THE EFFECT OF VARIATION IN IONIC STRENGTH ON THE APPARENT FIRST AND SECOND DISSOCIATION

CONSTANTS OF CARBONIC ACID.

BY A. BAIRD HASTINGS AND JULIUS SENDROY, JR.

(From the Hospital of The Rockefeller Institute for Medical Research.)

(Received for publication, June 22, 1925.)

The biological and chemical importance of carbonic acid has stimulated much work upon its first dissociation constant with the immediate object of using this constant for the calculation of the hydrogen ion concentration of biological solutions. Fewer inves- tigations have had as their object the determination of the second dissociation constant.

The mathematical treatment of the behavior of solutions of strong electrolytes by Milner (1) and by Debye and Htickel (2) has brought to the empirical results obtained by Lewis (3) and his collaborators, and Brdnsted and La Mer (4) a theoretical explana- tion which seems adequate. Experimental work has indicated that the deviation in the behavior of strong electrolytes in moder- ately dilute solution from their behavior in infinitely dilute solu- tion can be approximately expressed in the following manner.

If a = the activity

Y = “ (‘ coefficient, and c = “ molal concentration of an ion, then a =yc. Further, that the activity coefficient is related to the ionic

concentration of the solution by the empirical formula - log y = @P -\ii where

@ = an empirically deternined constant which has a value of approxi- mately 0.50

v = the valence and P = the ionic strength and is defined as f ZCU*.

The theoretical considerations of Debyc and Hiickel led to an expression, which under certain limiting conditions, corresponded well with the above empirical equation.

445

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446 Dissociation Constants of Carbonic Acid

In view of these developments and the desirability of knowing the activity of bicarbonate and carbonate ions for subsequent studies, we have endeavored to determine the effect of varying ionic strength on the apparent dissociation constants of carbonic acid.

FIRST DISSOCIATION CONSTANT.

The first dissociation constant of carbonic acid has been accu- rately determined by numerous investigators. Of the determina- tions based on conductivity measurements those of Walker and Cormack (5) and of Kendall (6) are probably the most reliable. The results of these workers, recalculated by Warburg (7), give the value for K, as 3.11 X lo-’ at 18”; pK1 = 6.507. Kendall’s value for pK1 is 6.656 at 0” and 6.460 at 25”. This represents a change in pK1 per degree of 0.0078. Assuming that this tempera- ture coefficient is valid up to 38”, the pK1 of carbonic acid would be 6.36 at body temperature. Hasselbalch (8) in 1916, determined pK1’ electrometrically in bicarbonate solutions ranging in con- centration from 0.005 to 0.05 N. The method for measuring electrometric pH in COz-containing solutions at that time con- tained certain errors, which Warburg has pointed out in recalcu- lating Hasselbalch’s values for pK1’. These values are consis- tently about 0.09 lower than Warburg’s and about the same amount lower than ours. The reason for the discrepancy appar- ently lies in the differences in technique employed in the measure- mentof pH values. The most accurate determinations of the first dissociation constant by electrometric measurement appear to be those of Warburg. These results we have recalculated, using the solubility coefficient for COZ found in this laboratory, and have included in Fig. 1 with our own results.

The mass law equation for the ionization of carbonic acid in terms of activity is

aH’ x aHCOz’ = K, aHd203

In the logarithmic form this becomes

(1) log aH. + log aHCO,, - log ctHrcol = log KI

where (Y represents the activity of each individual component. For anoo; one may write y1 [HCO,‘] where yl denotes the activ-

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A. B. Hastings and J. Sendroy, Jr. 447

ity coefhcient of the HC03’ ion. Furthermore, in the presence of a relatively large amount of NaHC03 one may write y1 [NaHC03] instead of 71 [HCOX’], assuming for the moment, that NaHC03 behaves as a strong electrolyte. Since the activity of a gas is proportional to its tension, one may write

won %o* aHzCOs = 760 ’ ,==

62

6.’ ,

)I5 0.20 0.25 0.30 035 040 045 050

I I I I

6.4 1 /-i-k--

0 0.05 0.10 c

FIG. 1. Values of pIi,’ are plotted as ordinates and of ~‘i, as abscissm. Points marked o and @I are from data given in the present paper. Those marked q indicate data recalculated from Warburg’s experiments. The line represents values of pK,’ calculated as pKI’ = 6.33 - 0.5 4;.

where Pcoz denotes the tension of CO2 with which the solution is in equilibrium and aoo, the solubility of CO2 in the solution at that temperature. In conformity with the recent recommendations of Sorensen and Linderstrom-Lang (9) we shall write pan for - log

cxrr* With the above substitutions Equation 1 then becomes

(2) - P aH + log INaH% - log Pcot - log 760 ;cO”b224

= log K, - log yt

(3) We shall let pKI’ = - log KI’ = pK1 + log yl

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448 Dissociation Constants of Carbonic Acid

Our cxperiments,have been performed with the object of deter- mining the relation between pK,’ (and consequently log n) and the ionic strength of the solution.

EXPERIMENTAL.

The experiments to be reported here represent the results of the determination of pK,’ in solutions of eight different ionic strengths. Each of these experiments was repeated with a freshly prepared solution.

The sixteen different solutions whose compositions are given in Table I were prepared from NaHCOs and NaCl of a high degree of purity. Each was rotated for 30 minutes in a water bath at 38” with hydrogen and COZ at a tension previously calculated to give the desired pH. The saturation was repeated for a second 30 minute period in order to insure equilibrium. The liquid phases were then separated by the technique described in a previous paper (10) and the following analyses were made.

