A Slide Rule for Carbonate Equilibria and Alkalinitv in Water Sumdies II J
A. A. HIRSCH, State Department of Education, Baton Rouge, La.
C
(OH-), as p. p. m. of CaCOs =
came the realization that the various ionic forms may be present in ordinarily unsuspected p H zones and more than two species of carbonate or alkalinity may coexist in the same water sample. Even from the standpoint of conventional stoichiometry the determination of carbon dioxide by titration is open to errors ( I ) and the phenolphthalein-methyl orange method has been shown to be imperfect (4).It is well known that hydroxide alkalinity by titration does not necessarily agree with the calculated value based on the p H of the water. True evaluation and differentiation of the various types of carbonate ions in solution are important wherever carbonate equilibria play a role, as in a n accurate statement of ionic content, softening calculations, corrosion control, and boiler v a l e conditioning a t low pressures. Distribution of the various carbonate forms as dependent on pH value was first calculated from ionization constants and the principle of electroneutrality by McKinney (W), who formulated the fraction each form represented of the total carbonate determined by evolution or by titration through the pH interval from 8.0 to 4.5. By somewhat similar reasoning but adhering to the A. P. H. A. definition of total alkalinity, Langelier and DeMartini (I) developed equilibrium equations to give the molar concentration of each carbonate and alkalinity ion as a function of pH and total alkalinity. Since most water plant laboratories employ the usual A. P. H . A. titration for total alkalinity rather than determine total carbonates by evolution, the latter set of equations is more applicable to treatment plant data and hence was selected for the basis of the slide rule described below. This total alkalinity equation assumes absence of salts of other weak acids; this assumption is usually valid except in the case of phosphate-treated boiler waters. Moore ( 3 ) modified Langelier and DeMartini's equations, substituting routine plant units instead of fundamental molal concentrations, and dreiv a series of graphs showing separately each form of carbonate and alkalinity ion plotted against total alkalinity. Deviation between phenolphthalein-methyl orange titration results and the calculated alkalinity pattern by means of equilibrium methods is especially marked a t high pH values. Moore's versions of Langelier and DeMartini's equations, n-ith substituted ionization constant values for carbonic acid R,= 4.54 X lo-', KP = 5.61 X lo-", and for water, K , = 1 x 10-1'. a t 25" C. are as follows:
CO?. ab p p. m.of COz =
x
10-10 (H +) Alk = total alkalinity, as p. p. m. of CaCOs (H+) = hydrogen-ion concentration, moles per liter
OIKCIDENT with the application of mass action relations to the determination of carbonate in water supplies
5
(4)'
I n considering the effects of temperature and ionic strength on the validity of the above equations Moore noted that errors due to usual water temperatures lower than 25" C. and presence of other salts were compensatory in direction; he concluded that for the temperature range 15' to 25" C., pH below 10.0, and dissolved solids less than 500 p. p. m., all corrections might be neglected and the equations and his graphs used in the form given. Accordingly, the slide rule based on these equations will likewise be subject to the same limitations. For use beyond this range the underlying equations should be suitably corrected, as explained by Moore, for variations in the values of the ionization constants with temperature and dissolved salts. I n the first three equations the recurring function Alk
10-14
11.22 x 10-11 (H+) represents the total alkalinity titer corrected for the ions of water and the second dissociation constant of carbonic acid; these corrections are most important a t low alkalinities and a t pH values differing from neutrality. For convenience let
Then Equations 1, 2 , and 3 may be concisely rewritten: COZ, as p. p. m. of COZ = 9.70 X 1Olo(H+) X F (Alk, pH) (6)
(HC08-), as p. p. m. of CaC08 = 50,000 X F (Alk, pH) (7)
Since the individual carbonate concentrations involve only (H+) and F (Alk,pH), this set of equations is well adapted for special slide rule design. The object of the slide rule described below is t o provide a mechanical solution to Equations 4, 6 , 7, and 8, so that the concentration of Alk 10-14 various carbonate and alkalinity species can be calculated expeditiously from the total 9.70 X 1 O l o (HT) alkalinity and p H value. I n most cases a 11.22 x 10-11 single setting of the rule suffices to define the -I(H+) carbonate and alkalinity distribution in a Alk 10-14 water sample,
