the hydrogen-ion concentration of …the hydrogen-ion concentration of natural waters 47 acid...
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THE HYDROGEN-ION CONCENTRATION OFNATURAL WATERS, i. THE RELATION OFpH TO THE PRESSURE OF CARBON DIOXIDE
BY J. T. SAUNDERS(From the Zoological Laboratory, Cambridge.)
(Received 16th March 1926.)
THE evidence that the variations which occur in the hydrogen-ion concentrationof a natural water have any direct effect on the inhabitants living under naturalconditions is scanty and not very convincing. On the other hand, there is goodevidence to show that many animals are tolerant of the changes in hydrogen-ionconcentration of their native habitat. These variations can hardly be related todistribution, epidemics of conjugation and the like, for these are known to occurat very different values of the hydrogen-ion concentration. Occasionally it can beshown that the variations are sufficiently extreme to cause the total extinction ofcertain species, but this will only be in very small pools. It is true, of course, thatprofound changes can be produced in biological reactions in the laboratory byaltering the hydrogen-ion concentration of the medium in which the reaction istaking place, but these changes are nearly always greatly in excess of the naturalchanges occurring in the normal environment. It appears to me that the realimportance of the measurement of the hydrogen-ion concentration of a naturalwater is that it can be used as an accurate measure of the carbon dioxide producedby the animals and of the photosynthetic activity of the plants. But to use themeasure of the hydrogen-ion for this purpose we must know something of theunderlying principles involved in the measurement and must not merely be contentwith matching the colour produced by the addition of an indicator with the colourof a buffer solution prepared by a rule of thumb method.
The object of this paper is to show that the hydrogen-ion concentration of anatural water depends on (1) the concentration of the dissolved alkaline andalkaline earth carbonates and bicarbonates, (2) the concentration of the dissolvedcarbon dioxide, (3) the temperature, and (4) the concentration of dissolved salts(neutral salts) other than alkaline and alkaline earth carbonates and bicarbonateswhich may be present in the solution. If we know the values of (i), (2), (3) and (4)these can be substituted in a very simple equation which will give us thethe hydrogen-ion concentration.
Neglecting for the present the effect of temperature and of neutral salts andassuming that a natural water behaves in every respect as a mixture of a weak
The Hydrogen-ion Concentration of Natural Waters 47
acid (carbonic acid) with the salt of a strong base, then by applying the law of massa n we can show that
where H" = hydrogen-ion,
ka = dissociation constant of the acid,
HA = undissociated acid,
A = dissociated acid,
and the brackets [ ] denote the concentration, thus [H'] denotes the concen-tration of the hydrogen-ion.
Now, since the dissociation constant of carbonic acid is very small the un-dissociated residue, HA, will be very nearly equal to the total concentration ofthe acid. Further, it is characteristic of the alkaline and alkaline earth salts thatthey are highly dissociated in solution, so that by far the greatest portion of the acidions, A, are supplied by the dissociation of this salt. If the salt is present in verysmall concentrations, as is the case in natural waters, it will be almost entirelydissociated so that the concentration of the salt may be substituted for [A] inequation (i), which then becomes
. i . i . [salt]
In natural waters the salt in equation (2) will be the carbonates and bicarbonatesof alkaline and alkaline earth metals, the concentration of which may be con-veniently written, following Hasselbalch, [Bik], and the acid will be carbonic acidwhich will be written [CO2]. If the dissociation of the salt is not total, as we haveassumed it to be, then the concentration of the salt must be multiplied by the• • *• * u- u • t. *• salt ionised , . , , , _lonisation constant which is the ratio -. = r—r and is denoted by 8.
total concentration 01 saltIntroducing S into equation (2) this becomes
/ - H ^ l o g S - l o g ^ + l o g ^ (4).
If for the expression log S — log ka we write pKx, then we have
(5),
which is the well-known equation of Hasselbalch.The values of ka and S at different concentrations of carbonates and bicar-
bonates have not been accurately measured, but the value of ̂ ^ can easily be founde^™:imentally by saturating solutions of bicarbonate of known concentration withcaroon dioxide at a known temperature and pressure *and then measuring thepH.The equivalent concentration of carbon dioxide was calculated by Hasselbalch
48 J. T. SAUNDERS
from measurements of the pressure of carbon dioxide in a mixture of this gas withhydrogen, with which mixture the bicarbonate solution was saturated. Pure ^ ^ ) rat i8°C. and 917 mm. pressure was calculated by Hasselbalch to react as anacid of o-oi normal concentration. For this purpose Hasselbalch used Bohr'stables of solubility of carbon dioxide. I have also used these tables, but for thesolubility of carbon dioxide in sea-water I have used Krogh's results. Hasselbalchnext assumes, following Henderson, (1) that only bicarbonates are present in thesolution, which is true providing that thepH of the solution does not exceed 8-50,and (2) that the carbon dioxide dissolves in the dilute solution of bicarbonates insame proportions as in distilled water or in a water free from bicarbonates.
Both Parsons and Michaelis have pointed out that Hasselbalch has departedfrom the usual method of expressing the concentration of the dissolved carbondioxide. Hasselbalch regarded carbonic acid as a divalent acid and has expressedthe concentration in terms of normality, whereas the usual custom in physicalchemistry is to use molar concentrations in such equations. If, then, we use molarconcentration instead of normality,
pKt (Hasselbalch) = pKx (Parsons and Michaelis) + -3010.Warburg has pointed out that the constant pKx needs further modification
and that equation (5) has only mathematical significance whereas in order to renderthe equation true both mathematically and actually it is necessary to introduce theconception of activity as formulated by G. N. Lewis. If the hydrogen-ion con-centration of a solution is determined by measuring the potential difference betweena hydrogen-platinum electrode and the solution we make use of Nernst's equationin the form
F — F~ °'S77 + 0-0002 {t— 18) *• ''
where E is the measured potential, Eo is a constant depending on the electrode usedfor comparison, and t is the temperature in degrees centigrade of the solution. Forthe o-i N calomel electrode Sorensen, on the basis of conductivity experiments,obtained for Eo the value of 03777 volts. Bjerrum and Gjaldhaek from calculationsbased on the activity coefficient obtained for EQ the value 0-3348. So that if we useBjerrum's Eo then
pH (Bjerrum) will be equal to/>H (Sorensen) + 0-048 (7)
within the limits of the experimental error of the measurements recorded later inthis paper. The meaning of this last statement is that the concentration of thehydrogen-ion is not equal to the activity of the hydrogen-ion but that
1*117 an~ CH (Sorensen) (8).
If we also take into consideration the apparent activity coefficient of carbonicacid, Fa (CO2), which will be the reciprocal of the absorption coefficient, and writethe equation using molecular concentrations we then have
v v tnr\ \ v o l u m e % dissolved CO2 ,an = K ^ . (CO2) y o l u m e % c o m b . n e d C O (9);
The Hydrogen-ion Concentration of Natural Waters 49
ushig the same method of expressing the concentration of combined and dissolvedC^Pon dioxide the Hasselbalch equation may be written
,T volume % dissolved CO» , .CH=K1—r ^ rp j - ~ 7 T X 2 ( iO),1 volume % combined CO2
v '
which may be written in logarithmic form as
/>H (Sorensen) = pK± (Hasselbalch) + log vol. % comb. CO2 -
log vol. % diss. CO2 — -3010 (11).
Warburg's equation (9) above in logarithmic form is
pa (Bjerrum) = pKx + log vol. % comb. CO2 — log vol. % diss.CO 2 - log^ a (CO 2 ) (12).
