Chemistry of natural waters—V

Water Research Pergamon Press 1971. Vol. S, pp. 933-941. Printed in Great Britain

CHEMISTRY OF NATURAL WATERS--V HARDNESS P. H . KEMP Natal Regional Laboratory, National Institute for Water Research (S.A. Council for Scientific and Industrial Research), Durban, South Africa (Received 19 November 1970) Abstract--The definition and measurement of total, temporary and permanent hardness are discussed. A stoicbeiometric division of the total hardness into carbonic and non-carbonic components (called other names by other authors) is also made. After consideration of the values of the relevant solubility products, it is shown that the non-carbonic hardness is approximately equal to the permanent hardness, apart from about 25 mg l - i of CaCOa which remains in solution after the deposition of the temporary hardness. Natural waters with TDS less than about 70 mg I- t therefore do not show temporary hardness. There is no invariable relationship between the temporary and permanent hardnesses of a water and its scaleforming powers as measured by the pHj.

a A e c' e2 e2' e h H i I kc kco km k=° kw K1 K2 Kc Km m m' m2 m2" n OH p pH, s z Z ~,

NOTATION Molar concentration of carbonic acid (total) Total alkalinity in mg 1-1 of CaCO3 Molar concentration of calcium hydroxide (total) Value of e after deposition of temporary hardness Molar concentration of divalent calcium ions Value of c2 after deposition of temporary hardness Total alkalinity in equiv. 1-1 Molar concentration of hydrochloric acid Molar concentration of hydrogen ions Total hardness in equiv. 1-1 Total hardness in mg 1- t of CaCOa Solubility product of Ca(OH)2, 5.38 x 10 -6 at 25°C Solubility product of CaCOa, 5"12 × 10 -9 at 25°C Solubility product of Mg(OH)2, 1.38 × 10 -11 at 25°C Solubility product of MgCOa, 1.62 × 10 -5 at 25°C Ionic product of water, 1.01 x 10 -1'* at 25°C First dissociation constant of carbonic acid, 4-43 × 10 -7 at 25°C Second dissociation constant of carbonic acid, 4.69 × 10 -11 at 25°C Second dissociation constant of calcium hydroxide, 3-1 × 10 -2 at 25°C Second dissociation constant of magnesium hydroxide, 2"6 × 10 -a at 25°C Molar concentration of magnesium hydroxide (total) Value of m after deposition of temporary hardness Molar concentration of divalent magnesium ions Value of ra2 after deposition of temporary hardness Molar concentration of sodium hydroxide Molar concentration of hydroxyl ions Molar concentration of potassium hydroxide Saturation pH value Molar concentration of sulphuric acid (total) Soda alkalinity in equiv. 1-1 Soda alkalinity in mg 1-1 of CaCO3 Activity coefficient of univalent ions. INTRODUCTION

HISTORICALLY, t h e h a r d n e s s o f a w a t e r was a p a r a m e t e r d e s i g n e d to m e a s u r e t h e s o a p destroying powers of that water, arising from the precipitation of insoluble fatty acid salts, a n d was e a r l y s h o w n t o b e r e l a t e d t o t h e c a l c i u m a n d m a g n e s i u m c o n c e n t r a t i o n s . 933

934

P.H. KEMP

Metals such as iron, aluminium, manganese, strontium and zinc can also contribute to hardness, but these are not usually of any significance in natural waters. The total hardness of a natural water, measured in equiv. 1-1, may therefore be defined as: i---- 2(c + m )

(1)

using our usual notation where c and m are respectively the total molar concentrations of calcium and magnesium hydroxide, the forms in which, following RtccI (1952), calcium and magnesium are conceived to be present. On heating many waters, as is well known, a precipitate of calcium and magnesium compounds is formed so that, when this is filtered off, the hardness of the water is found to have decreased. That part of the total hardness which is removed by heating is termed the temporary hardness while that part which remains in solution is termed the

permanent hardness. The total alkalinity of a natural water in equiv. 1-1 has been shown (Part I) to be: e=

2c + 2 m - ~ - n

q-p--2s--h

(2)

where the symbols on the right denote molar concentrations according to our usual notation. Comparing this with (1) we can thus write: e = i -~- z