(HCOa-), as p. p. m. of CaCOs = 50,000 X
Alk
(COS--),asp. p. m.of CaC03 =
5.G1
x
(H +)
Description
x 10-11 (H+)
11.22
x
[m11.22 (H+) - m1 (3) x 10-11 10-14
+
4-
(H+) 943
In the carbonate equilibria slide rule (Figure 1) the stock contains a logarithmic grrt h of F (Alk, pH) and a scale for bicarbonate alialinity. Incidentally, all carbonate forms are read off from this particular scale. The slide consists of two pH scales, the upper for deter&ing carbon dioxide and the lower for evaluating
944
I N D U S T R I A L A N D ENGINEERING CHEMISTRY
Vol. 14, No. 12
normal carbonate alkalinity, Along the lower pH scale is also drawn the scale for hydroxide alkalinity, as this latter quantity is de ndent sole1 on the pH value. the F pH) graph, the alkalinity parameters a t very low alkalinities exhibit an ogive type of curvature; a t total alkalinities above 50 p. p. m. as calcium carbonate the curvature at low pH due to hydrogen ions becomes masked by the comparatively larger alkalinity. This curl a t low pH and low total alkalinity indicates that true bicarbonate alkalinity is somewhat greater than the apparent bicarbonate value by titration, as at high (H+) a head start is already covered toward the end point. A t high pH these curves sweep decidedly toward the left, since the value of F (Alk, pH) rapidly decreases with increasing hydroxyl concentration. A convenient way to construct this rule for personal use is to photostat Figure 1 to desired size, cut out the scales, and mount them on stiff material like 6-mm. (0.25-inch) plywood. A guide strip is fastened to the top of the F (Alk, pH) plot along which a triangle, used as an index or hairline, can be moved in alignment as with a tee square.
E
(-46,
Method of Use BICARBONATE ALKALINITY.To find bicarbonate alkalinity, simply set the index at a point in the F (Alk, pH) lot corresponding to the given pH and totay alkalinity; read the bicarbonate alkalinity, as p. p. m. of calcium carbonate, directly below on the (HCOB-) scale. Using E uation 7 the numerical value of F (Alk, pqd) may be readily calculated by dividing the bicarbonate alkalinity by 50,000. CARBONDIOXIDE. With the index still in position in the F (Alk, pH) plot, shift the slide so the iven pH figure on the upper scale of the sfde comes under the index. 0 posite the C02arrow is read the p. p. m. orcarbon dioxlde on the (HCOs-) scale. NORMAL CARBONATE.In a manner analogous to the above, carbonate alkalinity is found with the index still fixed a t the appropriate point in the F (Alk, pH) graph' in this case shift the slide so that the p d figure in the lower sliding scale falls under the hairline. Normal carbonate alkalinity as p. p. m. of calcium carbonate, is read afong the (HCOs-) scale opposite the (Cod--) arrow. HYDROXIDE ALKALINITY.Simply note the hydroxide alkalinity, as p. p. m. of calcium carbonate, on the superposed scale above the prevailing pH on the (Cos--) scale. Example I. Given a well water with H 6.6 and total alkalinity of 300 p..m., l n d carbon dioxide, bicarbonate alEalinity, and normal carbonate alkalinity. In the F (Alk, pH) plot set the index where the 300 p. p. m. total alkalinity line crosses the pH 6.6 ordinate. Under the hairline read (HCOs-) equals 300 . p. m. as CaCOa. Above 50 p. p. m. totaralkalinity (Hi) as high as molar is without effect on the value of F (AZk, pH); hence above this total alkalinity @l alkalinity is in the bicarbonate form in slightly acid waters. Set the upper pH sliding scale so the pH value 6.6 comes under the hairline; read where the C01 arrow points on the (HCOa-) scale that carbon dioxide equals 147 p. p. m. of CO2. Ezample 8. Given a lime-treated water having a pH of 9.8 and total alkalinity equal to 40 p. p. m. of CaCOs, find bicarbonate alkalinity, normal carbonate alkalinity, and hydroxide alkalinity. In the F (Alk, H) graph set the index .at H 9.8 and totctfalkalinity 40; immediate& beneath, on the (HCOs-) scale, read bicarbonate alkalinity
.
w
rn8
L -
P
ANALYTICAL EDITION
December 15, 1942
FOR RATIOS O F CONCENTRATIONS TABLE1. FORMULAS
VARIOUS CARBONATE
9.70 X 1010 X F ( A l k , pH)
X F ( A l k , pH)
5.16 X 1 0 7 or 1 0 p ~ n.zm -
.......................