Now pKi- logFa(CO2) = pK1' (Warburg) (13),
and as at 180 C. the absorption coefficient of carbon dioxide is 0-927, log Fa (CO2)will be 0-033.
From the equations (10), (11), (12) above we can easily see the relationshipbetween the various brands of pKx.
At 180 C.
pKx (Hasselbalch) = />KX (Parsons and Michaelis) + -3010 (14)
= pKx' (Warburg) + -219 (15).
The general relationship between Kx (Hasselbalch) and K/ (Warburg) is givenby the equation
, 1793^! (Hasselbalch) .K = - — ; ( l 6 )
It must of course be pointed out, as Warburg has already done, that theserelationships will not be satisfied unless we use Sorensen's Eo in calculating the pKt
of Hasselbalch or Parsons and Michaelis, while Bjerrum's Eo must be used in calcu-lating the pKy of Warburg.
The complete equation for the relation between the pH of a solution of alkalinecarbonates saturated with carbon dioxide at varying pressures is given by Warburg as
a a - K ' F (CO) v o l u m e % dissolved CO2aa - Kj ta (CU)
where K '̂ is a constant which bears the same relations to k%, the second dissociationconstant of carbonic acid as K/ does to ky, the first dissociation constant, so thatat considerable dilutions we may put
= A 8 = i x 10-10 (18).
Then K ' - h.Fa (H C Qa')m? ^2 -^2 Fa(CO3')As the value of K?' is small its effect on the equation below pH 8-50 is negligibleand it will only begin to exceed the experimental error when the pH exceeds 8-90.
BJEBMVi
50 J. T. SAUNDERS
For the experimental determination of pKt Hasselbalch used solutions ofsodium bicarbonate saturated with carbon dioxide and hydrogen at known press^A.Warburg, in addition to using sodium bicarbonate, used potassium bicarbonateas well, and his experiments covered a wider range of pressure and concentrationthan Hasselbalch's. As is well known, Hasselbalch found that the value of />Kjwas constant over a wide range of pressures of carbon dioxide provided that theconcentration of the sodium bicarbonate remained constant. Other workers haveconfirmed this and Warburg further showed that the value of pK1 was dependenton the concentration of the neutral salts, such as sodium chloride, present inaddition to the bicarbonate. The pH of the solutions in equilibrium with carbondioxide at a known pressure was measured by means of the hydrogen electrode.Warburg has criticised the technique employed by Hasselbalch and has shownthat the wire electrode making minimal contact with the solutions gives readingswhich are not quite constant and are 008 to o-iopH below the correct value. Aswill be seen later in this paper, I, too, fell into this same error.
The question now arises as to whether natural waters can be treated as simplesolutions of bicarbonates and carbonates. Are the equations given above applicablein their entirety to natural waters or are there other substances present which willprevent their direct application? The bicarbonates present in natural waters arechiefly those of calcium and magnesium. Neither of these salts is very soluble andthe solubility is, as Schloesing showed as long ago as 1872, dependent on thepressure of carbon dioxide with which the solution is in equilibrium. As a resultof this in a normal hard water, which contains calcium bicarbonate to the extentof 0-002 normal, the water will be supersaturated with this salt when the pressureof carbon dioxide falls to 3/10,000 of an atmosphere, which is the normal pressureof carbon dioxide in fresh air. The calcium carbonate is not, however, thrown downas a precipitate immediately the pressure of carbon dioxide falls below the limitrequired to maintain it in solution. The solution will remain supersaturated fora long time and will behave to all intents and purposes as a solution of sodiumbicarbonate of the same strength. While there can hardly be a stronger base presentin a natural water than those commonly found, viz. calcium, magnesium, sodiumand potassium, there might quite well occur many stronger acids than carbondioxide. Analysis shows the presence of small quantities of phosphoric, silicic,boric and humic acids in some, but not in all, waters; but of these only phosphoricacid has a dissociation constant greater than that of carbonic acid and none of themoccur in quantities sufficient to affect the carbonic acid-bicarbonate equilibrium.There is, therefore, every possibility that natural waters will behave in the same wayas solutions of sodium bicarbonate do when the pressure of the carbon dioxide isrelatively great, but we must expect a difference when this pressure falls below thelimits necessary to maintain the carbonates of calcium and magnesium in solution.
The Hydrogen-ion Concentration of Natural Waters 51
EXPERIMENTAL APPLICATION OF THE HENDERSON-HASSELBALCHEQUATION TO NATURAL WATERS.
In order to test the validity of the application of the Henderson-Hasselbalchequation to natural waters, we must be able to measure accurately (1) the value of[Bik], (2) the pressure of carbon dioxide in order to ascertain the value of [CO2],(3) the pH at any given concentration of Bik or CO2. Particulars of the methodsof measuring these quantities will now be given.
Measurement of the concentration of bicarbonates. The concentration of thebicarbonates present in natural waters does not commonly exceed 0-05 normaland may be as low as 0-00005 normal. The easiest and the most accurate methodof measuring this concentration is to titrate the water with o-oi normal sulphuricacid using methyl orange as an indicator. The end point of the reaction is thecolour given when methyl orange is added to pure distilled water saturated withcarbon dioxide. This is a method which has been shown by Kiister to give veryaccurate results, the error being no more than 0-05 per cent, with proper precautions.The actual titration is performed by taking 5 c.c. of the water to be tested andplacing it in a test tube. In another test tube of similar bore is placed 5 c.c. ofdistilled water saturated with carbon dioxide. To the water in each of these testtubes is added a drop of methyl orange. The distilled water, when compared withthe water to be tested, should show a faint reddish tinge. Centinormal sulphuricacid is now added to the water to be tested until the colour matches that shownby the distilled water saturated with carbon dioxide. It will be necessary, whenmaking the final comparison, to increase the volume of the distilled water by anamount equal to that of the acid added, so that the depth of colour in both tubesis the same. The small quantity of water used does not diminish the accuracy ofthe titration, rather it increases it, for, if the two test tubes are held against a whitebackground, the end point is very clearly defined. When the water contains onlya very small concentration of bicarbonates, such as occurs in waters from districtswhere the soil is very poor in lime, it will be necessary to use 25 c.c. for the titration.If a boiling tube is used instead of a test tube the end point can be controlled inthe same way as before. In practice if the concentration of bicarbonates in thesolution is less than o-ooi normal 25 c.c. of the water should be used. The distilledwater saturated with carbon dioxide plays a very important part in the titrationby providing us with a constant colour for the end point of the reaction. Ordinarylaboratory distilled water prepared by a continuous still is generally useless and itwill be found that the colour of methyl orange does not change when this wateris saturated with carbon dioxide. Good distilled water should show a distinctchange of colour, after adding methyl orange and saturating it with alveolar air bybreathing into the water. But the most important point of all is for the observer,to accustom his eyes to seeing the colour change and so obtaining an accurate
£tch of the two test tubes at the end point of the reaction. Accurate and con-ent results with this method cannot be expected immediately the experiment is
attempted. The following table shows the accuracy of the method. Pure dry sodiumcarbonate prepared in the usual way by heating sodium bicarbonate (I used Kahl-
52 J. T. SAUNDERS
baum's for analysis) was weighed and dissolved in distilled water to give a solutiono-i normal. From this solution all the other solutions were prepared by dilThe ooi normal sulphuric acid used for the titration was standardised againNaOH, which had been carefully standardised by titrating weighed amounts ofrecrystallised potassium phthalate dissolved in water. Both pipettes and buretteswere checked as to accuracy by weighing the quantity of water delivered. Aftercalibration the volume delivered by these could be measured with an error of nomore than o-oi c.c.
