DERIVATION OF EQUATION EXPRESSING THE SATURATION INDEX OF A NATURAL WATER

29.3 DERIVATION OF EQUATION EXPRESSING THE SATURATION INDEX OF A NATURAL WATER

In accord with the discussion of saturation index in Section 7.2.6.1 , we have K s ′= ( Ca 2 + )( CO 2 3 − ) (29.24)

Assume that salts of weak acids other than carbonic acid are absent. Then when a water is titrated, equivalents of added acid equal equivalents of

carbonate and bicarbonate, plus OH − or minus H + , depending on pH of the water:

( alk )( + H + ) → 2 ( CO 2 −

3 )( + HCO 3 )( + OH ) (29.26)

462 APPENDIX

where (alk) = alkalinity, or titratable equivalents of base per liter (by titrating with acid) obtained by using methyl orange as indicator. Concentrations of (H + ) and (OH − ) are small between pH values 4.5 and 10.3 and may be neglected. From Eq. (29.25) , we obtain

From Eq. (29.26) ,

Equation (29.29) was derived by Langelier [3] on the assumption that K ′ s and K 2 ′ are based on concentrations (moles/liter) rather than on activities. For example, referring to (29.24) , if K s is the true activity product, then K s =′ K s γ ± 2 , where γ ± refers to the mean ion activity coeffi cient for CaCO 3 . The activity coeffi cient was approximated by Langelier using the Debye – H ü ckel theory,

2 − log γ = 0.5 z 1/2 μ , where μ is the ionic strength and z is the valence. Hence, concentrations of CO 2 3 − and HCO − 3 obtained by titration can be equated to

corresponding concentrations of these species in K ′ s and K 2 ′ . Accordingly, K ′ s and K ′ 2 vary not only with temperature, but also with total dissolved solids because of the effect of ionic strength of a solution on activities of specifi c ions.

Substituting (29.29) into (29.27) , we obtain

and substituting (29.30) into (29.24) , we get

( alk )

( Ca )

=′ K s (29.31)

( H + ) 12 + K 2 ′ /H ( + )

Taking logarithms of both sides and using the notation log 1/ α = p α , we obtain

DERIVATION OF EQUATION EXPRESSING THE SATUR ATION INDEX

2 K T A B L E 29.2. 2 Values of

log 1 + ′

as a Function of pH s

pH s : 10.3 10.0 9.7 9.4 9.1 2 K

( s + ′+ Ca ) + p ( alk ) log + ⎛ 1 + ⎞ (29.32) ⎝⎜ ( H + ) s ⎠⎟

where pH s is the pH of a given water at which solid CaCO 3 is in equilibrium with its saturated solution. The last term is ordinarily small and can be omitted when pH s < 9.5. Based on the value K

. 4 8 10 × − ′= 11 2 , typical values as a function of pH s in the alkaline range at 25 ° C are given in Table 29.2 . Values of ( p K 2 ′− p K s ′ ) decrease with increasing temperature as follows: 0 ° C, 2.48; 20 ° C, 2.04; 25 ° C, 1.96; 50 ° C, 1.54. In the presence of other salts (e.g., NaCl, Na 2 SO 4 , or MgSO 4 ), the increasing ionic strength of the solution depresses the activity of other ions in solution. This effect increases values of ( p K 2 ′− p K s ′ ). For example, at 25 ° C, at a total dissolved - solids content of 100 ppm, the value is 2.13, and for 500 ppm it is 2.19.

A nomogram for obtaining pH s of a water at various temperatures and dissolved - solids content was constructed by C. Hoover [4] . A chart for the same purpose as prepared by Powell, Bacon, and Lill [5] is reproduced in Fig. 29.4 . To use the chart, we must know the alkalinity of a water and calcium ion con-

centration calculated as ppm CaCO 3 , total dissolved solids in ppm, and the temperature. The saturation index is then the algebraic difference between the measured pH of a water and the computed pH s :

Saturation index = pH measured − pH s (29.33) To calculate the saturation index at above - room temperature, the actual pH of

water at the higher temperature should be used. This can be estimated from the room temperature value by using Fig. 29.5 [5] , which gives values for two waters of differing alkalinity.

464 APPENDIX

Figure 29.4. Chart for calculating saturation index (Powell, Bacon, and Lill [5] ). ( “ Ca ” and “ alkalinity ” are expressed as ppm CaCO 3 , and temperature is expressed in ° F.)

DERIVATION OF EQUATION EXPRESSING THE SATUR ATION INDEX

Figure 29.5. Values of pH of water at elevated temperatures (Powell, Bacon, and Lill [5] ). ( a ) For water at 25 ppm alkalinity (methyl orange end point). ( b ) For water of 100 ppm alkalin- ity (methyl orange end point).

466 APPENDIX

Figure 29.5. Continued

DERIVATION OF POTENTIAL CHANGE ALONG A CPP