The evidence that the variations which occur in the hydrogen-ion concentration of a natural water have any direct effect on the inhabitants living under natural conditions is scanty and not very convincing. On the other hand, there is good evidence to show that many animals are tolerant of the changes in hydrogen-ion concentration of their native habitat. These variations can hardly be related to distribution, epidemics of conjugation and the like, for these are known to occur at very different values of the hydrogen-ion concentration. Occasionally it can be shown that the variations are sufficiently extreme to cause the total extinction of certain species, but this will only be in very small pools. It is true, of course, that profound changes can be produced in biological reactions in the laboratory by altering the hydrogen-ion concentration of the medium in which the reaction is taking place, but these changes are nearly always greatly in excess of the natural changes occurring in the normal environment. It appears to me that the real importance of the measurement of the hydrogen-ion concentration of a natural water is that it can be used as an accurate measure of the carbon dioxide produced by the animals and of the photosynthetic activity of the plants. But to use the measure of the hydrogen-ion for this purpose we must know something of the underlying principles involved in the measurement and must not merely be content with matching the colour produced by the addition of an indicator with the colour of a buffer solution prepared by a rule of thumb method.

The object of this paper is to show that the hydrogen-ion concentration of a natural water depends on (1) the concentration of the dissolved alkaline and alkaline earth carbonates and bicarbonates, (2) the concentration of the dissolved carbon dioxide, (3) the temperature, and (4) the concentration of dissolved salts (neutral salts) other than alkaline and alkaline earth carbonates and bicarbonates which may be present in the solution. If we know the values of (1), (2), (3) and (4) these can be substituted in a very simple equation which will give us the value of the hydrogen-ion concentration.

Neglecting for the present the effect of temperature and of neutral salts and assuming that a natural water behaves in every respect as a mixture of a weak acid (carbonic acid) with the salt of a strong base, then by applying the law of mass action we can show that
where
and the brackets [ ] denote the concentration, thus [H.] denotes the concentration 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 of the acid. Further, it is characteristic of the alkaline and alkaline earth salts that they are highly dissociated in solution, so that by far the greatest portion of the acid ions, A, are supplied by the dissociation of this salt. If the salt is present in very small concentkations, as is the case in natural waters, it will be almost entirely dissociated so that the concentration of the salt may be substituted for [A] in equation (1), which then becomes
In natural waters the salt in equation (2) will be the carbonates and bicarbonates of alkaline and alkaline earth metals, the concentration of which may be con-veniently written, following Hasselbalch, [Bik], and the acid will be carbonic acid which will be written [CO2]. If the dissociation of the salt is not total, as we have assumed it to be, then the concentration of the salt must be multiplied by the ionisation constant which is the ratio and is denoted by δ.
Introducing δ into equation (2) this becomes
Or
If for the expression log δ – log ka we write pK1, then we have
which is the well-known equation of Hasselbalch.

The values of ka and δ at different concentkations of carbonates and bicarbonates have not been accurately measured, but the value of pK1 can easily be found experimentally by saturating solutions of bicarbonate of known concentration with carbon dioxide at a known temperature and pressure ‘and then measuring the pH. The equivalent concentration of carbon dioxide was calculated by Hasselbalch from measurements of the pressure of carbon dioxide in a mixture of this gas with hydrogen, with which mixture the bicarbonate solution was saturated. Pure water at 18°C. and 91·7 mm. pressure was calculated by Hasselbalch to react as an acid of o-oi normal concentration. For this purpose Hasselbalch used Bohr’s tables of solubility of carbon dioxide. I have also used these tables, but for the solubility of carbon dioxide in sea-water I have used Krogh’s results. Hasselbalch next assumes, following Henderson, (1) that only bicarbonates are present in the solution, which is true providing that the pH of the solution does not exceed 8·50, and (2) that the carbon dioxide dissolves in the dilute solution of bicarbonates in same proportions as in distilled water or in a water free from bicarbonates.

Both Parsons and Michaelis have pointed out that Hasselbalch has departed from the usual method of expressing the concentration of the dissolved carbon dioxide. Hasselbalch regarded carbonic acid as a divalent acid and has expressed the concentration in terms of normality, whereas the usual custom in physical chemistry is to use molar concentkations in such equations. If, then, we use molar concentration instead of normality,
Warburg has pointed out that the constant pK1 needs further modification and that equation (5) has only mathematical significance whereas in order to render the equation true both mathematically and actually it is necessary to introduce the conception of activity as formulated by G. N. Lewis. If the hydrogen-ion concentration of a solution is determined by measuring the potential difference between a hydrogen-platinum electrode and the solution we make use of Nernst’s equation in the form
where E is the measured potential, E0 is a constant depending on the electrode used for comparison, and t is the temperature in degrees centigrade of the solution. For the o-i N calomel electrode Sorensen, on the basis of conductivity experiments, obtained for E0 the value of 0·3777 volts. Bjerrum and Gjäldhaek from calculations based on the activity coefficient obtained for E0 the value 0·3348. So that if we use Bjerrum’s E0 then
within the limits of the experimental error of the measurements recorded later in this paper. The meaning of this last statement is that the concentration of the hydrogen-ion is not equal to the activity of the hydrogen-ion but that
If we also take into consideration the apparent activity coefficient of carbonic acid, Fa (CO2), which will be the reciprocal of the absorption coefficient, and write the equation using molecular concentkations we then have
Using the same method of expressing the concentration of combined and dissolved carbon dioxide the Hasselbalch equation may be written
which may be written in logarithmic form as
Warburg’s equation (9) above in logarithmic form is
Now
and as at 18°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 relationship between the various brands of pK1.