Table I.
c.c. ofsolutiontaken fortitration
S55
S
5
5
25
25
25
c.c. of•oi N H2SO4required toneutralise
50-022502
5005025-022-502521231-250-500522-502521-23125•25•27•27
Normality asdetermined by
titration (1)
•10004•5004
•01004
•00502
•00248
•00102
•0010
•000497
•000105
Normality asdetermined byweighing (2)
•1000•0500
•0100
•0050
•0025
•0010
•0010
•0005
•0001
Log nor-mality (1)
l-oo1 7 0
2-OO
3 7 O
3 3 9
3 0 1
3-00
4-70
4-02
Log nor-mality (2)
l-ooi-70
2 00
3 7 O
3 4O
3-OO
3 0 0
4 7 0
4 0 0
Measurement of the pressure of carbon dioxide. If the water be shaken with,or have bubbled through it, a mixture of carbon dioxide and air at atmosphericpressure, the proportion of carbon dioxide in the mixture and hence the pressurecan easily be ascertained by withdrawing samples of the mixture and analysingthem in a Haldane apparatus.
Measurement of the pH. Hasselbalch and Warburg used the hydrogen electrodeand saturated the bicarbonate solutions with mixtures of hydrogen and carbondioxide. If we are going to test natural waters under natural conditions they mustbe saturated with air and carbon dioxide, which will, of course, preclude the useof the hydrogen electrode. Under these circumstances the colorimetric methodusing the indicators recommended by Clark and Lubs appears to be the best avail-able. The choice of these indicators depends on (1) their excellent virage, permittingconsiderable accuracy in comparison, (2) the fact that only very small quantitiesof the indicator require to be added to the solutions to be tested. The indicatorsrecommended by Michaelis, while admittedly very convenient, do not allow ^same accuracy of comparison to be attained. With the indicators of Clark and LubsI find that the pH as measured colorimetrically will not differ from the value
The Hydrogen-ion Concentration of Natural Waters 53
measured electrometrically by more than 0-02. Accurate estimation of the/>H by^wimetric methods depends in the first place on the accurate matching of thetint of the indicator which has been added to a solution of unknown pH with thetint of the same indicator added to buffer mixtures of known pH. This matchingis a matter of practice. At first it will not be possible to distinguish a differencein tint unless the />H of the two mixtures differs by not less than 0-05, very soon,however, the differences in tint caused by a difference of only 002pH becomeeasily distinguishable (see Saunders, 1923).
It is well known that colorimetric method is subject to certain "errors" whichmust be taken into account if results comparable with the electrometric methodare to be attained. If j for example, we add an indicator to a buffer solution andmatch the tint produced against that produced in another buffer solution the pHof which has been measured by the hydrogen electrode, then we may say (but itwill not always be correct) that thepH ofboth solutions is the same. If, now, we 70proceed to measure (assuming this to bepossible) the pH of the first buffer mix-ture by means of the hydrogen electrode 60we may perhaps find that the pH is not m
the same as the pH of the buffer which <°it matched colorimetrically. There is, in n 50
fact, an "error" in the colorimetric $measurement. This "error" or differ- 3ence between the measurements obtainedby the colorimetric method and thehydrogen electrode may be due toseveral causes or a combination ofthese causes. If we know the causes ofthese "errors" it will be possible tomake the proper allowance for them andso to bring the results obtained colori-metrically into accord with those obtainedelectrometrically.
One of these "errors," the "error"due to the presence of proteins in thesolution need not concern us here indealing with natural waters. Natural
0 . . Fig. 1. pH displacement by temperature of thewaters do not contain protein in solution indicators brom-thymol blue, phenol red, cresolin Sufficient Concentration to affect the red, and thymol blue (alkaline range). In order. ,. „ • r • c j j to obtain the real pH of a solution at a temperatureindicator. Even an infusion of dead a b o v e o r b e l o w l6» c w h e n c o m p a r e d with aleaves or hay Such as is commonly used buffer mixture of known pH at 16° C. the valuesc^^> ,. r r> • .. • of the abscissae marked with a + sign must bef » C culture of Paramecium contains a d d e d ) a n d those marked with a - sign subtracted,no more than I*O grammes per litre of from the pH of the buffer mixture, which theprotein as measured by the refracto- s o l u t i o n m a t c h e s in t in t-
£4001<uac£30
J20
10
v X Brom-Thymol
A Phenol Red
Blue
O Cresol Red
• Thymol Bluefelk)I I
+0-1 0-0 -0-1 -0-2pH displacement
-0-3
54 J- T. SAUNDERS
meter, an amount which will be very small when expressed in molecular con-centration. ^ B
Another " error " is that due to temperature. By this we mean that an indicatormay show different tints when added to two buffer mixtures of similar compositionandpH, but differing in temperature. We can, of course, very easily avoid this errorby doing all our experiments at the same temperature. It restricts us, however,to a temperature at or near i8° C. for not only do most indicators change their tint,but the buffer mixture used for comparison is itself liable to considerable variationin pH with changes of temperature. This displacement of the indicator exponenthas been measured by Kolthoff for certain indicators between 18° C. and 700 C.I have measured it for the indicators I have used by making use of Walbum'srecords of the changes in thepH of certain buffer mixtures when these are heated.Walbum found that all the mixtures of Sorensen's phosphates suffered no appreci-able change in pH as measured by the hydrogen electrode at temperatures betweenio° C. and 700 C. Mixtures of Sorensen's phosphates were prepared of suitablepH so as to coincide with that portion of the range of the indicator, where the viragewas strongest, and each of these portions was divided into two after the additionof indicator. One portion was heated and the other was maintained at a temperatureof io° C. The tint of the indicator in the heated portion was then matched againsta mixture at 10° C , thepH of which was known. In this way I was able to deter-mine the heat "error" of brom-thymol blue, phenol red and cresol red. For deter-mining the heat "error" of thymol blue I used Sorensen's borate-HCl mixtures,as the pH of these had been measured at different temperatures by Walbum. Thedetails of these comparisons are given in Table II below. The results are plottedin Fig. 1.
It appears, therefore, that brom-thymol blue is very little affected by changesof temperature, while phenol red, cresol red and thymol blue are most affected butbehave in practically the same manner. Kolthoff gives the displacement of the pHbetween 180 C. and 70° C. as being 0-4 for thymol blue and 0-3 for phenol red.
The last " error," which it is necessary to take into account, is the salt " error."If the composition of the buffer mixture used for the comparison differs very muchin the concentration of salts from that of the mixture whose pH is to be ascertained,the pH of the buffer mixture which the unknown matches is not the pH of theunknown. If, however, we know the concentration of the salts in both the buffermixture of known pH and in the mixture of unknown pH, then, from a colorimetriccomparison, we can easily ascertain the pH of the unknown. I have already pub-lished an account of the method of estimating the salt error in the case of cresol red,but, as I have reason to believe that the method is applicable to all the sulphon-phthalein indicators, I have thought it worth while to republish (in a more convenientform) the curve given in my previous paper and briefly to summarise the method.This curve is printed as Fig. 2 of this paper.