(3)

where z is defined by: z = n +p--2s--h

(4)

and is termed the soda alkalinity of the water. If z > 0, this will be a true net alkalinity and the difference e -- z = i represents that portion of the total alkalinity that can stoicheiometrically be considered as due to calcium and magnesium bound to carbonic acid. It is convenient to refer to this as the carbonic hardness of the water. If z < 0, the soda alkalinity is strictly a net acidity, and then - - z equiv. 1-1 of the total calcium and magnesium can stoicheiometrically be considered to be bound to sulphuric and hydrochloric acids while the remainder of the total hardness may similarly be considered bound to carbonic acid. In this case the carbonic hardness will be the remainder i + z = e equiv. 1- t while the balance of the total hardness, equal to - - z = i - e equiv, l - j , may be called the non-carbonic

hardness. These relationships are summarized in the upper part of TABLE 1. It is common practice to report hardness and alkalinity results in the units mg 1-1 of CaCO3, and in terms of these units the relationships are as in the lower part of TABL~ 1. These purely stoicheiometric concepts have given rise to a confused nomenclature in the literature, other authors referring to the carbonic and non-carbonic hardness by other names in a contradictory fashion. It is therefore essential to be quite sure just what a particular author means when he refers to any particular kind of hardness. M E A S U R E M E N T OF P E R M A N E N T HARDNESS It is noteworthy that the concepts of permanent and temporary hardness are rather vague. To make them precise, operational definitions are necessary, specifying just

Chemistry of Natural Waters--V

935

TABLE1. HARDNESSANDALKALINn~RELATIONSHIPS Units of equiv. 1-1 Total alkalinity • Total hardness i = 2(c + m) Soda alkalinity z = e -- i z<0

z>0 Carbonic hardness = e -- z = i Non-carbonic hardness = 0

Carbonic hardness = e Non-carbonic hardness ---- --z = i -- e

Practical units of nag 1- t of CaCO~ Total alkalinity Total hardness Soda alkalinity A>I

Carbonic hardness = I Non.carbonic hardness = 0

A = 5e x 10" I = 5i × I04 Z = A -- I A
Carbonic hardness = A Non-carbonic hardness = I -- A

what is intended by "heating" and "filtering" the water. For these the following procedure used by the present author to determine hardnesses may be invoked. A sample of the water whose total hardness has already been determined (e.g. by versenate titration) is boiled under reflux for 30 min and then, after removal from the source of heat, allowed to cool for 30 min during which time it is kept loosely covered. It is then filtered (Whatman No. 42 paper), the first runnings being used to rinse out the receiver and then rejected. The total hardness of the filtrate is determined, this being taken as the permanent hardness of the sample, and the temporary hardness is finally found by subtraction. The difficulty with this or any other procedure is that, although the heating and cooling are standardized so as to minimize evaporation losses, the precipitate at the time of filtration is unlikely to be in equilibrium with the water, from the point of view of both temperature and dissolved carbonic acid. Moreover the conditions of the determination, provided they are such as to minimize experimental errors, necessarily depart from those relating to practical water engineering, e.g. the conditions within any form of water heater, where the temporary hardness is likely to assume importance through its bearing on the deposition of scale. Most writers suppose that the effect of heating the water is to drive off carbon dioxide so that bicarbonates are converted to less soluble carbonates that will precipitate if their concentration is great enough. This will be complicated by alterations in the values of the relevant solubility products at changing temperature, although reliable information on the extent of these alterations appears to be quite lacking. Unfortunately a rigid theory of hardness is difficult to discuss because of the uncertainties involved in the operational definitions and because of the complexity of the mathematical relationships concerned. Before attempting the dissussion it is necessary to establish the values of the relevant solubility products. w.R. 5/10--r

936

P.H. KEMP

SOLUBILITY PRODUCTS At 25°C the solubility product kca of calcium carbonate is 5.12 × 10 -9 (see Part IV). Published values of the solubility in water of magnesium carbonate at 25°C vary somewhat, but 0.08 g 1-1 appears to be a fair estimate and, by the same procedure as used for kca, gives the value of the corresponding solubility product km~ as 1"62 × 10 -s. This compares favourably with the value 1 × 10 -s given by FAIR and GEYER (1954). The solubility product of calcium hydroxide is defined as: (5)

k~ ---- c2(0H)276 = c2kw276/H 2

in our usual notation. Under ideal conditions the hydrogen ion concentration of a solution of this base is given by: H

kw -H

c (kw + 2KcH) (kw + KcH)

(6)

This expands to a cubic equation in H. The solubility of calcium hydroxide at 25°C being 1.59 g 1-1 (WEAST et al., 1964), the principles outlined in Part I show that the value of H, corrected for non-ideality, can be found by solving the quadratic: 2cK~H 2 -- kw(K~ -- c72)H - kw 2 :

0.