At 18°C.
The general relationship between K1 (Hasselbalch) and K1 (Warburg) is given by the equation
It must of course be pointed out, as Warburg has already done, that these relationships will not be satisfied unless we use Sörensenos E0 in calculating the pK1of Hasselbalch or Parsons and Michaelis, while Bjerrum’s E0 must be used in calculating the pK1 ′of Warburg.
The complete equation for the relation between the pH of a solution of alkaline carbonates saturated with carbon dioxide at varying pressures is given by Warburg as
where K2is a constant which bears the same relations to, the second dissociation constant of carbonic acid as K1does to k1, the first dissociation constant, so that at considerable dilutions we may put
Then
As the value of K2 is small its effect on the equation below pH 8·50 is negligible and it will only begin to exceed the experimental error when the pH exceeds 8·90.

For the experimental determination of pK1 Hasselbalch used solutions of sodium bicarbonate saturated with carbon dioxide and hydrogen at known pressures. Warburg, in addition to using sodium bicarbonate, used potassium bicarbonate as well, and his experiments covered a wider range of pressure and concentration than Hasselbalch’s. As is well known, Hasselbalch found that the value of pK1 was constant over a wide range of pressures of carbon dioxide provided that the concentration of the sodium bicarbonate remained constant. Other workers have confirmed this and Warburg further showed that the value of pK1 was dependent on the concentration of the neutral salts, such as sodium chloride, present in addition to the bicarbonate. The pH of the solutions in equilibrium with carbon dioxide at a known pressure was measured by means of the hydrogen electrode. Warburg has criticised the technique employed by Hasselbalch and has shown that the wire electrode making minimal contact with the solutions gives readings which are not quite constant and are 0·08 to 0·10 pH below the correct value. As will 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 simple solutions of bicarbonates and carbonates. Are the equations given above applicable in their entirety to natural waters or are there other substances present which will prevent their direct application? The bicarbonates present in natural waters are chiefly those of calcium and magnesium. Neither of these salts is very soluble and the solubility is, as Schloesing showed as long ago as 1872, dependent on the pressure of carbon dioxide with which the solution is in equilibrium. As a result of this in a normal hard water, which contains calcium bicarbonate to the extent of 0-002 normal, the water will be supersaturated with this salt when the pressure of carbon dioxide falls to 3/10,000 of an atmosphere, which is the normal pressure of carbon dioxide in fresh air. The calcium carbonate is not, however, thrown down as a precipitate immediately the pressure of carbon dioxide falls below the limit required to maintain it in solution. The solution will remain supersaturated for a long time and will behave to all intents and purposes as a solution of sodium bicarbonate of the same strength. While there can hardly be a stronger base present in a natural water than those commonly found, viz. calcium, magnesium, sodium and potassium, there might quite well occur many stronger acids than carbon dioxide. 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 phosphoric acid has a dissociation constant greater than that of carbonic acid and none of them occur in quantities sufficient to affect the carbonic acid-bicarbonate equilibrium. There is, therefore, every possibility that natural waters will behave in the same way as solutions of sodium bicarbonate do when the pressure of the carbon dioxide is relatively great, but we must expect a difference when this pressure falls below the limits necessary to maintain the carbonates of calcium and magnesium in solution.

In order to test the validity of the application of the Henderson-Hasselbalch equation 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 methods of measuring these quantities will now be given.

Measurement of the concentration of bicarbonates

The concentration of the bicarbonates present in natural waters does not commonly exceed 0·05 normal and may be as low as 0·00005 normal. The easiest and the most accurate method of measuring this concentration is to titrate the water with 0·01 normal sulphuric acid using methyl orange as an indicator. The end point of the reaction is the colour given when methyl orange is added to pure distilled water saturated with carbon dioxide. This is a method which has been shown by Küster to give very accurate 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 and placing it in a test tube. In another test tube of similar bore is placed 5 c.c. of distilled water saturated with carbon dioxide. To the water in each of these test tubes is added a drop of methyl orange. The distilled water, when compared with the water to be tested, should show a faint reddish tinge. Centinormal sulphuric acid is now added to the water to be tested until the colour matches that shown by the distilled water saturated with carbon dioxide. It will be necessary, when making the final comparison, to increase the volume of the distilled water by an amount equal to that of the acid added, so that the depth of colour in both tubes is the same. The small quantity of water used does not diminish the accuracy of the titration, rather it increases it, for, if the two test tubes are held against a white background, the end point is very clearly defined. When the water contains only a very small concentration of bicarbonates, such as occurs in waters from districts where 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 in the same way as before. In practice if the concentration of bicarbonates in the solution is less than 0·001 normal 25 c.c. of the water should be used. The distilled water saturated with carbon dioxide plays a very important part in the titration by providing us with a constant colour for the end point of the reaction. Ordinary laboratory distilled water prepared by a continuous still is generally useless and it will be found that the colour of methyl orange does not change when this water is saturated with carbon dioxide. Good distilled water should show a distinct change of colour, after adding methyl orange and saturating it with alveolar air by breathing 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 match of the two test tubes at the end point of the reaction. Accurate and consistent results with this method cannot be expected immediately the experiment is attempted. The following table shows the accuracy of the method. Pure dry sodium carbonate prepared in the usual way by heating sodium bicarbonate (I used Kahlbaum’s for analysis) was weighed and dissolved in distilled water to give a solution 0·1 normal. From this solution all the other solutions were prepared by dilution. The 0· 01 normal sulphuric acid used for the titration was standardised against NaOH, which had been carefully standardised by titrating weighed amounts of recrystallised potassium phthalate dissolved in water. Both pipettes and burettes were checked as to accuracy by weighing the quantity of water delivered. After calibration the volume delivered by these could be measured with an error of no more than 0·01 c.c.