In order to allow for the salt "error" it is necessary first to ascertain the ^mality of the metallic kations in the buffer solution used for the comparison. Thisis very simple as the solution used for this purpose will always be of known com-position and the normality can be calculated from the formula for its preparation.
The Hydrogen-ion Concentration of Natural Waters 55Next it is necessary to know the normality for metallic kations of the solution whoseJ^Hs to be found. In the case of fresh-waters where the carbonates and bicarbonatesform by far the largest proportion of the dissolved salts, it is sufficiently accurateto assume that the concentration of these, which is determined by titration in themanner indicated above, represents the concentration of all the dissolved salts. Inthe case of brackish or sea-water the concentration of the metallic kations can bederived from the density which is easily measured by the floating hydrometer byassuming that all the density is due to NaCl. With mineral waters it may be necessaryto resort to chemical analysis, but here again a hydrometer and the assumptionthat all the density is due to NaCl is usually sufficiently accurate.
Table II.
( I )
Composition ofbuffer mixture heated
or cooled
Equal parts ofSorensen's phos-phates
12-6 c.c. primary+ 7 4 c.c. secon-dary (Sorensen'sphosphates)
5-70 c.c. primary+1430 c.c. sec-
ondary (Soren-sen's phosphates)
1 "7 c.c. primary +183 c.c. secon-dary (Sorensen'sphosphates)
IS c.c. of Soren-sen's borate + sc.c.ofo-i JVHC1
( 2 )
Indicatorused
Brom-thymolblue
Brom-thymolblue
Phenolred
Cresolred
Thymolblue
(3)
Temperaturein degreescentigrade
16377O
163567
14354570
104060
1016305070
(4)
Buffer mixtureused to matchthe mixture
heated to tem-perature in
col. (3)
Sorensen'sphosphates
Sorensen'sphosphates
Palitzsch'sborax-boricacid
Palitzsch'sborax-boricacid
Palitzsch'sborax-boricacid
(5)
pH of buffermixture at i6°C.which matchesin tint the mix-ture heated to
the temperaturein col. (3)
6816856-92
6-506-526-59
7-257367-457'57
7-827878018 1 3
do d
o do
do
00
(6)
Change in pHin buffer due to
heating orcooling (from
Walbum'stables)
Nil
»
+ 003OOO
-009-0-19-031
(7)
Apparentdifference inpH of the
two mixturesdue to the
heating of theindicator
O-OO+ 0 0 4+ O-II
OOO+ O-O2+ 0-09
OOO+ O-II+ O-20+ O32
-OOSo-oo
+ O-I4+ O-26
-OO3OOO
+ 0-09+ 0-19+ 0-31
Fig. 2 shows graphically the pH at which a buffer mixture, to which NaCl isadded in varying proportions, remains constant in tint on the addition of cresol redas an indicator. The method of using this curve is fairly obvious. For example, somefresh-water known to be 0-004 normal for bicarbonates matches in tint Sorensen'sphosphate mixture of pH 7-80 when cresol red is added. The normality of metallickations in the phosphate buffer mixture is 0-125. According to the curve a mixtureof />H 797 and 0-125 normal for NaCl will match in tint a mixture of/>H8i6^^0-004 normal. We must therefore add 0-21 to 7-80 in order to obtain the realor electrometric pH of the fresh-water. On the other hand, if sea-water, which isvery nearly o-6 normal for NaCl, matched exactly th*e tint of Sorensen's phosphatemixture of pH 7-80, then from the curve it is seen that a mixture of pH 7-80 and o-6
The Hydrogen-ion Concentration of Natural Waters 57
•
mal for NaCl will match exactly a buffer mixture of pH 7-97 and 0-125 normal,must, therefore, subtract 0-17 from 7-80 in order to obtain the real or electro-
metric pH of the sea-water. If, therefore, the normality of the metallic kationsin the solution of unknown />H exceeds that of the buffer mixture with which itcompares in tint we must subtract the correction from thepH of the buffer mixturein order to obtain the real pH; on the other hand, if the normality in the solutionof unknown pH is less than that of the buffer then we must add the correction tothe pH of the buffer mixture. The difference in pH between solutions of differentnormal concentration matched in tint will be the same for the whole range of pHcovered by the sulphonphthalein indicators.
It has sometimes been assumed, but without justification, that it is unnecessaryto apply any correction when the concentration of the dissolved salts in the solutionof unknown pH is very small, as is the case in most fresh-waters. Actually, as wehave just seen, the amount of the correction to be applied depends on the differencein the normality of the metallic kations in the solutions compared. There is a veryconsiderable difference in this concentration both in the case of fresh- and sea-water, but the correction to be applied for fresh-water will be of opposite sign tothat used for sea-water and it may, moreover., be considerably larger.
The last "error" which concerns us here is that caused by the addition of theindicator to a mixture which is very weakly buffered. In the case of natural waters,when the concentration of the bicarbonates falls below o-ooi normal the additionof the indicator may make an appreciable difference to the pH, so that the pHmeasured is not the pH of the water but the pH of the water after the addition ofthe indicator. At concentrations of bicarbonate exceeding o-ooi the pH of thewater will not be changed to any measurable extent by the addition of the indicator.When the indicator is added in the acid form we may make an approximate allow-ance for the effect by the use of the equations given by Michaelis in his book (DieWasserstoffionenkonzentration, 1922 edition), pp. 40 and 41. The indicators ofClark and Lubs are, however, added in the form of the sodium salt of the indicatorwhich is a weak acid, the effect of the addition of the indicator in this form can beestimated as follows. The hydrogen-ion concentration in the solution before theaddition of the indicator will be represented by the equation
and, after the indicator is added, by the equation
rxj-i _ ^1 [ac'd] + 2̂ [indicator]•• •* [salt']
where kx = first dissociation constant of carbonic acid,2̂ = dissociation constant of indicator acid,
[acid] = molecular concentration of carbonic acid,[indicator] = molecular concentration of indicator,
[salt] = molecular concentration of alkali before the indicator is added,[salt'] = molecular concentration of alkali after the indicator is added.
58 J. T. SAUNDERS
If the concentration of the alkali in the solution whose />H is to be founo-oooi M, the carbonic acid is 0-00005 M and the indicator after addition ^ Psolution 0-00003 M, then, if brom-thymol blue, the dissociation constant of whichis 1 x io~7, be used, we find by substituting these values in the equations (20)and (21) above, that the pH of the solution before the addition of the indicator is6-824, anc* after the addition it is 6-858. So, in this case, the observed pH will be0-03 greater than that of the pH of the solution. If phenol red (dissociation con-stant 1-2 x. io~8) were used instead of brom-thymol blue in the case stated abovethe/>H observed would be 6-934, or o-i 1 too much. Variation in the added indicatorof the ratio of the concentration of indicator acid to the.concentration of alkali will,of course, vary the error due to the addition of the indicator. It will be possibleso to adjust this ratio that, at a given concentration of alkali and carbonic acid, theaddition of indicator will not alter the^>H of the solution to which it is added. Butif it be added to any other concentration of alkali and carbonic acid, this indicatorwill alter thepH. by varying amounts.