CO:, as p. p. rn. of COz (HCOa-), as p. p. rn. of CaCOi
1.94 X 106(H+) or 10d*m-PH
(COz--), a s p . p. m. of CaCOa (OH-), a s p . p. m. of CaCOa
1.728 x I O : ~ ( H +or) ~io*a.*a'- *pH 1.94 X lOZo(H+)r X F (Alk, pH)
AND
Numerators (HCOs-1, as p. p. m. of CaCO;
COX,as p. p. m. of COS
Bases ( H + ) , moles per liter
Alk
OF
945
(H +)
.................... 8.91 X lOg(H+) or 109.9s- DH 1 X lO"(H+) X F ( A l k , pH)
ALKALINITY SPECIES (COa--), a s p . p. m. of CaCO; lo-' X F ( A l k , pH) (H+)Z 5.78 x 10-17 or 1 0 2 p H - i i . ~ : ~ (H + ) Z 1.122 x lo-" or lopH- 8.W (H +)
...................
1.122 X 10' X F ( A l k , pH)
total alkalinity, as p. p. m. of CaCOs. 10-14
F ( A l k , pH) =
e uals 21.7 p. p. m. as CaCOa. Move the slide so pH 9.8 on the (80d--)scale falls under the index; opposite the (COS--) arrow read, along the (HCOs-) scale, normal carbonate alkalinity equals 15.1 p. p. m. as CaCOs. Directly under the hairline, reading along the (OH-) scale, is found hydroxide alkalinity equals 3.2 p. p. m. as CaCOa. Usually p H values fall decidedly below 8.0 in the cases of streams and ground waters, so that there is generally no need to determine normal carbonates. Treated or adjusted waters, on the contrary, are often considerably above p H 8.0, so carbon dioxide is vanishingly small. Hence, a single setting of the index and the slide will ordinarily furnish all the necessary distribution data. A notable exception is in the case of waters in the p H range 8.0 to 9.0 and having a very high total alkalinity; here carbon dioxide and normal carbonate alkalinity may be present together in appreciable amounts and use of the rule requires two settings of the slide, one on each p H scale. Ail scales may readily be used for determining various forms beyond the p H ranges provided on the slide merely by using a p H value one or two integral units above or below as needed to obtain a readable scale figure which is then multiplied or divided by 10 or 100 as the case may require. Thus p H 7.9 is slightly outside the range of the (CO,--) scale, but a p H 8.9 setting may be used if the indicated normal carbonate alkalinity is divided by 10. I n this manner a t p H 7.9 and total alkalinity of 1000 p. p. m., the normal carbonates are found to be 8.9 p. p. m. as calcium carbonate. A similar method may be employed for extending the p H range in calculating carbon dioxide.
Ratio and Partition of Species Formulas for the ratio of concentrations of dual species, found from the set of Equations 4,6, 7, and 8, are tabulated in Table I. Ratios between the concentrations of carbonate forms are dependent solely on the p H of the solution; ratios involving (H+) and hydroxide alkalinity together with a carbonate form depend on total alkalinity, as F (Alk, pH), as well as on pH. A semilogarithmic plot of some important concentrsltion ratios against p H is given in Figure 2. Logarithms of ratios between carbonate species are linear with p H but the ratio (OH-)/(C03--) depends also on total alkalinity, and is given for three total alkalinity levels, 10, 50, and 250 p. p. m. of calcium carbonate, respectively. At a given p H value high alkalinities lower the ratio of hydroxide to normal carbonate alkalinities. Particular p H values, a t which any two species may be present in numerically equal amounts, may be found either directly from Figure 2 or by solving the equations in Table I set to unity. These special p H values are shown in Table 11.
TABLE11. SPECIAL pH VALUES FOR EQUALITYBETWEEN VARIOUSCARBONATE AND ALKALINITY SPECIES Equality COz, a s p . p. m. of COz = (HCOa-), as p. p. m. of CaCOa COI, a s p p m. of COz COa--), a s p . p. m. of CaCOa COY, asp: p: m. of coz = [OH-), a s p . p. m. of c a c o a
PH 6.288 8.119
-
(HCOa-), aa p. p. m. of CaCOs
10
8.29
250
8.97 9.95
, a s p . p. m. of CaCOa
:Yo;--) OH-), asp. p. m. of CaCO;
Z Alk 10
(CO;--), asp. p. m. of CaCO;
-
250 (OH-), a8 p. p. m. of CaCOa From Figure 2
9.79
10.72
10 9.71 50 10.67 {250 11.36
IO
0 6
4
2
I
B .6 .4
0 .2
k
Fs .I .06 .04
.02
.oI
5
6
7
9
IO
FIGURE 2. RATIOSOF VARIOUSCARBONATE AND ALKALINITY SPECIES COYa8 P.P. m. COS,all other form an p. p. m. CsCO,.