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 atmospheric pressure, the proportion of carbon dioxide in the mixture and hence the pressure can easily be ascertained by withdrawing samples of the mixture and analysing them in a Haldane apparatus.

Measurement of the pH

Hasselbalch and Warburg used the hydrogen electrode and saturated the bicarbonate solutions with mixtures of hydrogen and carbon dioxide. If we are going to test natural waters under natural conditions they must be saturated with air and carbon dioxide, which will, of course, preclude the use of the hydrogen electrode. Under these circumstances the colorimetric method using the indicators recommended by Clark and Lubs appears to be the best available. The choice of these indicators depends on (1) their excellent virage, permitting considerable accuracy in comparison, (2) the fact that only very small quantities of the indicator require to be added to the solutions to be tested. The indicators recommended by Michaelis, while admittedly very convenient, do not allow the same accuracy of comparison to be attained. With the indicators of Clark and Lubs I find that the pH as measured colorimetrically will not differ from the value measured electrometrically by more than 0·02. Accurate estimation of the pH by colorimetric methods depends in the first place on the accurate matching of the tint of the indicator which has been added to a solution of unknown pH with the tint of the same indicator added to buffer mixtures of known pH. This matching is a matter of practice. At first it will not be possible to distinguish a difference in tint unless the pH of the two mixtures differs by not less than 0·05, very soon, however, the differences in tint caused by a difference of only 0·02 pH become easily distinguishable (see Saunders, 1923).

It is well known that colorimetric method is subject to certain “errors” which must be taken into account if results comparable with the electrometric method are to be attained. If, for example, we add an indicator to a buffer solution and match the tint produced against that produced in another buffer solution the pH of which has been measured by the hydrogen electrode, then we may say (but it will not always be correct) that the pH of both solutions is the same. If, now, we proceed to measure (assuming this to be possible) the pH of the first buffer mix-ture by means of the hydrogen electrode we may perhaps find that the pH is not the same as the pH of the buffer which it matched colorimetrically. There is, in fact, an “error” in the colorimetric measurement. This “error” or differ-ence between the measurements obtained by the colorimetric method and the hydrogen electrode may be due to several causes or a combination of these causes. If we know the causes of these “errors” it will be possible to make the proper allowance for them and so to bring the results obtained colori-metrically into accord with those obtained electrometrically.

One of these “errors,” the “error” due to the presence of proteins in the solution need not concern us here in dealing with natural waters. Natural waters do not contain protein in solution in sufficient concentration to affect the indicator. Even an infusion of dead leaves or hay such as is commonly used for the culture of Paramecium contains no more than 1·0 grammes per litre of protein as measured by the refracto-meter, an amount which will be very small when expressed in molecular concentration.

Another “error “is that due to temperature. By this we mean that an indicator may show different tints when added to two buffer mixtures of similar composition and pH, but differing in temperature. We can, of course, very easily avoid this error by 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 variation in pH with changes of temperature. This displacement of the indicator exponent has been measured by Kolthoff for certain indicators between 18° C. and 70° C. I have measured it for the indicators I have used by making use of Walbum’s records of the changes in the pH of certain buffer mixtures when these are heated. Walbum found that all the mixtures of Sörensenos phosphates suffered no appreciable change in pH as measured by the hydrogen electrode at temperatures between io° C. and 70° C. Mixtures of Sörensenos phosphates were prepared of suitable pH so as to coincide with that portion of the range of the indicator, where the virage was strongest, and each of these portions was divided into two after the addition of indicator. One portion was heated and the other was maintained at a temperature of 10° C. The tint of the indicator in the heated portion was then matched against a mixture at 10° C., the pH of which was known. In this way I was able to determine the heat “error” of brom-thymol blue, phenol red and cresol red. For determining the heat “error” of thymol blue I used Sörensenos borate-HCl mixtures, as the pH of these had been measured at different temperatures by Walbum. The details of these comparisons are given in Table II below. The results are plotted in Fig. 1.

Table I.
graphic
graphic
Table II.
graphic
graphic
Fig. 1.

pH displacement by temperature of the indicators brom-thymol blue, phenol red, cresol red, and thymol blue (alkaline range). In order to obtain the real pH of a solution at a temperature above or below 16° C. when compared with a buffer mixture of known pH at 16° C. the values of the abscissae marked with a + sign must be added, and those marked with a -sign subtracted, from the pH of the buffer mixture, which the solution matches in tint.

Fig. 1.

pH displacement by temperature of the indicators brom-thymol blue, phenol red, cresol red, and thymol blue (alkaline range). In order to obtain the real pH of a solution at a temperature above or below 16° C. when compared with a buffer mixture of known pH at 16° C. the values of the abscissae marked with a + sign must be added, and those marked with a -sign subtracted, from the pH of the buffer mixture, which the solution matches in tint.