Here it might be as well to point out the futility of attempting to measure thepH of distilled water by the use of indicators. When an indicator is added to puredistilled water it is diluted and the pH which is thus measured is the pH of thediluted indicator and may be quite different from that of the distilled water towhich it has been added. The use of brom-thymol blue adjusted by the additionof NaOH to a certain colour before it is added to the distilled water has beenrecommended. This recommendation is based upon the fact that the measurementsgiven by this indicator after adjustment compare with the hydrogen electrodemeasurements. What in effect has been done is to adjust the indicator so that whenit is diluted on being added to distilled water the pH of the indicator so diluted isapproximately that of the distilled water as measured by the hydrogen electrode.But if such an indicator gives a correct reading for pure distilled water it will ceaseto do so if the distilled water contains a very small quantity of carbon dioxide insolution. A small quantity of carbon dioxide will cause a relatively great increasein the hydrogen-ion concentration in the distilled water, but this effect will bealmost completely masked on the addition of the indicator. For example, let ussuppose that brom-thymol blue in the form of the sodium salt is added to distilledwater which contains carbonic acid to the extent of o-ooooi molecular. The hydro-gen-ion concentration of such a solution will be v i x io~5 x 3 x io"7ori-73X io~6
(or pH 5 76). The indicator in the form in which it is added is a buffer mixtureformed by the base and the weak acid indicator, further it is adjusted beforeaddition to a green colour so that the pH of the indicator as added must be about6-8o with the alkali and indicator present in equal concentration. When it is addedto the distilled water the indicator is diluted to a concentration of 000003 mole-cular. The dissociation constant of brom-thymol blue is I-I x io~7, then, sub-stituting in equation (21) we have
..„., _ 3 x io~7 x -ooooi + 1 x io~7 x -00003*• * -00003
whence pH = 6-70.
The Hydrogen-ion Concentration of Natural Waters 59
Thus the effect of the indicator, when added to distilled water containing a^ quantity of dissolved CO2, is to cause an error of i-o in estimating the pH,an error so large as to make the indicator method useless.
Having thus outlined the methods applicable to natural waters for measuringpH, [Bik] and [CO3] in equation (5), the validity of the application of the equationitself to these waters can now be tested experimentally. For this purpose a verysimple apparatus may be used (Fig. 3). A CO2-air mixture is prepared by breathinginto a large carboy. This mixture taken from the carboy is drawn through the testtubes by an aspirator until equilibrium is reached. The amount of the CO2 in themixture bubbling through the test tubes is measured by withdrawing samples by
* IV n)
asbiitral/
Fig. 3. Diagram showing the construction of the apparatus used for bringing the solutions intoequilibrium with a CO2-air mixture and for measuring the pH of the solutions.
the three-way tap and analysing these in a Haldane apparatus. The pressure of theCO2 is obtained by readings of the barometer and of a mercury manometer attachedto the aspirator (hot shown in the diagram). The attainment of equilibrium isshown by the indicator added to the water in the test tubes maintaining a constantcolour with continued bubbling. Natural waters often take a long time to reachequilibrium whereas with solutions of sodium bicarbonate it reached very rapidly,
he pH at equilibrium exceeds 8-5 equilibrium is reached very slowly and theat which the final equilibrium is reached greatly increases as the pH exceeds
this value. The effect of salt is also to render the attainment of the final equilibriuma much slower process. The necessity for this prolonged bubbling when the pH
60 J. T. SAUNDERS
exceeds 8-5 is a well-known fact. Sorensen, who bubbled hydrogen through sodibicarbonate found that equilibrium was not reached even after 24 hours, ^came to the conclusion that it was impossible to record electrometrically the pHof a solution of bicarbonates during the transition from bicarbonates to carbonates.If, however, instead of pure hydrogen, we use a mixture of air and carbon dioxideand if the pressure of carbon dioxide is so small that the pH at equilibrium is 9-0,then I find that stable equilibrium is reached despite the fact that the mixture nowcontains both carbonates and bicarbonates. It takes a very long time to reach thisequilibrium. If we have two solutions of sodium bicarbonate, the one o-oi normaland the other 0-005 normal, and bubble through both solutions at the same ratefresh air at atmospheric pressure containing three parts per ten thousand of carbondioxide final equilibrium is reached in the weaker solution in 30 minutes, whereasit takes six hours before the final equilibrium is reached in the stronger solution.In effect then the formation of carbonates from bicarbonates when the pressureof carbon dioxide in equilibrium with the solution is reduced is an extremely slowprocess. It is doubtful if the reaction is ever complete if the carbon dioxide pressurebe reduced to zero. Generally speaking, equilibrium is reached fairly rapidlywhatever the pressure of carbon dioxide, provided that the pH of the final equili-brium does not exceed 8-50. Above 8-50 the rate at which equilibrium is reachedfalls off rapidly, and this rate is further slowed down by the presence of neutral salts.In sea-water it is reached very slowly and Warburg has noticed that the presenceof sugar added to a solution of sodium bicarbonate increases very considerably thetime taken to reach equilibrium.
The results of the methods outlined above are summarised in Table III. TheHCOg' normality in column (2) is determined by titration. The Na' normalityis given in column (3) and immediately below it, in brackets, is the cube root ofthis normality. The Na' normality will be the same actually as the HCO3' nor-mality in the sodium and calcium bicarbonate solutions. In the natural fresh-waters I have assumed that it is also the same except where the contrary is stated incolumn (3). In Cambridge tap-water the sum of the normalities of the NaCl, KC1,CaSO4 and MgSO4 present in solution only amounts to 00006, so that the tap-water is actually 0-0050 normal for all metallic kations, a difference which will bewithout influence in estimating the salt error of the indicator. But in the softerwaters, such as those from Plymouth and Manchester, published analyses show thatthe error involved in making the assumption that the Na' normality is the same asthe HCOg', may be as much as 1000 per cent. This appears to be a very large error,but, as can be seen from Fig. 2, it will not cause an error at this dilution of morethan 0-05 pH in estimating the salt error of the indicator. Such an error in esti-mating the pH is about the same as the experimental error in these very dilutesolutions. Below the double line in the table where the Na' normality is shownin column (3) as exceeding the HCO3' normality, it was determined in the case ofsea-water from the density and in the other cases by the addition of ^ Bquantities of pure, dry NaCl.
The pressure of CO2 in column (4) is derived from the proportion of CO2
The Hydrogen-ion Concentration of Natural Waters 61
the mixture passing over the test tubes, which consists of air, water vapourCO2, the proportion being determined by the Haldane apparatus. The pro-
portion of CO2 multiplied by the total pressure of the mixture gives the pressureof CO2. The total pressure of the mixture is derived from readings of the baro-metric pressure. As the mixture is drawn through the test tubes by the aspiratorpurhp, the pressure in each tube will be less than that of the preceding one. Whenthe mixture is drawn through six tubes in series the decrease in pressure asindicated by the manometer attached to the aspirator pump is 30-5 mm. In calcu-lating the pressure, allowance may be made for this drop but it is of small im-portance. If six tubes are used in series, the pressure in the first will be, say,760 - 5 mm., and the pressure in the last test tube 730 mm. The correction for thelowering of pressure in the last tube of six in series to be applied in equation (5)will be log f§§ , which is — -015, a difference of pH which is barely detectable bycolorimetric methods. Without creating any serious error, and with a great gain inconvenience, we can reckon the total pressure in all the tubes of the series to be
(•-!»')•
where B is the barometric pressure and n is the number of the tubes in the series.I have never used more than eight tubes in series, as a rule the number was four.The values of pKx' in column (9) are obtained by substituting the values given incolumns (2), (4) and (8) in equation (5), the normality of the CO2 being obtained
from the pressure by multiplying the pressure by (see p. 48). From these
values of pKt (Hasselbalch), pKx' (Warburg) is obtained by equation (16). Incalculating the averages in column (10), I have omitted the values enclosed insquare brackets, [ ], as these particular values were sufficiently divergent fromthe mean to indicate the possibility of a serious error in the experiment to whichthey relate.
