the galvanic couple and, accordingly, the entire speed of the corrosion process is significantly reduced.

It is obvious that the slight tendency of water to release solid calcium carbonate can significantly reduce the corrosion rate of steel pipes of water supply systems. But there are processes where even a slight tendency of water to release solid calcium carbonate is highly undesirable. For example, reverse osmosis desalination of water is used. If the water does not contain organics and the water is safe for microbiological indicators (artesian water), the maintenance of conditions under which there is no possibility of precipitation of solid calcium carbonate from the water will allow to lead the process of reverse osmosis desalination of water without chemical washing at all (at least for technical purposes).

It is necessary to ask whether there can be carbonate in water as an ion form rather than solid calcium carbonate, provided that the number of calcium ions in water is greater than or equivalent to the amount of carbonate ion. The presence of a carbonate ion in water is probably due to the hydration properties of water and the ionic strength of the ionic composition of water, i.e. its salt content. Let’s imagine what happens to the carbonate system of water. If initially water that does not contain any ions is saturated with carbon dioxide (CO2), then part of the gas will dissolve in water to form a hydrogen ion (H+) and bicarbonate (HCO3).

СО2+Н2О=H++ НСО3    (1)

The dissociation constant of equation (1) is 4.45* 10-7.

Accordingly, it is possible to calculate the pH value of water if it contains only carbon dioxide.

рН=Lg((4,45*10-7со2)1/2)

Ссо2 – concentration of carbon dioxide, mol/l.

Example:

The water contains 0.1 mmol / l of carbon dioxide or 4.4 mg/l (ppm).

Then

рН= lg((4,45*10-7со2)1/2)=5,17

Or in terms of hydrogen ion concentration

Н=1/105,17=0,00676 mmol/l.

Accordingly, the percentage of dissociation of carbon dioxide in water is:

(0.00676/0.1)*100=6.76 %

If the water contains 1.0 mmol / l of carbon dioxide or 44 mg/l (ppm), we get:

рН=4.67

Н=0.02137 mmol/l

The percentage of dissociation of carbon dioxide into hydrogen ion and bicarbonate is:

(0.02137/1)*100=2.13%

As can be seen, the increase of the concentration of carbon dioxide in water decreases the percentage of its dissociation into ions.

As a result of the saturation of water with carbon dioxide, “excess” hydrogen ions which are not bound to the water molecule are formed in water and, accordingly, if solid calcium carbonate is added to such water, the process of dissolution of calcium carbonate will begin to form calcium ions and bicarbonate ions (equation 2).

СаСО3+2Н++2НСО3=Са2++(НСО3)2    (2)

As can be seen, the process involves 2 moles of carbon dioxide and one mole of calcium carbonate. As a result, the “excess” hydrogen of carbon dioxide binds into bicarbonate and is called semi-bound carbon dioxide. That is, this hydrogen ion cannot participate in the process of steel corrosion as a depolarizer.

The process (2) is equilibrium and can proceed in both directions when the concentration of the ions included in equation (2) change. So, when distilling carbon dioxide (heating water), the process will go to the left side and solid calcium carbonate will be released from the water.

It is obvious that the existence of carbonate in water is possible only as solid in the composition of calcium carbonate if we consider that initially only calcium carbonate was dissolved in such water. Nevertheless, in such water, the existence of the carbonate ion is observed and its concentration can be determined by the product of the solubility of calcium carbonate. In this case the concentration is extremely low. Probably, it can be assumed that after the dissolution of calcium carbonate in water, ions are formed and, accordingly, the intensity of the electromagnetic field formed by ions (ionic strength ) increases, which allows a small amount of carbonate to exist in the form of ions when the concentration of carbon dioxide in water is below equilibrium. It is necessary to take into account the hydrating properties of water. It is the decrease in the effect of water hydration (decrease in water viscosity) that can explain the decrease in the value of KspСаСО3 with the increase in water temperature.

If you neglect the activity of calcium ions and carbonate, then according to KspСаСО3=4.44*10-9 [1] with equal concentrations of calcium and carbonate, in water at 25 C it will be only 4 mg/l (ppm) of carbonate.