It appears, therefore, that brom-thymol blue is very little affected by changes of temperature, while phenol red, cresol red and thymol blue are most affected but behave in practically the same manner. Kolthoff gives the displacement of the pH between 18° 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 much in 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 the unknown. If, however, we know the concentration of the salts in both the buffer mixture of known pH and in the mixture of unknown pH, then, from a colorimetric comparison, we can easily ascertain the pH of the unknown. I have already published 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 convenient form) the curve given in my previous paper and briefly to summarise the method. This curve is printed as Fig. 2 of this paper.

Fig. 2.

Graph for finding the salt error of cresol red when any standard buffer mixture of known pH is matched in tint with a solution of different normal concentration. The normal concentkations of the metallic kations are plotted as abscissae and the pH at which a constant tint is produced when cresol red is added to buffer mixtures of varying concentration are plotted as ordinates. The differences in pH between solutions of different normal concentration but matching in tint apply not only to the particular case from which this curve is constructed but to the whole range covered by the sulphonphthalein indicators. Detailed directions for using this graph are given in the text. (Plotted from Wells’ values, Journ. Amer. Chem. Soc. xLII. 2160.)

Fig. 2.

Graph for finding the salt error of cresol red when any standard buffer mixture of known pH is matched in tint with a solution of different normal concentration. The normal concentkations of the metallic kations are plotted as abscissae and the pH at which a constant tint is produced when cresol red is added to buffer mixtures of varying concentration are plotted as ordinates. The differences in pH between solutions of different normal concentration but matching in tint apply not only to the particular case from which this curve is constructed but to the whole range covered by the sulphonphthalein indicators. Detailed directions for using this graph are given in the text. (Plotted from Wells’ values, Journ. Amer. Chem. Soc. xLII. 2160.)

In order to allow for the salt “error” it is necessary first to ascertain the normality of the metallic kations in the buffer solution used for the comparison. This is very simple as the solution used for this purpose will always be of known composition and the normality can be calculated from the formula for its preparation.

Next it is necessary to know the normality for metallic kations of the solution whose pH is to be found. In the case of fresh-waters where the carbonates and bicarbonates form by far the largest proportion of the dissolved salts, it is sufficiently accurate to assume that the concentration of these, which is determined by titration in the manner indicated above, represents the concentration of all the dissolved salts. In the case of brackish or sea-water the concentration of the metallic kations can be derived from the density which is easily measured by the floating hydrometer by assuming that all the density is due to NaCl. With mineral waters it may be necessary to resort to chemical analysis, but here again a hydrometer and the assumption that all the density is due to NaCl is usually sufficiently accurate.

Fig. 2 shows graphically the pH at which a buffer mixture, to which NaCl is added in varying proportions, remains constant in tint on the addition of cresol red as an indicator. The method of using this curve is fairly obvious. For example, some fresh-water known to be 0·004 normal for bicarbonates matches in tint Sörensenos phosphate mixture of pH 7·80 when cresol red is added. The normality of metallic kations in the phosphate buffer mixture is 0·125. According to the curve a mixture of pH 7 97 and 0·125 normal for NaCl will match in tint a mixture of pH 816 and 0·004 normal. We must therefore add 0·21 to 7 80 in order to obtain the real or electrometric pH of the fresh-water. On the other hand, if sea-water, which is very nearly 0·6 normal for NaCl, matched exactly the tint of Sörensenos phosphate mixture of pH 7·80, then from the curve it is seen that a mixture of pH 7·80 and 0·6 normal for NaCl will match exactly a buffer mixture of pH 7·97 and 0·125 normal. We 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 kations in the solution of unknown pH exceeds that of the buffer mixture with which it compares in tint we must subtract the correction from the pH of the buffer mixture in order to obtain the real pH ; on the other hand, if the normality in the solution of unknown pH is less than that of the buffer then we must add the correction to the pH of the buffer mixture. The difference in pH between solutions of different normal concentration matched in tint will be the same for the whole range of pH covered by the sulphonphthalein indicators.