I have already shown (Saunders, 1923) that by the application of these methodsthe value of pK1 (Hasselbalch), using a mixture of CO2 and air to saturate the sodiumbicarbonate solution and measuring the pH by indicators, is the same as that foundby the electrometric method within the limits of experimental error. But my valueof pKx agreed with the value given by Hasselbalch, whereas Warburg has pointed outthat, owing to an error in his technique, Hasselbalch's values of pKt are 008-0-10 toolow. How, then, does the same error occur in the colorimetric technique? Theanswer is that it does not occur. When I was measuring the value of pK.1 by colori-metric methods, my results were consistently 0-08 higher than the figures given byHasselbalch. I was much puzzled by this, especially as I had taken the greatestcare in the preparation of my buffer mixtures. I therefore checked the pH of my
mixtures with a hydrogen electrode and I found that the electrode measure- 'gave a pH valuexfor the buffer mixtures which was 0-08 pH less than the
stated value. This appeared to me to explain the discrepancy. I accepted the hydro-gen electrode measurements as being correct and corrected the buffer mixtures
Tab
le 1
11.
Buf
fer
mix
ture
us
ed f
or
com
- pa
riso
n --
SO
~. ph
os.
,>
~H
of bu
ffer
m
ixtu
re
mat
chin
g ti
nt
of
solu
tion
6.79
6.
79
Ave
rage
va
lue
of
PK,'
--
6.50
Na'
no
rmal
ity
and
(y&
Pre
ssur
e of
C0
2 in
m
m.
Hg
S
ourc
e an
d na
ture
of
wat
er
or
solu
tion
Cor
rect
ed
pH
of
solu
tion
H
CO
,' n
orm
al~
ty
pK,'
Indi
cato
r
Sod
ium
bic
arbo
nate
P.R
. P
al.
bora
x
Ply
mou
th t
ap-w
ater
S
or.
pho
s.
Man
chea
ter
tap-
wat
er
So
r. p
hos.
Sod
ium
bic
arbo
nate
C
.R.
Pal
. bo
rax
C.R
. P
al.
bora
x
Cal
cite
dis
solv
ed i
n di
stil
led
wat
er
C.R
. 9
)
1)
Pal
. bo
rax
,I
I *
Cam
brid
ge t
ap-w
ater
dil
uted
w
ith
dist
ille
d w
ater
C
.R.
Pal
. bo
rax
C.R
. 1
)
Pal
. bo
rax
The Hydrogen-ion Concentration of Natural Waters
sb
M r f - M O O O X " C O OsOO i-< ^J-COroO O s t ^ N in\O CO CO Os N
<M c* M MvO *© t^OO OO J vi\O O O O
) M *-* i-t O OsiOO CO M *-* i-t O OsOO r^GO O N ^ - (00 00 CO 00 00 00
O O Os Os O TJ- T|- Tf OsOO Os COOO '
o 2
o .
u • ft
6
os - -o«O OH'
m-us
OH
MOOCOONinOOONO^M r~ o •+ Tj-oo « » o n
ooN M M TJ-MOO O M •<)• T j - O VO M N O
•H N CO CO CO ̂ " ̂ " tOGO M M N N N M f > f > c > T j - T h -^-00
8^ 8? 8£
J
f
4-> bo
S eav
u c
2
Tab
le I
11 (
cont
inue
d).
(4)
Pre
ssur
e of
CO
, in
m
m.
Hg
Buf
fer
mix
ture
us
ed f
or
com
- pa
riso
n
~H
of bu
ffei
m
ixtu
re
mat
chin
g ti
nt o
f so
luti
on
7'77
Na'
S
ourc
e an
d na
ture
of
wat
er
HC
O;
1 nomdiF
or
sol
utio
n n
om
allt
y
and
(dc)
C
orre
cted
p
H o
f so
luti
on
Ave
rage
va
lue
of
PK;
pK,'
Indi
cato
r
Sod
ium
bic
arbo
nate
.o
o5
('171
)
Pon
d w
ater
fro
m N
ewnh
am,
,007
8 ne
ar C
ambr
idge
('1
99)
C.R
. P
al.
bora
x
Pal
. bo
rax
9, , ,
B.-
T.B
. C
.R.
C.R
. &
P.R
.
Sod
ium
bic
arbo
nate
P
al.
bora
x
C.R
. ,, 9
>
Pal
. bo
rax
3,
2 9
Pal
. bor
ax
9,
--
Cam
brid
ge t
ap-w
ater
dil
uted
a0
023
'010
, w
ith
NaC
l so
luti
on
(.z
I 6)
C
.R.
C.R
. &
P.R
.
C.R
. 1
,
Pal
. bo
rax
Pal
. bo
rax
,*
C.R
. ,,
C.R
. P
al. b
orax
bica
rbon
ate
and
NaC
1 '2
28
1 C
.R.
Pal
. bo
rax
--
Pal
. bo
rax
.228
I C
.R.
-- S
olut
ion
of c
alci
te a
nd N
aCl
1 C
.R.
Pal
. bo
rax
Pal
. bo
rax
Pal
. bor
ax
Dil
uted
Cam
brid
ge ta
p-w
ater
an
d " S
hore
's s
ea s
alt"
Sod
ium
bic
arbo
nate
and
NaC
l
22
8
1 C
.R.
Pal
. bor
ax
>I
, , 9,
Sam
ple
of
sea-
wat
er f
rom
5
mil
es S
.S.W
. of
B
olt
Tai
l,
Ply
mou
th.
Col
lect
ed
Jan.
4t
h, 1
922
and
exp.
don
e Ja
n.
24th
, 19
22
.228
C
.R.
22
8
,, S
ea-w
ater
fro
m o
utsi
deB
reak
- w
ater
, P
lym
outh
. C
olle
cted
an
d ex
p. d
one
on A
pril
I ~
th,
1922
Pal
. bor
ax
99
.jo
o
I C
.R.
'300
9
Pal
. bo
rax
9 9
Sea
-wat
er
from
L
owes
toft
, O
ct.
16th
, 192
2
66 J. T. SAUNDERS
accordingly. But I have now no doubt, after reading Warburg's criticism^Hasselbalch's technique, that the hydrogen electrode which I used was atand that the buffer mixtures were of the stated values. The electrode used to checkthe pH of my buffer mixtures was a platinum wire, the hydrogen was bubbledthrough the buffer mixture in an open dish, and minimal contact was made withthe liquid, all of which are conditions which would favour the electrode beingdepolarised by traces of oxygen. If, then, thepH of my buffer mixtures, preparedexactly according to the directions given from chemicals which I was careful topurify myself by several recrystallisations, are accepted as correct, then^K^ (Hassel-balch) is o-o8 too low and my measurements made by colorimetric methods agreevery closely with those made by Warburg.
•8 TO V2 1-4 V6 V8
6-50
6-40
6-30
6^20
6-10
6-00
5-90
A,o * > 4-
• ^- V
6-50
6-30
6-10
5*90
5-70
5-50
5-30
5-10
1-0 1-2
Fig. 4. Relation of pK^ to the concentration of Na. The cube roots of the normal concentration ofNa- (and other metallic kations where and when present) are plotted as abscissae and the corre-sponding values oi pKi as ordinates. The marks x are the values taken from Table III of thispaper, and the marks O, A, + are values taken from Warburg's paper. To all these marks the bottomand left hand scales apply. The points marked W, to which the upper and right hand scales apply,are values derived from Wilke's results. The points marked W are not absolute values but are allrelative to the point marked •.