(7)

The method of successive approximations finally gives 7 : 0.817 and H ~ 4.24 × 10-13 so that kc : 5.38 × 10 - 6 . Published values of the solubility of magnesium hydroxide in water at 25°C again vary somewhat, but a fair estimate is 0.0096 g 1. - 1 By the same procedure as just used, this gives the solubility product km as 1.38 × 10-11, which compares well with the 1.2 × 10-11 given by FAIR and GEYER(1954). THEORETICAL TREATMENT OF HARDNESS To account quantitatively for the temporary and permanent hardnesses of a natural water it must be assumed that the precipitate after the water has been boiled and cooled is in an equilibrium state, for without this assumption no account is possible at all. The assumption very conveniently obviates the need for any knowledge of the values of the solubility products or other constants at elevated temperatures. It must be further supposed that, while the effect of boiling the water is to drive off excess carbon dioxide until the bicarbonates present are stoicheiometrically converted to carbonates, there is no uptake of carbon dioxide again during the subsequent cooling. The result is that the whole process of the deposition of temporary hardness may be considered as proceeding at room temperature, the carbonic acid content a of the water decreasing in the cold until it becomes equal to e/2, whereon precipitation reactions occur in virtue of the raised pH value. For simplicity we may also suppose that conditions are ideal. The error involved here is doubtless small in comparison with those arising from the other assumptions. Now when a = e/2 the water may, in view of its raised pH, deposit the carbonates and hydroxides of calcium and magnesium, depending upon conditions arising from the solubility product relations. These conditions may be written: for Mg(OH)2 :

m >

kmH (kw -~- KmH) kw 2 Km

(8)

Chemistry of Natural Waters---V

937

k,H (k~, + KcH) for Ca(OH)2 :

c >

for M g C 0 3 :

m >

k~ 2 Kc

(9)

k=o (kw + KmH) (H 2 + K , H + KaK2) aKt K2K,,,H

k,a (kw + K,H) (H 2 + KxH + K1K2) aK1K2K~H The value of H is given approximately by: for CaCO3:

c >

1 n = ~ [kw + (kw2 + 4aK2kw) ~]

(10)

(11)

(12)

(see Part I), the error becoming greater with larger calcium and magnesium concentrations. Using this equation, however, provides a rough guide to the limiting values of c and m that must be attained before hydroxides and carbonates precipitate, and these approximate values are shown in TABLE2 for various values ofa. It may thus be deduced that most natural waters will precipitate calcium carbonate and magnesium hydroxide; calcium hydroxide will not be precipitated, but some waters of high TDS or unusually excessive magnesium content may perhaps deposit magnesium carbonate. This last possibility can be discounted to some extent since equation (12) leads to an underestimate of H a n d hence to an underestimate of the limiting concentration values. Consequently we need only be concerned with the precipitation of calcium carbonate a n d magnesium hydroxide. The soda alkalinity of the water will remain throughout the whole process at z equiv. 1-1, being unchanged by the loss of carbon dioxide and the precipitation of the temporary hardness. The calcium and magnesium concentrations, initially c and m, TABL~2. LIrarrxNoVALUESoF m and c

a 10-2 10-a 10-'* 10-s 10-4

m for Mg(OI-I) 2 1.13 1.17 2-58 1.50 1-33

x x × x x

10-5 10-4 10-a 10-1 10

c for Ca(OH)2 3-03 4.05 9.62 5.78 5.26

x x x x

10 102 10" 10~

m for MgCOa 2-87 x 2.94 x 6-50 x 3.81 x 3.53 x

10-a 10-2 10-t 10 10a

e for CaCOa 6.20 x 8.23 x 2.00 x 1.20 x 1.11

10-7 10-s 10-4 10-2

will fall to c' and m' moles 1-1 respectively, while the carbonic acid content will become a' = (c' + m' + z/2) moles 1-1, greater with z positive than with z negative. The ideal condition for saturation with calcium carbonate will then be:

c'(c' + m' + z/2) K~K1K2H kca = c2' a2' : (kw + KcH) ( n 2 + K1H + K1K2)'

(13)

That for saturation with magnesium hydroxide will be:

k= =

m2' kw2 m'kw2 K,. H2 = H(kw + K,H)"

(14)

938

P.H. KEMP

The hydrogen ion concentration will ideally be given by:

H _ _k~' = (c' -Frn' -Fz/2) (K1Hq-2K1K2) + c'(k,,, q- 2K~H) H (H 2 q- K1H --k K1K2) (k~, -t- KcH)

rn'(k,~ + 2KmH) +

(k., -k K m H )

- - z.