It has sometimes been assumed, but without justification, that it is unnecessary to apply any correction when the concentration of the dissolved salts in the solution of unknown pH is very small, as is the case in most fresh-waters. Actually, as we have just seen, the amount of the correction to be applied depends on the difference in the normality of the metallic kations in the solutions compared. There is a very considerable 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 to that 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 the indicator to a mixture which is very weakly buffered. In the case of natural waters, when the concentration of the bicarbonates falls below 0·001 normal the addition of the indicator may make an appreciable difference to the pH, so that the pH measured is not the pH of the water but the pH of the water after the addition of the indicator. At concentkations of bicarbonate exceeding 0·001 the pH of the water 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 (Die Wasserstoffionenkonzentration, 1922 edition), pp. 40 and 41. The indicators of Clark and Lubs are, however, added in the form of the sodium salt of the indicator which is a weak acid, the effect of the addition of the indicator in this form can be estimated as follows. The hydrogen-ion concentration in the solution before the addition of the indicator will be represented by the equation
and, after the indicator is added, by the equation
If the concentration of the alkali in the solution whose pH is to be found 0·0001 M, the carbonic acid is 0·00005 M and the indicator after addition to the solution 0·00003 M, then, if brom-thymol blue, the dissociation constant of which is 1× 10-7, 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 is 6·824, and after the addition it is 6·858. So, in this case, the observed pH will be 0·03 greater than that of the pH of the solution. If phenol red (dissociation constant 1·2 ×10-8) were used instead of brom-thymol blue in the case stated above the pH observed would be 6·934, or 0·11 too much. Variation in the added indicator of 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 possible so to adjust this ratio that, at a given concentration of alkali and carbonic acid, the addition of indicator will not alter the pH of the solution to which it is added. But if it be added to any other concentration of alkali and carbonic acid, this indicator will alter the pH by varying amounts.
Here it might be as well to point out the futility of attempting to measure the pH of distilled water by the use of indicators. When an indicator is added to pure distilled water it is diluted and the pH which is thus measured is the pH of the diluted indicator and may be quite different from that of the distilled water to which it has been added. The use of brom-thymol blue adjusted by the addition of NaOH to a certain colour before it is added to the distilled water has been recommended. This recommendation is based upon the fact that the measurements given by this indicator after adjustment compare with the hydrogen electrode measurements. What in effect has been done is to adjust the indicator so that when it is diluted on being added to distilled water the pH of the indicator so diluted is approximately 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 cease to do so if the distilled water contains a very small quantity of carbon dioxide in solution. A small quantity of carbon dioxide will cause a relatively great increase in the hydrogen-ion concentration in the distilled water, but this effect will be almost completely masked on the addition of the indicator. For example, let us suppose that brom-thymol blue in the form of the sodium salt is added to distilled water which contains carbonic acid to the extent of 0·00001 molecular. The hydrogen-ion concentration of such a solution will be or 1·73×10-6 (or pH 5·76). The indicator in the form in which it is added is a buffer mixture formed by the base and the weak acid indicator, further it is adjusted before addition to a green colour so that the pH of the indicator as added must be about 6·8o with the alkali and indicator present in equal concentration. When it is added to the distilled water the indicator is diluted to a concentration of 0·00003 molecular. The dissociation constant of brom-thymol blue is 1·1 × 10-7, then, substituting in equation (21) we have
whence
Thus the effect of the indicator, when added to distilled water containing a small quantity of dissolved CO2, is to cause an error of 1·0 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 measuring pH, [Bik] and [CO3] in equation (5), the validity of the application of the equation itself to these waters can now be tested experimentally. For this purpose a very simple apparatus may be used (Fig. 3). A CO2-air mixture is prepared by breathing into a large carboy. This mixture taken from the carboy is drawn through the test tubes by an aspirator until equilibrium is reached. The amount of the CO2 in the mixture bubbling through the test tubes is measured by withdrawing samples by the three-way tap and analysing these in a Haldane apparatus. The pressure of the CO2 is obtained by readings of the barometer and of a mercury manometer attached to the aspirator (hot shown in the diagram). The attainment of equilibrium is shown by the indicator added to the water in the test tubes maintaining a constant colour with continued bubbling. Natural waters often take a long time to reach equilibrium whereas with solutions of sodium bicarbonate it reached very rapidly, if the pH at equilibrium exceeds 8·5 equilibrium is reached very slowly and the rate at 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 equilibrium a much slower process. The necessity for this prolonged bubbling when the pH exceeds 8·5 is a well-known fact. Sorensen, who bubbled hydrogen through sodium bicarbonate found that equilibrium was not reached even after 24 hours, and he came to the conclusion that it was impossible to record electrometrically the pH of 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 dioxide and 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 now contains both carbonates and bicarbonates. It takes a very long time to reach this equilibrium. If we have two solutions of sodium bicarbonate, the one 0·01 normal and the other 0·005 normal, and bubble through both solutions at the same rate fresh air at atmospheric pressure containing three parts per ten thousand of carbon dioxide final equilibrium is reached in the weaker solution in 30 minutes, whereas it takes six hours before the final equilibrium is reached in the stronger solution. In effect then the formation of carbonates from bicarbonates when the pressure of carbon dioxide in equilibrium with the solution is reduced is an extremely slow process. It is doubtful if the reaction is ever complete if the carbon dioxide pressure be reduced to zero. Generally speaking, equilibrium is reached fairly rapidly whatever the pressure of carbon dioxide, provided that the pH of the final equilibrium does not exceed 8·50. Above 8·50 the rate at which equilibrium is reached falls 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 presence of sugar added to a solution of sodium bicarbonate increases very considerably the time taken to reach equilibrium.

Fig. 3.

Diagram showing the construction of the apparatus used for bringing the solutions info equilibrium with a CO2-air mixture and for measuring the pH of the solutions.

Fig. 3.

Diagram showing the construction of the apparatus used for bringing the solutions info equilibrium with a CO2-air mixture and for measuring the pH of the solutions.

The results of the methods outlined above are summarised in Table III. The HCO3′ normality in column (2) is determined by titration. The Na normality is given in column (3) and immediately below it, in brackets, is the cube root of this normality. The Na normality will be the same actually as the HCO3’ normality in the sodium and calcium bicarbonate solutions. In the natural freshwaters I have assumed that it is also the same except where the contrary is stated in column (3). In Cambridge tap-water the sum of the normalities of the NaCl, KC1, CaSO4 and MgSO4 present in solution only amounts to 0·0006, so that the tapwater is actually 0·0050 normal for all metallic kations, a difference which will be without influence in estimating the salt error of the indicator. But in the softer waters, such as those from Plymouth and Manchester, published analyses show that the error involved in making the assumption that the Na normality is the same as the HCO3′, 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 more than 0·05 pH in estimating the salt error of the indicator. Such an error in estimating the pH is about the same as the experimental error in these very dilute solutions. Below the double line in the table where the Na normality is shown in column (3) as exceeding the HCO3′ normality, it was determined in the case of sea-water from the density and in the other cases by the addition of weighed quantities of pure, dry NaCl.