The results of the experiments recorded in Table III are expressed graphi-cally in Fig. 4. Following Warburg I have plotted the values of pKx' as ordinates
o 1 —
and as .abscissae « where c is the concentration of Na' expressed as normal. Thevalues found by Warburg are plotted to the same scale and indicated by the marksused by him.
It will be seen that my determinations of the values of />KX' at both higher ^lower concentrations of Na' than those used by Warburg for his experiments allfall on the same straight line. I have also calculated the value of pK±' from the
The Hydrogen-ion Concentration of Natural Waters 67oiiservations of Wilke. These appear to show that the relationship ceases to be a
light line one at concentrations greater than i-o molecular.The value of pKx' in Table III is the value at i8° C. The value of pKj1
changes with temperature. Julius Thomsen, by thermodynamic methods, calcu-lated that the heat of reaction, that is the change in the constant pK-i per degreecentigrade, should be 0-0065. According to Hasselbalch's and Warburg's experi-ments the change is 0-0055. My experiments give results which are almost identicalwith those of Hasselbalch and Warburg. In order to measure the thermal incre-ment I prepared two solutions, one of calcium bicarbonate by dissolving calcitein distilled water saturated with CO2 and another of sodium bicarbonate. Thesolutions were adjusted so as to be of the same equivalent concentration, viz.0-0017. Using the apparatus shown in Fig. 3 fresh air from outside the buildingwas drawn through the solutions, four test tubes being run in series. The first two
Table IV.
( 1 )
Solution
Sodium bicarb.Do. +NaCl
Sodium bicarb.Do. +NaCl
Calcium bicarb.Do. +NaCl
Calcium bicarb.Do. +NaCl
( 2 )
HCO3'normality
•00096•00096
•00112•00106
•00072•00072
•00180•00174
(3)
Na*normality
•00096I-2O
•00112I-2O
•00072•30
•0018•60
(4)
re
•099i-o6o
•1041-060
•090•670
•123•775
(S)
pHoibuffer
mixturesused for
com-parison
8098-25
8168-27
7-91797
8-31840
(6)
/>Hofsolutions
8-32806
8-37801
8-15786
8-Si821
(7) (8)
Difference in pH(and also in pK.^)
after correcting forsolubility of CO2 and
any difference inHCO3' normality
found
•52
•5°
•33
•36
calculated
•si
•51
•31
•38
were maintained as controls at a temperature of 18° C , while the temperature ofthe last two was varied by immersing them in a large bath of water. Both thecalcium and the sodium bicarbonate solutions behaved exactly alike. The indicatorwas cresol red and the buffer mixture used for the comparison was Palitzsch'sborax-boric acid. The pH of the buffer mixture, which the control tubes matchedexactly in tint, was 8-30. From equation (5) we see that when the temperature ofthe solution is varied the solubility coefficient of CO2 will vary and the effect oftemperature on the pH will be measured by the difference between the logarithmsof the coefficient of solubility at the different temperatures, provided that the
:ssure of CO2 and the concentration of HCO3' remains the same and providedthat pKi does not vary. It will be seen from Table IV that when the colori-
metric method is used the difference in the pH between the two solutions atdifferent temperatures appears to correspond almost exactly with the difference
5-2
68 J. T. SAUNDERS
in the logarithms of the coefficient of solubility of CO2 at these temperatures, andthat^K/ remains constant. But this appearance is illusory only, for it is produd^by the indicator exponent itself changing in a similar manner. If we introduce tnecorrection due to the displacement of the indicator exponent by heat (see Fig. i),then we have pKx' changing in a manner exactly similar to the indicator. We havealready measured this change, which is a displacement of 0-385 pK between o-o and70° C. or 0-0055 pH per degree centigrade. The effect of this change in the value
i with temperature is to reduce to some extent the effect which changes ofilibi
pi ptemperature would otherwise have on a solution of bicarbonates in equilibriumwith CO2. The change in pH due to changes in temperature in a solution ofbicarbonates of a given concentration in equilibrium with a given pressure of CO2
is shown graphically in Fig. 5. Approximately an alteration of the temperatureby i° causes an alteration in^>H of o-oi.
50
30
20
10
• - *
• ' • /
J////
11
1 ///
///
/
/////
•1 pn • 2
Fig. 5. The broken line shows the change in pH which would occur in a solution of bicarbonates,if we were to assume that p K / remained constant, when the concentration of the bicarbonates andthe pressure of carbon dioxide remain constant but the temperature changes. The continuous lineshows the actual change, the difference between the two lines is the change in the value of pK^with temperature. The pH at i8°C. is taken as zero, values to right of this line indicate an increasein the pH, those to the left a decrease.
The following conclusions may be drawn from the results given in Table IIIand Fig. 4: (1) that solutions of calcite behave in the same way as solutions of puresodium bicarbonate; (2) that natural waters which usually contain a mixture of thebicarbonates of calcium and magnesium in varying proportions also behave inthe same way as solutions of pure sodium bicarbonate; (3) that the value of />KX'is determined by the equivalent concentration of the kations present, not only thosederived from the ionisation of the bicarbonate itself but also those derived fromthe ionisation of any neutral salts that may be present in the solution; (4) that, forany given concentration of sodium ions, the value of pKx' is the same nowhether the bicarbonate be that of sodium, calcium, magnesium, or a mixof these; (5) that the value of pKt' changes with temperature.
It must be pointed out as a remarkable fact that, as recorded in Table III,
The Hydrogen-ion Concentration of Natural Waters 69
equation (5) holds in the case of the Cambridge tap-water even when the pressureo^B02 falls as low as the average pressure of this gas in the atmosphere and the />Hin consequence reaches nearly 9-00. Now equation (5) applies only when bicar-bonates alone are present in the solution, and we must therefore conclude that inthe tap-water this is the case even though the pH has reached this high value. Wehave already seen (p. 60) that the formation of carbonates from bicarbonates is anextremely slow process. Further, we find that equilibrium in the case of theCambridge tap-water saturated with fresh air is reached only after from 2 to 3hours' continuous bubbling of air through the test tubes. At the end of this timethe pH. indicated by equation (5) is reached. If the bubbling be continued thesolution remains at this pH for an hour or two longer and then the pH commencesto fall. If the water be titrated immediately thepH has reached the maximum valuethe equivalent concentration of HCO3' will be found to be unchanged, but when thepH falls, the equivalent concentration of HCO3' also falls. This fall in the equivalentconcentration is due, of course, to the fact that the carbonates of Ca and Mg areonly very slightly soluble and are precipitated from the solution soon after they areformed. Bicarbonates are therefore converted into carbonates but only very slowlywhen the pressure of CO2 is reduced. This formation of carbonates from bicar-bonates appears, as might be expected, to be proportional to the concentration. Ifthe concentration is relatively large (0-0078 normal) we see from Table III thatneither the pH calculated from equation (5) nor (17) is reached when we bubblefresh air through this water and this is obviously due to the carbonates formingand precipitating too quickly. On the other hand, if the equivalent concentrationof Ca and Mg bicarbonates is reduced the formation of carbonates from bicar-bonates, when the solution is exposed to the atmosphere, may be so slow thatpractically no formation of carbonates is found to occur. A solution of CaHCO3
of an equivalent concentration of 0-0020 normal will remain for an almost indefinitetime in equilibrium with the pressure of CO2 in the atmosphere without thecarbonates forming in sufficient quantities to be precipitated. This fact is ofimportance because it determines the maximum value of the HCO3' concentrationin the surface waters of large lakes and probably to some extent also in the sea.In large lakes, where the water supply is derived from calcareous sources theequivalent concentration of HCO3' of the surface water rarely exceeds 0-0030normal and is usually in the neighbourhood of 0-0020 normal. In the sea theequivalent concentration of HCO3' varies, within narrow limits, from 0-0023 intropical to 0-0026 normal in temperate regions.