On the condition that there is a carbonate ion in water you can calculate the pH value of such water using the following equation:

рН=10,328+lg (CO3/НСО3)  (3)

where,

10,328-negative logarithm of the dissociation constant of carbon dioxide in the second stage;

CO3, HCO3-concentration of carbonate and bicarbonate, mol/l.

You need to understand what is required to determine the pH of water when it is in a state of carbon balance, i.e. water is saturated with calcium carbonate and a further increase in the concentration of calcium carbonate and (or) a decrease in the concentration of carbon dioxide will lead to the beginning of the process of formation of solid phase of calcium carbonate. Conversely, a decrease in the concentration of calcium carbonate and (or) an increase in the concentration of carbon dioxide will lead to the beginning of the corrosion process with hydrogen depolarization.

This pH value of water in the determination of Langelier saturation index is defined as pHS. If the current pH of the water is less than the pHS, then obviously there is free carbon dioxide in such water and the water is corrosive. If the current pH of water is greater than pHS, then such water is prone to deposition of solid calcium carbonate, because KspСаСО3 at this current pH is greater than 4.4*10-9.

As a result, the whole question comes down to the definition of pHS water in which the amount of calcium and carbonate corresponds to KspСаСО3 which is equal to 4.4*10-9.

To calculate pHS, the author offers the following method:

  1. The concentration of the carbonate ion is determined on the basis of the product of the solubility of calcium carbonate.

The product of solubility, taking into account the coefficients of ion activity, is determined as follows

KspСаСО3са*fсасо3*fсо3

where,

Сса Ссо3 – calcium and carbonate concentration, mol/l;

fса fсо3 – activity coefficients of calcium and carbonate. It is determined by figure 1.

Given that calcium and carbonate are divalent ions, we assume that fса=fсо3, respectively

KspСаСО3са* Ссо3*f2

Then,

Ссо3= KspСаСО3/(Сса* f2)   (4)

  1. Then the pHS is calculated by equation (3), taking into account the activity coefficients.

рНs=10,328+lg ((Ссо3* fсо3)/(Снсо3*fнсо3))     (5).

Снсо3 – bicarbonate concentration in water, mol/l;  

fнсо3 – the coefficient of activity of bicarbonate. It is determined by figure 1.

To verify this, the author calculated the pHS according to this technique and Langelier index using the R. O. S. A. 9.0 program, as well as using the nomogram presented in [2].

The results of the calculations are presented in table 1.

Table 1 contains the following data.

In line 1, the calculation of the pHS value by three methods for the following water composition is presented: Ca-1.25 mmol/l; HCO3-2.5 mmol/l. On the basis of these values, the salt content (S), the ionic strength (I) are calculated, and the activity coefficients for the ions in figure 1 are determined. In line 2: sodium chloride (NaCl) – 10 mmol/l is added to the water. Accordingly, the salinity and ionic strength increase and the activity coefficients decrease. In line 3: 500 mmol/l (NaCl) is added to the water and the salinity of the water is 29.5 g/l (ppt). In this case, the pH value calculated by Langelier index gives incorrect values and R. O. S. A. suggests using Stiff & Davis index. The pHS value calculated according to Stiff & Davis index is represented in the denominator in the column “R. O. S. A. 9.0” and, respectively, in the numerator of the same column are the pHS values calculated according to Langelier index.

In lines 4,5,6 the saturation index for Ca – 5 mmol/l, HCO3 – 10 mmol/l with the addition of sodium chloride in the same quantities as in lines 2,3 is calculated.

In lines 7,8,9,10 the saturation index for Ca – 10 mmol/l, HCO3 – 20 mmol/l with addition of sodium chloride is calculated. Only in line 9 is the calculation made with the addition of 100 mmol/l NaCl.

For each technique, the pHS value is calculated for 3 water temperatures: 15 0C, 25 0C, 35 0C.

The calculation results presented in table 1 cover a fairly wide range of calcium bicarbonate concentrations and total salinity.