Table III.
graphic
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Table 4.
graphic
graphic
The pressure of CO2 in column (4) is derived from the proportion of CO2 the mixture passing over the test tubes, which consists of air, water vapour and CO2, the proportion being determined by the Haldane apparatus. The proportion of CO2 multiplied by the total pressure of the mixture gives the pressure of CO2. The total pressure of the mixture is derived from readings of the barometric pressure. As the mixture is drawn through the test tubes by the aspirator pump, the pressure in each tube will be less than that of the preceding one. When the mixture is drawn through six tubes in series the decrease in pressure as indicated by the manometer attached to the aspirator pump is 30·5 mm. In calculating the pressure, allowance may be made for this drop but it is of small importance. 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 the lowering of pressure in the last tube of six in series to be applied in equation (5) will be log 7 3 07 5 5, which is — ·015, a difference of pH which is barely detectable by colorimetric methods. Without creating any serious error, and with a great gain in convenience, 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 pK1 in column (9) are obtained by substituting the values given in columns (2), (4) and (8) in equation (5), the normality of the CO2 being obtained from the pressure by multiplying the pressure by (seep.48). From these values of pK1 (Hasselbalch), pK1 ′ (Warburg) is obtained by equation (16). In calculating the averages in column (10), I have omitted the values enclosed in square brackets, [ ], as these particular values were sufficiently divergent from the mean to indicate the possibility of a serious error in the experiment to which they relate.

I have already shown (Saunders, 1923) that by the application of these methods the value of pK1 (Hasselbalch), using a mixture of CO2 and air to saturate the sodium bicarbonate solution and measuring the pH by indicators, is the same as that found by the electrometric method within the limits of experimental error. But my value of pK1 agreed with the value given by Hasselbalch, whereas Warburg has pointed out that, owing to an error in his technique, Hasselbalch’s values of pK1 are 0·08–0·10 too low. How, then, does the same error occur in the colorimetric technique? The answer is that it does not occur. When I was measuring the value of pK1 by colorimetric methods, my results were consistently 0·08 higher than the figures given by Hasselbalch. I was much puzzled by this, especially as I had taken the greatest care in the preparation of my buffer mixtures. I therefore checked the pH of my buffer mixtures with a hydrogen electrode and I found that the electrode measuremeants gave a pH value for the buffer mixtures which was 0·08 pH less than the stated value. This appeared to me to explain the discrepancy. I accepted the hydrogen electrode measurements as being correct and corrected the buffer mixtures accordingly. But I have now no doubt, after reading Warburg’s criticismHasselbalch’s technique, that the hydrogen electrode which I used was at fault and that the buffer mixtures were of the stated values. The electrode used to check the pH of my buffer mixtures was a platinum wire, the hydrogen was bubbled through the buffer mixture in an open dish, and minimal contact was made with the liquid, all of which are conditions which would favour the electrode being depolarised by traces of oxygen. If, then, the pH of ‘my buffer mixtures, prepared exactly according to the directions given from chemicals which I was careful to purify myself by several recrystallisations, are accepted as correct, then pK1 (Hasselbalch) is 0·08 too low and my measurements made by colorimetric methods agree very closely with those made by Warburg.

The results of the experiments recorded in Table III are expressed graphically in Fig. 4. Following Warburg I have plotted the values of pK1as ordinates and as.abscissae where c is the concentration of Na expressed as normal. The values found by Warburg are plotted to the same scale and indicated by the marks used by him.

Fig. 4.

Relation of pK1to the concentration of Na. The cube roots of the normal concentration of Na-(and other metallic kations where and when present) are plotted as abscissae and the corresponding values of pK1 as ordinates. The marks x are the values taken from Table IIIof this paper, and the marks ○, △, + are values taken from Warburg’s paper. To all these marks the bottom and 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 all relative to the point marked *.

Fig. 4.

Relation of pK1to the concentration of Na. The cube roots of the normal concentration of Na-(and other metallic kations where and when present) are plotted as abscissae and the corresponding values of pK1 as ordinates. The marks x are the values taken from Table IIIof this paper, and the marks ○, △, + are values taken from Warburg’s paper. To all these marks the bottom and 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 all relative to the point marked *.

It will be seen that my determinations of the values of pK1at both higher and lower concentkations of Na than those used by Warburg for his experiments all fall on the same straight line. I have also calculated the value of pK1 from the observations of Wilke. These appear to show that the relationship ceases to be a straight line one at concentkations greater than 1·0 molecular.

The value of pK1in Table III is the value at 180 C. The value of pK1 changes with temperature. Julius Thomsen, by thermodynamic methods, calculated that the heat of reaction, that is the change in the constant pK1 per degree centigrade, should be 0·0065. According to Hasselbalch’s and Warburg’s experiments the change is 0·0055. My experiments give results which are almost identical with those of Hasselbalch and Warburg. In order to measure the thermal increment I prepared two solutions, one of calcium bicarbonate by dissolving calcite in distilled water saturated with CO2 and another of sodium bicarbonate. The solutions 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 building was drawn through the solutions, four test tubes being run in series. The first two were maintained as controls at a temperature of 18 ° C., while the temperature of the last two was varied by immersing them in a large bath of water. Both the calcium and the sodium bicarbonate solutions behaved exactly alike. The indicator was cresol red and the buffer mixture used for the comparison was Palitzsch’s borax-boric acid. The pH of the buffer mixture, which the control tubes matched exactly in tint, was 8-30. From equation (5) we see that when the temperature of the solution is varied the solubility coefficient of CO2 will vary and the effect of temperature on the pH will be measured by the difference between the logarithms of the coefficient of solubility at the different temperatures, provided that the pressure of CO2 and the concentration of HCO3 ′’ remains the same and provided also that pK1 does not vary. It will be seen from Table IV that when the colorimetric method is used the difference in the pH between the two solutions at different temperatures appears to correspond almost exactly with the difference in the logarithms of the coefficient of solubility of CO2 at these temperatures, and that pK1 remains constant. But this appearance is illusory only, for it is produced by the indicator exponent itself changing in a similar manner. If we introduce the correction due to the displacement of the indicator exponent by heat (see Fig. 1), then we have pK1changing in a manner exactly similar to the indicator. We have already measured this change, which is a displacement of 0-385 pH between 00 and 70° C. or 0·0055 Per degree centigrade. The effect of this change in the value of pK1 with temperature is to reduce to some extent the effect which changes of temperature would otherwise have on a solution of bicarbonates in equilibrium with CO2. The change in pH due to changes in temperature in a solution of bicarbonates of a given concentration in equilibrium with a given pressure of CO2 is shown graphically in Fig. 5. Approximately an alteration of the temperature by 1° causes an alteration in pH of 0·01.