It has often been suggested that the difficulty in raising the pH of sea-waterby bubbling through it mixtures containing CO2 at very low pressures is due tothe presence of acids other than carbonic. It is difficult to prove the presence ofthese acids and analysis has never revealed them in anything like sufficient quantitiestoproduce the effect required. It is much more probable that the presence of the^Bum chloride in the quantities in which it is present in sea-water is amplysufficient to account for these difficulties. The addition of sugar will also extendvery considerably the time taken to reach equilibrium in a solution of sodiumbicarbonate and here there can be no question of the presence of any other acid than
70 J. T. SAUNDERS
carbonic. If very small pressures of CO2 are used then there is the possibili^^ofsome of the carbonates being thrown out of solution. The effect in this case^Blbe that thepH at equilibrium will be lower than if all the bicarbonate had remainedin solution. It appears to me hardly necessary to drag in these extra acids, proofof the existence of which is lacking, in order to escape from what appears to be adifficulty, when this difficulty can be explained by simple physical means.
Shipley and McHaffie have put forward the hypothesis that in very dilutesolutions carbonates are never fully transformed into bicarbonates. This is the exactopposite to the explanation which I have just suggested. But the experimentalwork on which this hypothesis of Shipley and McHaffie is based appears to me tobe open to serious criticism. Shipley and McHaffie noticed that when solutionsof NajCOg are titrated with HC1 using the hydrogen electrode to determine the endpoints of the reactions, the ratio of acid required for the first end point to thatrequired for the final end point is less than the expected ratio of 1/2 when thesolutions are very dilute. Down to o-ooi iV NagCOg the ratio scarcely departs fromthe expected ratio by more than the experimental error. But at a concentration of0-0005 N the ratio becomes 1/3, and, using CaCO3 instead of NagCOg, the ratiobecomes 1/3-5 a t a concentration of 0-00032 N. This departure from the expectedratio does not become at all obvious until the dilution is very considerable, whenthe experimental error may be very large and the difficulties of obtaining consistentresults are very great. Shipley and McHaffie's experiments show a fairly regulardecrease in this ratio with dilution, and this has led the authors to put forward thesuggestion (1) that at great dilutions the bicarbonate is never formed, and (2) thatthe second dissociation constant of carbonic acid, k2, increases with the dilutionof the solution. They found that in the equation
[H-][C03']**- [HCO3']
the product [H'] [C03'] is a constant the value of which was determined as being5 x io~13, so that when [HCO3'J is very small k2 is large. As a result of this, whena certain dilution is reached, k^ will be the same as kx and there will appear to beonly one end point for the titration. It appears to me that these authors have notentirely excluded the possibility of their solutions remaining contaminated withC02. Merely bubbling hydrogen through the solutions will not remove all tracesof CO2 produced by the added acid, except, perhaps, after a very long time. Thesetraces of CO2 are, at the dilutions used, quite sufficient to account for the diver-gences from the expected ratio. These experiments are, in fact, only anotherexample of the difficulty, first pointed out by Sorensen, of raising the pH of asolution of bicarbonates to the theoretical value by bubbling pure hydrogen throughthe solution.
The difficulty of obtaining proper equilibrium with low pressures of CO2
in solutions of bicarbonates and also in sea-water is probably responsible fo r^^error in Henderson and Cohn's, and also in McClendon, Gault and Mulhollai^Pwork. None of these workers found any constancy in the value of pK^, either insea-water or in simple solutions of sodium bicarbonate. McClendon's results areonly presented in graphical form, which makes them a little difficult to criticise,
The Hydrogen-ion Concentration of Natural Waters 71
moreover in the graph showing the relation of the pressure of CO2 to the/>H of sea-\^Br (p. 36, Carnegie Institute Publ. No. 251, 1917) he omits to mention what isthe equivalent concentration of HCO3', although he tells us elsewhere that it mayvary from 0-0023 to 0-0025 normal. The value of />K/ for sea-water (I haveassumed it to be -0025 N) as determined by readings taken from McClendon's graphsvaries from 5-78 at/>H 8-oo to 6-oi at/>H 7-00, at/>H 6-oo it again changes to 5-90.This result, to my mind, clearly shows the graphs to be erroneous and that theerrors are due to not obtaining proper equilibrium in the solutions. Legendre(1925) has reproduced these graphs from McClendon in his book. A few pagesearlier in this book we find the Hasselbalch equation is stated but Legendre hasfailed to point out that this equation will not fit with McClendon's results.
With the exceptions just referred to, my results can be shown to be in sub-stantial agreement with those of other workers. We can derive the dissociationconstant of carbonic acid from Fig. 4 in the following manner:
Since /.K1' = /.K1 + logF0(HC03') (13)
and since the apparent activity constant may be put as equal to the real activityconstant at very considerable dilutions and further since the activity coefficientapproaches unity as the concentration approaches zero, so that, at infinite dilution
The extrapolation of the line in Fig. 4 to infinite dilution gives the value ofas being 6-52, whence kt is 3-02 x io~7 at 180 C. This is practically identical withWalker and Cormack's average value.
For the line drawn in Fig. 4 the equation
logP1o(HCQB') = o - S 3 ^ (23)
appears to hold. We again suppose that at considerable dilution the apparentactivity coefficient is equal to the real activity coefficient. Now the logarithm of thisreal activity coefficient can be represented according to Bjerrum by the expression— )8 v c The value of /? in this expression is, according to Debye and Hiickelfor a uniunivalent electrolyte, 0-495, which is a close approximation to the value0-530 in the equation (23) above.
SUMMARY.1. The Henderson-Hasselbalch equation is shown to be entirely applicable to
natural waters.2. The value of pK±' is dependent on the normal concentration of the metallic
kations present in the solution, including those derived from any neutral salts. Therelation between pKx' and this concentration can be represented by a straight linefor concentrations up to i-o normal. The equation which expresses this relation is
pKt' = 6-52 - 0-53 sfc,c is the normal concentration of metallic kations.
3. Methods for measuring accurately the^>H by colorimetric methods are given.From the/>H thus measured the pressure of carbon dioxide with which the solutionis in equilibrium can be calculated with great accuracy.
72 J. T. SAUNDERS
4. By combining the results obtained the pR (corrected, if necessaryerror by the curve on p. 56) of a solution of bicarbonates of normal c o n c e ^ ^(Bik) as determined by the method described on p. 51, is related to the pressure ofCO2 in mm. Hg (/>CO2) with which the solution is in equilibrium by the equation
*H = 1070-o-53*£+log-J.^-.
5. Bicarbonates are transformed into carbonates at a very slow rate when thepressure of carbon dioxide in the solution is reduced. The slow rate at which thisprocess occurs accounts for many natural waters having larger amounts of calciumand magnesium bicarbonates held in solution than can be accounted for by thepressure of carbon dioxide with which the solution is in equilibrium.
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