(15)

It is not permissible to attempt any simplification of these equations because H must be of the order of 10-lo to 10 -a, just in the range where H z and the products K1H, KcH and K,,,H are comparable in magnitude with kw and K1Kz, and the values of c' and m' at this stage are quite unknown. Consequently the equations must be left in their full and exact forms. In principle the three equations can be solved for the three unknowns H, c' and m' for any known value of z, but the working is evidently extremely complex and will not lead to any simple algebraic relation linking the variables with z. It is possible, however, to substitute (13) and (14) into (15) so as to obtain a relation between H, z and c' that can be expanded to a polynomial of the fourth degree in H. Using the general method of handling such equations described in Part I, according to the probable values of H and z, this gives a simpler equation linking H, z and c' from which an approximate value for c' can be obtained and which can also be used in (14) to find an approximate value for m'. This procedure shows that: (a) ifz > 0, then c' is of the order of 10 -4 and m' is very small, of the order of 10-13. (b) ifz < 0, then c' is approximately equal to --z/2 and m' is again very small, of the order of 10 -is. Since the permanent hardness of the original water is 2 (c' + m'), it is apparent that in each case the permanent hardness will be approximately equal to the non-carbonic hardness, with an error of about 20 mg 1-1 as CaCO3 in case (a) and presumably about the same in case (b). An appeal to experiment becomes necessary to settle this, if fully detailed calculations are not attempted. In Table 3 are given the analyses of fourteen synthetic waters which were prepared for hardness determinations, using the method specified above. The experimental hardness results are given in TABLE4 together with the calculated values of carbonic and non-carbonic hardness. The results for three samples (Nos. 5, 6 and 13) are clearly anomalous. Discounting these, the plot of the experimental hardnesses against the known non-carbonic hardnesses (FIG. 1) can be represented fairly well by a straight line with a positive intercept of 25 mg 1-1 CaCO3. This agrees very well with the value of about 20 m 1-1 just deduced and which was also cited by TAYLOR(1958). It confirms that the error involved in taking the non-carbonic hardness as the permanent hardness is not affected appreciably by the magnitude or sign of the soda alkalinity. In practical units, we therefore have very closely: permanent hardness ----- non-carbonic hardness q- 25 mg 1-1 CaCOa temporary hardness = carbonic hardness --25 mg 1-i CaCOa. The cause of the anomalous experimental results for samples 5, 6 and 13 is not known. FIGURE 1 indicates that the precipitation of the temporary hardness has been far from complete in these three instances, and TABLE 3 shows that these samples are the three

pH value

7.38 6"68 6"83 6-40 8"42 6"99 8"13 7"12 7"89 7.00 6.77 6"46 7"13 7.09

Sample no.

1 2 3 4 5 6 7 8 9 10 11 12 13 14

26"5 26"4 5"3 5"1 13 "2 13"0 2"7 2"9 52"9 53 "3 10"7 10.4 67-8 16"3

Ca (mg I- ~) 21.1 21 "0 4"5 4"6 57"0 57"1 11"3 11"4 41"1 41-0 8"2 8-4 63"5 17"6

Mg (mg 1- ~) 114.0 114"0 22"8 22"8 54"8 54"8 11"0 11"0 70-8 70.8 14-2 14.2 35"2 13.3

Na (rag 1- ~) 79.7 78" 1 15"9 15"4 353 "6 352"1 82"2 82"2 132"7 137-2 31 "4 31 "9 514-3 123-9

Total alkalinity (mg 1- t CaCOa) 80"0 80"0 16"0 16"0 Nil Nil Nil Nil 160-0 160.0 32-0 32-0 Nil Nil

SO. (mg 1- J)

TABLE 3. WATER SAMPLES USED FOR HARDNESS DETERMINATIONS

185"3 185"3 37"1 37"1 12"0 12-0 2"4 2-4 127"9 127"9 25.6 25-6 12.1 12-1

CI (mg 1- t)

153.2 152"8 31"9 31"7 267" 8 267"6 53"5 54"7 301 "3 302.3 60-5 60.5 431-2 113.0

Total hardness (nag 1- ~ CaC03)

T "<

.~

~.