Fig. 5.

The broken line shows the change in pH which would occur in a solution of bicarbonates, if we were to assume that pK1remained constant, when the concentration of the bicarbonates and the pressure of carbon dioxide remain constant but the temperature changes. The continuous line shows the actual change, the difference between the two lines is the change in the value of pK1 with temperature. The pH at 18°C. is taken as zero, values to right of this line indicate an increase in the pH, those to the left a decrease.

Fig. 5.

The broken line shows the change in pH which would occur in a solution of bicarbonates, if we were to assume that pK1remained constant, when the concentration of the bicarbonates and the pressure of carbon dioxide remain constant but the temperature changes. The continuous line shows the actual change, the difference between the two lines is the change in the value of pK1 with temperature. The pH at 18°C. is taken as zero, values to right of this line indicate an increase in the pH, those to the left a decrease.

The following conclusions may be drawn from the results given in Table III and Fig. 4 : (1) that solutions of calcite behave in the same way as solutions of pure sodium bicarbonate ; (2) that natural waters which usually contain a mixture of the bicarbonates of calcium and magnesium in varying proportions also behave in the same way as solutions of pure sodium bicarbonate; (3) that the value of pK1is determined by the equivalent concentration of the kations present, not only those derived from the ionisation of the bicarbonate itself but also those derived from the ionisation of any neutral salts that may be present in the solution ; (4) that, for any given concentration of sodium ions, the value of pK1is the same no matter whether the bicarbonate be that of sodium, calcium, magnesium, or a mixture of these; (5) that the value of pK1changes with temperature.

It must be pointed out as a remarkable fact that, as recorded in Table III, equation (5) holds in the case of the Cambridge tap-water even when the pressure of CO2 falls as low as the average pressure of this gas in the atmosphere and the pH in consequence reaches nearly 9·00. Now equation (5) applies only when bicarbonates alone are present in the solution, and we must therefore conclude that in the tap-water this is the case even though the pH has reached this high value. We have already seen (p. 60) that the formation of carbonates from bicarbonates is an extremely slow process. Further, we find that equilibrium in the case of the Cambridge tap-water saturated with fresh air is reached only after from 2 to 3 hours’ continuous bubbling of air through the test tubes. At the end of this time the pH indicated by equation (5) is reached. If the bubbling be continued the solution remains at this pH for an hour or two longer and then the pH commences to fall. If the water be titrated immediately the pH has reached the maximum value the equivalent concentration of HCO3 ′ will be found to be unchanged, but when the pH falls, the equivalent concentration of HCO3 ′also falls. This fall in the equivalent concentration is due, of course, to the fact that the carbonates of Ca and Mg are only very slightly soluble and are precipitated from the solution soon after they are formed. Bicarbonates are therefore converted into carbonates but only very slowly when the pressure of CO2 is reduced. This formation of carbonates from bicarbonates appears, as might be expected, to be proportional to the concentration. If the concentration is relatively large (0·0078 normal) we see from Table IIIthat neither the pH calculated from equation (5) nor (17) is reached when we bubble fresh air through this water and this is obviously due to the carbonates forming and precipitating too quickly. On the other hand, if the equivalent concentration of Ca and Mg bicarbonates is reduced the formation of carbonates from bicarbonates, when the solution is exposed to the atmosphere, may be so slow that practically 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 indefinite time in equilibrium with the pressure of CO2 in the atmosphere without the carbonates forming in sufficient quantities to be precipitated. This fact is of importance because it determines the maximum value of the HCO3 ′ concentration in 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 the equivalent concentration of HCO3 ′ of the surface water rarely exceeds 0-0030 normal and is usually in the neighbourhood of 0·0020 normal. In the sea the equivalent concentration of HCO3 ′ varies, within narrow limits, from 0·0023 in tropical to 0-0026 normal in temperate regions.

It has often been suggested that the difficulty in raising the pH of sea-water by bubbling through it mixtures containing CO2 at very low pressures is due to the presence of acids other than carbonic. It is difficult to prove the presence of these acids and analysis has never revealed them in anything like sufficient quantities to produce the effect required. It is much more probable that the presence of the sodium chloride in the quantities in which it is present in sea-water is amply sufficient to account for these difficulties. The addition of sugar will also extend very considerably the time taken to reach equilibrium in a solution of sodium bicarbonate and here there can be no question of the presence of any other acid than carbonic. If very small pressures of CO2 are used then there is the possibility some of the carbonates being thrown out of solution. The effect in this case will be that the pH at equilibrium will be lower than if all the bicarbonate had remained in solution. It appears to me hardly necessary to drag in these extra acids, proof of the existence of which is lacking, in order to escape from what appears to be a difficulty, when this difficulty can be explained by simple physical means.