In calculating the pHS value of equation (5), the following assumptions are used:

  1. The activity coefficients for calcium and carbonate are assumed to be the same. Also, the influence of temperature on the ion activity coefficient is not taken into account due to its insignificant influence.
  2. The data for KspСаСО3 are taken from [1] for temperatures 25 0C, 50 0C, 150 0C, 200 0 The values of KspСаСО3 for 15 0C and 35 0C are obtained by the author independently, assuming that their dependence corresponds to the dependence of the solubility of calcium carbonate on temperature.

KspСаСО3=5,6*10-9 at a temperature of 15 C;

KspСаСО3=4,4*10-9 at a temperature of 25 C;

KspСаСО3=3,2*10-9 at a temperature of 35 C;

  1. The calculation of the pHS value using the nomogram (third column) is limited to the salinity of water ranging to 15 g/l (ppt) and HCO3 not more than 10 mmol/l.

Table 1. Calculated values of pHs depending on temperature and salinity by three different methods

п/п Water composition and design parameters Proposed (universal) method R.O.S.A. 9.0 Simplified method (nomogram)
15 0C 25 0C 35 0C 15 0C 25 0C 35 0C 15 0C 25 0C 35 0C
1 Са – 1,25 mmol/l, НСО3 – 2,5 mmol/l, S – 203 mg/l (ppm), I=0,00375 mol/l, f(НСО3)=0,935, f(Са)=f(CO3)=0,77 7,72 7,62 7,48 7,87 7,64 7,44 7,71 7,58 7,45
2 Са – 1,25 mmol/l, НСО3 – 2,5 mmol/l, NaCl-10 mmol/l,     S – 788 mg/l (ppm), I=0,01375 mol/l, f(НСО3)=0,89, f(Са)=f(CO3)=0,63 7,83 7,73 7,59 7,94 7,74 7,5 7,86 7,73 7,6
7,4 7,21 7,02
3 Са – 1,25 mmol/l, НСО3 – 2,5 mmol/l, NaCl-500 mmol/l,  S – 29,5 g/l (ppt), I=0,51 mol/l, f(НСО3)=0,55, f(Са)=f(CO3)=0,12 8,76 8,66 8,53 8,03 7,81 7,6      
8,88 8,69 8,5
4 Са – 5 mmol/l, НСО3 – 10 mmol/l,  S – 812 mg/l (ppm), I=0,015 mol/l, f(НСО3)=0,88, f(Са)=f(CO3)=0,63 6,63 6,53 6,4 6,73 6,52 6,3 6,68 6,52 6,4
6,23 6,04 5,85
5 Са – 5 mmol/l, НСО3 – 10 mmol/l, NaCl- 10 mmol/l, S – 1400 mg/l (ppm), I=0,025 mol/l, f(НСО3)=0,87, f(Са)=f(CO3)=0,56 6,69 6,58 6,45 6,76 6,53 6,33 6,73 6,6 6,46
6,44 6,25 6,05
6 Са – 5 mmol/l, НСО3 – 10 mmol/l, NaCl- 500 mmol/l, S – 30 g/l (ppt), I=0,53 mol/l, f(НСО3)=0,55, f(Са)=f(CO3)=0,12 7,56 7,45 7,32 6,84 6,6 6,4      
7,68 7,49 7,31
7 Са – 10 mmol/l, НСО3 – 20 mmol/l,  S- 1620 mg/l (ppm), I=0,03 mol/l, f(НСО3)=0,85, f(Са)=f(CO3)=0,55 6,11 6,00 5,88 6,16 5,94 5,74      
5,91 5,72 5,53
8 Са – 10 mmol/l, НСО3 – 20 mmol/l, NaCl- 10 mmol/l,    S – 2,2 g/l (ppt), I=0,04 mol/l, f(НСО3)=0,83, f(Са)=f(CO3)=0,5 6,16 6,05 5,92 6,18 5,95 5,74      
6,02 5,83 5,64
9 Са – 10 mmol/l, НСО3 – 20 mmol/l, NaCl- 100 mmol/l, S – 7,5 g/l (ppt), I=0,131 mol/l, f(НСО3)=0,73, f(Са)=f(CO3)=0,33 6,4 6,29 6,15 6,21 5,98 5,77      
6,5 6,31 6,12
10 Са – 10 mmol/l, НСО3 – 20 mmol/l, NaCl- 500 mmol/l, S – 30,8 g/l (ppt), I=0,54 mol/l, f(НСО3)=0,55, f(Са)=f(CO3)=0,12 6,96  6,85  6,72 6,23 6,01 5,8      
7,09 6,9 6,72

Having analyzied the obtained pHS values, we can draw the main conclusion that at a water temperature of 25 C and an ionic strength (I) of not more than 0.03, the pH values calculated by three methods are almost equal (lines 1,2,4,5,7).