Temperature 25' C.
INDUSTRIAL AND ENGINEERING CHEMISTRY
946
Vol. 14, No. 12
FIGURE3 (Right). DISTRIBUTION IN TERNARY SYSTEM Carbon dioxide, p. p. m. COI, and carbonate alkalinities, as p. p. m . CaCOa. Temperature 25' C.
Such relations are useful in judging by inspection the accuracy of carbon dioxide and alkalinity figures in a water analysis. Thus at p H about 6.3 the reported carbon dioxide should equal the total alkalinity as calcium carbonate; below p H 6.3 the carbon dioxide will exceed the alkalinity; at p H near 7.3 the carbon dioxide should be one tenth the total alkalinity. Similarly, bicarbonate and normal carbonate alkalinities as calcium carbonate are equal at p H 9.95 and analogous relation again hold within this set of alkalinities; however, the usual phenolphthalein-methyl orange titration at this p H will show a bicarbonate content considerably smaller than the normal carbonates. From a consideration of the underlying equilibria it has been evident since McKinney's (2) theoretical development that or three alkalinity forms such as COp(HCO3-)-(CO3--) (HCO3-)-(CO3--)-(OH-) may be present simultaneously in the same water. Distribution of carbonates in the system C02-(HC03-)-(C0,--) as affected by p H only is indicated in Figure 3. Normal carbonates do not appear in significant amounts below p H near 7.5. On the other hand partition within the system (HCO3-)-(CO,--)-(OH-) depends on total alkalinity as well as on pH. By using the special slide rule to determine concentrations of various species at total alkalinity levels of 10, 50, and 250 p. p. m. of calcium carbonate, data for the triangular plot (Figure 4) was obtained. The
% of CO, and Total Alkalinity
higher the total alkalinity the greater is the fraction of normal carbonate possible. Tie lines between points of equal p H converge at the (OH-) vertex because a value for total a k a linity completely as hydroxide exists a t any p H as defined in Equation 4. A similar diagram has been developed by the Public Health Service Committee for revision of drinking water standards to replace the present limit on caustic alkalinity (6). As a matter of theoreticalinterest the possibility of having all four carbonate and alkalinity forms present simultaneously may be investigated with the use of either fundamental Equations l,2,3, and 4 or the various ratiosof Table I.
r n
20
80
A triple point may be found where carbon dioxide, as p. p. m. of carbon dioxide, normal carbonates, and hydroxide alkalinities, as p. p. m. of calcium carbonate are numerically equal. By equating COI equal to (COS--), in Equations 1 and 3 the required pH is found; the total alkalinity is then calculated by equating 3 and 4, substituting the value of (H+) just found. The calculated values are: pH 8.119 and total alkalinity 4.54 p. p. m. as calcium carbonate; the individual forms are each: CO, = (COa--) = (OH-) = 0.06 p. p. m. and (HCOI-) = 4.43 p. p. m. The first three species are below the limit of accurate detection. The same combination of values might also be found by trial and error manipulation of the special slide rule.
Literature Cited (1) DeMartini, F. E., J . Am. Water Work8 Assoc., 30, 85 (1938). (2) McKinney, D. S., IND. E N G . C H ~ M ANAL. ., ED.. 3. 192 (1931). (3) Moore, E. W.,.J. Am. Water Works Assoc., 31,51 (1939).
(4) schroerier, W. c., IND. ENG.CHEM., ANAL.E D . , 5, 389 (1933). ( 5 ) U. S. Public Health Service, Drinking Watm Standards, Part IV; Discussion of Chemical Requirements, Public Health Reports (in preas)
H
(1942).
FIGURE4. DISTRIBUTION IN TERNARY SYSTEM HCOa-COrOH alkalinities a8 p. p. m. CaCOa a t various total alkalinities. Temperature 25" C.
PREEENTED before the Division of Water, Sewage, and Sanitation Chemistry a t the 103rd Meeting of t h e AMERICAN CHEMICAL SOCIETY, Memphis, Tenn.