Shipley and McHaffie have put forward the hypothesis that in very dilute solutions carbonates are never fully transformed into bicarbonates. This is the exact opposite to the explanation which I have just suggested. But the experimental work on which this hypothesis of Shipley and McHaffie is based appears to me to be open to serious criticism. Shipley and McHaffie noticed that when solutions of Na2CO3 are titrated with HC1 using the hydrogen electrode to determine the end points of the reactions, the ratio of acid required for the first end point to that required for the final end point is less than the expected ratio of 1/2 when the solutions are very dilute. Down to 0·001 N Na2CO3 the ratio scarcely departs from the expected ratio by more than the experimental error. But at a concentration of 0·0005 N the ratio becomes 1/3, and, using CaCO3 instead of Na2COg, the ratio becomes 1/3·5 at a concentration of 0·00032 N. This departure from the expected ratio does not become at all obvious until the dilution is very considerable, when the experimental error may be very large and the difficulties of obtaining consistent results are very great. Shipley and McHaffie’s experiments show a fairly regular decrease in this ratio with dilution, and this has led the authors to put forward the suggestion (1) that at great dilutions the bicarbonate is never formed, and (2) that the second dissociation constant of carbonic acid, k2, increases with the dilution of the solution. They found that in the equation
the product [H’] [CO3 ′] is a constant the value of which was determined as being 5 × 10-13, so that when [HCO3 ′] is very small k2 is large. As a result of this, when a certain dilution is reached, will be the same as and there will appear to be only one end point for the titration. It appears to me that these authors have not entirely excluded the possibility of their solutions remaining contaminated with C02. Merely bubbling hydrogen through the solutions will not remove all traces of CO2 produced by the added acid, except, perhaps, after a very long time. These traces of CO2 are, at the dilutions used, quite sufficient to account for the divergences from the expected ratio. These experiments are, in fact, only another example of the difficulty, first pointed out by Sorensen, of raising the pH of a solution of bicarbonates to the theoretical value by bubbling pure hydrogen through the solution.

The difficulty of obtaining proper equilibrium with low pressures of CO2 in solutions of bicarbonates and also in sea-water is probably responsible for error in Henderson and Cohn’s, and also in McClendon, Gault and Mulholland's work. None of these workers found any constancy in the value of pK1, either in sea-water or in simple solutions of sodium bicarbonate. McClendon’s results are only presented in graphical form, which makes them a little difficult to criticise, moreover in the graph showing the relation of the pressure of CO2 to the pH of sea-water (p. 36, Carnegie Institute Publ. No. 251, 1917) he omits to mention what is the equivalent concentration of HCO3 ′, although he tells us elsewhere that it may vary from 0·0023 to 0·0025 normal. The value of pK1 for sea-water (I have assumed it to be ·0025 N) as determined by readings taken from McClendon’s graphs varies from 5·78 at pH 8·00 to 6·01 at pH 7·00, at pH 6·00 it again changes to 5·90. This result, to my mind, clearly shows the graphs to be erroneous and that the errors are due to not obtaining proper equilibrium in the solutions. Legendre (1925) has reproduced these graphs from McClendon in his book. A few pages earlier in this book we find the Hasselbalch equation is stated but Legendre has failed 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 substantial agreement with those of other workers. We can derive the dissociation constant of carbonic acid from Fig. 4 in the following manner:
and since the apparent activity constant may be put as equal to the real activity constant at very considerable dilutions and further since the activity coefficient approaches 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 of pK1as being 6·52, whence k1 is 3·02 × 10-7 at 18 ° C. This is practically identical with Walker and Cormack’s average value.
For the line drawn in Fig. 4 the equation
appears to hold. We again suppose that at considerable dilution the apparent activity coefficient is equal to the real activity coefficient. Now the logarithm of this real activity coefficient can be represented according to Bjerrum by the expression. The value of β in this expression is, according to Debye and Hiickel for a uniunivalent electrolyte, 0·495, which is a close approximation to the value 0·530 in the equation (23) above.

  1. The Henderson-Hasselbalch equation is shown to be entirely applicable to natural waters.

  2. The value of pK1 is dependent on the normal concentration of the metallic kations present in the solution, including those derived from any neutral salts. The relation between pK1and this concentration can be represented by a straight line for concentkations up to 1·0 normal. The equation which expresses this relation is
    where c is the normal concentration of metallic kations.
  3. Methods for measuring accurately the pH by colorimetric methods are given. From the pH thus measured the pressure of carbon dioxide with which the solution is in equilibrium can be calculated with great accuracy.

  4. By combining the results obtained the pH (corrected, if necessary for salt error by the curve on p. 56) of a solution of bicarbonates of normal concentration (Bik) as determined by the method described on p. 51, is related to the pressure of CO2 in mm. Hg (pCO2) with which the solution is in equilibrium by the equation
  5. Bicarbonates are transformed into carbonates at a very slow rate when the pressure of carbon dioxide in the solution is reduced. The slow rate at which this process occurs accounts for many natural waters having larger amounts of calcium and magnesium bicarbonates held in solution than can be accounted for by the pressure of carbon dioxide with which the solution is in equilibrium.

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