It can be noted that the pHS values obtained by the proposed method practically correspond to the pHS values obtained by the simplified method in the entire temperature range under consideration. The pHS values calculated using the ROSA program tend to be greater at 0.1-0.15 pH units at 15 0C and to be less at 0.05-0.1 pH units at 35 0C of the pH value calculated by the proposed method.

At the beginning of the calculation, The author pointed to the condition of applicability of KspСаСО3. If we focus on the values of pHS obtained using the R.O.S.A. program, the values of the product of solubility should be as follows:

KspСаСО3=7,2*10-9 at a temperature of 15 0C;

KspСаСО3=4,4*10-9 at a temperature of 25 0C;

KspСаСО3=3,0*10-9 at a temperature of 35 0C;

When using these values of KspСаСО3, the difference in the calculations of the pHS value according to the proposed method and ROSA in the entire temperature range under consideration was not more than 0.03 pH units.

Figure 2 shows graphs of the influence of temperature on KspСаСО3. Ksp1 is the dependence graph constructed using the solubility of calcium carbonate. Ksp2 is the dependence graph constructed on the basis of the condition of equality of pHS values calculated according to the ROSA program and the proposed method.

It should also be noted that at the ionic strength I > 0.04 (salinity about 2.2 g/l (ppt)) the pHS values calculated by the proposed method and with ROSA for Langelier index and Stiff & Davis index have a slight difference (line 8).

With an ionic strength of more than 0.1 (salinity of about 7 g/l (ppt)), the pHS values of  Langelier index calculated by ROSA begin to give obviously incorrect values, but the pHS values of Stiff & Davis index calculated by ROSA almost completely correspond to the pHS values calculated with the proposed method (line 9).

We can say that Langelier index can be used at the salinity of water up to 4.0-5.0 g/l (ppt). This value of salinity is a kind of a transition boundary from Langelier index to Stiff & Davis index.

For water with a salinity of about 30 g/l (ppt) (I > 0.5) (lines 3, 6, 10) the pHS values according to the proposed method are correlated with the pHS value for Stiff & Davis index with a discrepancy of not more than 0.15 pH units (as with Langelier index at low salinity).

When using the solubility product values according to the Ksp2 dependence (figure 2), the pHS values are practically the same for all three methods under consideration in the entire temperature and salinity range under consideration. The discrepancy is not more than 0.03 pH units.

To sum up, it should be said that the use of the pH value in the calculation, which corresponds to the state of saturation of calcium carbonate with water, according to the proposed method, has a relatively simple and understandable meaning. The calculation is quite simple and clear. To make the calculation it is only necessary to determine the value of KspСаСО3 depending on the water temperature in figure 2 (for the graph Ksp2) and the values of the activity coefficients of calcium ions, carbonate and bicarbonate depending on the ionic strength of the solution in figure 1.

It should be reiterated that the ionic strength is calculated as the half-sum of the ions taken in moles/l multiplied by the square of the valence of each ion. Or for water containing ions Ca, Mg, Na, HCO3, SO4, Cl:

I= 0.5*(Ca*4+Mg*4+Na*1+HCO3*1+SO4*4+Cl*1)

Next, determine the concentration of carbonate ion in water by the formula (4).

Then use the obtained data into equation (5).

This method of calculating the pH value of water saturated with calcium carbonate, can be called universal, because the results are quite reliable in the entire range of salinity (fresh, brackish and salty waters). At the same time, other indices work only in a certain range of water salinity.

Figure 1.

Figure 2.

Reference materials:

  1. Наумов Г.Б., Рыженко Б.Н., Ходаковский И.Л. Справочник термодинамических величин. – М.: Атомиздат, 1971.
  2. СНиП 2.04.02 – 84 «Водоснабжение. Наружные сети и сооружения»

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