O. o

~.

P. H. KEr,,~

940

c7 (~ 8

7..,

"~ ]00

5

7

Non-carbonic

I

I

I O0

200

hardness,

mg Lq C a C 0 3

FI6. I. Comparison of non-carbonic and permanent hardness. TABLE4. CALCULATEDAND EXPERIMENTALHARDNESSES(mg I-x CaCO3) Sample No.

Total hardness

1 2 3 4 5 6 7 8 9 10 11 12 13 14

153.2 152-8 31-9 31.7 267.8 267.6 53.5 54.7 301.3 302.3 60-5 60.5 431.2 113.0

Carbonic Non-carbonic Temporary hardness hardness hardness 79.7 78.1 15.9 15.4 267.8 267.6 53.5 54.7 132.7 137.2 31.4 31.9 431.2 113.0

73.5 74.7 16.0 16.3 0-0 0.0 0.0 0-0 168.6 165.1 29-1 28.6 0.0 0.0

36.9 41-2 0.2 --0.4 207.4 177.1 28.6 29.0 117.3 119-1 7.0 6.4 345.9 75.4

Permanent hardness 113.3 116.6 31.7 32-1 60.4 90.5 24.9 25.7 184-0 183-2 53.5 54.1 85-3 37.6

with the greatest m a g n e s i u m concentrations (exceeding 50 mg I-1), a n d there m a y therefore be a c o n n e c t i o n between the error of the p e r m a n e n t hardness calculation a n d the original m a g n e s i u m c o n t e n t of the water, particularly so as the solubility o f m a g n e s i u m hydroxide increases m u c h more greatly t h a n that o f calcium c a r b o n a t e with rising temperature. I n other words, the error involved in the a s s u m p t i o n that the precipitate is in a n equilibrium state is more serious when the initial m a g n e s i u m c o n c e n t r a t i o n is high.

Chemistry of Natural Waters--V

941

HARDNESS IN N A T U R A L WATERS The above discussion shows that no water will possess temporary hardness if its total hardness is less than 25 mg 1-1 as CaCOa. In practice, this comment applies to most natural waters with a TDS less than about 70 mg 1-1. For waters with a positive soda alkalinity, all except 25 mg 1-1 of the total hardness will be temporary so that, as the TDS increases, the temporary hardness rapidly approaches 100 per cent of the total. For waters with a negative soda alkalinity, however, the permanent hardness is much greater, and with increasing TDS the temporary hardness will never exceed a fraction of the total equal to A/I, which may be as low as 60 per cent or less depending on the water concerned. Scale-forming waters of low pH~ (see Part IV) necessarily have high alkalinity and high total hardness, but there is no invariable relationship between the PHs and the temporary and permanent hardnesses since the latter are conditioned by the soda alkalinity while the former is not. On the other hand, accepting that corrosiveness to metals is indicated by high values of the ratio (2s + h)/e, it is clear that waters of high permanent hardness (i.e. large values of s and h relative to e) should be corrosive, although the converse is not necessarily true. Acknowledgements--Much of the content of this series of papers is based upon a Ph.D. thesis accepted by the University of Natal. Grateful acknowledgementsare made to Professor J. W. BAVLESof that University, and to Dr. G. J. STANDER,Director of the National Institute for Water Research of the South African Council for Scientific and Industrial Research, for their guidance and advice during the writing of the thesis; also to the Natal Town and Regional Planning Commission for permission to make use, where necessary, of the results of river surveys carried out in Natal on their behalf. Thanks are also offered to the CSIR for permission to publish these papers.

REFERENCES FAIR G. M. and GEYERJ. C. (1954) Water Supply and Waste-Water Disposal Wiley, New York. Rx¢¢I J. E. (1952) Hydrogen Ion Concentration. Princeton University Press, Princeton, N.J. TAYLORE. W. (1958) The Examination of Waters and Water Supplies. (Thresh, Beale and Suckling), 7th edn, Churchill, London. W~AST R. C. et aL (1964) Handbook of Chemistry and Physics, 45th edn. Chemical Rubber Co., Cleveland, Ohio.