Samples of the gas phase were transferred to the Haldane- Henderson gas apparatus and the tension of CO2 was determined. The liquid phase, which was contained in filled glass bulbs over mercury, was analyzed for its CO2 content in the Van Slyke (11) manomctric gas apparatus. The pCYH was determined electro- metrically at 38” in the Clark electrode vessel using the cell

The potential of the cell, with 0.1 N HCI in place of the unknown solution, was determined at the beginning of each day’s experiment. This value of “eo”, calculated on the assumption that the po(H of 0.1 N HCI is 1.08, is given in each table. As pointed out by Cullen (12) this procedure serves as a daily cali- bration of the cell system used. The value 1.08, used as the pan of 0.1 N HCI, is taken from Lewis’ value for the activity of H’ in an HCl solution whose ionic strength is 0.1 I*.’ Values of p”H are calculated from the equation

P aH = E.M.F. (corrected to 1 atmosphere dry Hz) - r.

0.06164

‘Lewis and Rxnlall (3), p. 382

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hlubil

ity

mffir

ient

of CO

%

H&03

co

, Na

HCOa

1%

Na

HCOz

Hz

COs’

PKI

--.--

0.55

2 0.

552

rnM

pm1.

1.99

7 2.

127

?nM

per 1

. 12

.25

12.4

0

m&r

per

1.

10.2

5 10

.27

0.71

0 0.

684

6.26

7 6.

266

0.55

0 0.

550

3.80

3.

407

23.9

7 24

.12

20.1

7 20

.71

0.72

5 0.

784

6.26

5 6.

176

0.54

9 0.

549

5.76

5.

57

35.5

6 35

.62

29.8

0 30

.05

0.71

4 0.

732

6.24

3 6.

235

0.54

6 0.

546

5.07

5.

083

35.1

0 34

.96

30.0

3 29

.88

0.77

2 0.

768

6.26

4 6.

202

NO.

PC02

PUH

_ mm

.

1 61

.6

6.97

7 2

65.6

6.

950

3 11

7.6

6.99

0 42

10

5.4

6.96

0

5 17

8.6

6.95

7 6

172.

8 6.

967

7 15

8.1

6.97

6 8

158.

5 6.

970

9 15

2.3

6.97

0 10

15

2.4

6.97

5

11

145.

0 6.

970

12

150.

3 6.

962

13

140.

5 6.

970

14

140.

1 6.

984

15

136.

8 6.

950

16

131.

9 6.

980

- -

In

odd

num

bere

d ex

perim

ents

~0 =

0.

2364

; in

ev

en

num

bere

d ex

perim

ents

ED =

0.

2351

. *

Loss

of

CO

, fro

m

gas

phas

e.

Not

plotte

d in

Fi

g.

1.

0.54

4 0.

544

4.87

4.

87

35.1

2 34

.86

30.2

5 29

.99

0.79

3 0.

789

6.17

7 6.

186

0.54

1 0.

541

4.61

4.

775

34.7

5 34

.90

30.1

4 30

.12

0.81

5 0.

800

6.15

5 6.

162

0.53

8 0.

538

4.44

4.

427

34.8

6 34

.46

30.4

2 30

.03

0.83

6 0.

831

6.13

4 6.

153

0.53

2 0.

532

4.28

4.

122

34.6

0 34

.70

30.3

2 30

.58

0.85

0 0.

870

6.10

0 6.

110

-

TABL

E I.

The

pK1’

of

Carb

onic

Acid

at

58

” in

Sa

lt So

lutio

ns

of Va

rying

Io

nic

Stre

ngth

.

NaCl

P dr

nod

per

2.

0.00

0.

00

0.01

025

0.10

12

0.01

027

0.10

13

0.00

0.

0201

7 0.

1418

0.

00

0.02

071

0.14

39

0.00

0.

0298

0 0.

1726

0.

00

0.03

005

0.17

34

24.8

3 0.

0548

6 0.

2342

24

.83

0.05

471

0.23

43

49.6

5 0.

0799

0 0.

2827

49

.65

0.07

964

0.28

22

74.4

3 74

.43

99.3

4 99

.34

149.

00

149.

06

0.10

457

0.32

34

0.10

455

0.32

33

0.12

976

0.36

02

0.12

937

0.35

97

0.17

932

0.42

35

0.17

958

0.42

38

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450 Dissociation Constants of Carbonic Acid

In Table I are given the results of our analyses. The H,CQ is calculated from the CO2 tension. This is subtracted from the total COZ leading to the values for NaHC03.

In the ninth column are given the values of pKr’ calculated from these data by means of Equation 2. The results of our detcr- minations, together with those found by Warburg, are given in Fig. 1, in which the values pK1’ are plotted as ordinates and the di as abscissae.

It will be seen that most of the points lie on a straight line which intersects the ordinate at 6.33 and has a slope of 0.5. The equa- tion of this line is therefore

pK1’ = 6.33 - 0.5 d,i

Combining Equation 3 with this and substituting 6.33 for pK1 we obtain

-log y1 = 0.5 &

This empirical equation apparently fits the experiment,al data even when the ionic strength of the solution is as great as 0.180. Such correspondence would not have been expected from the t,heory of Dcbye and Huckel because it is known that in solutions whose ionic concentration is greater than 0.10 p, correction should be made for the dimensions of the ions present and for the altered dielectric properties of the solvent. This would cause the theoreti- cal line to bend toward the abscissa and our actual points lie slightly above such a curve. It is shown clearly, however, that the curve relating -logy1 and di is linear and has a slope of the order of magnitude predicted by the Debye and Hiickel theory in the region of moderately dilute solutions.

SECOND DISSOCIATION CONSTANT.

Determinations of the second dissociation constant of carbonic acid have been made by McCoy (13), Seyler and Lloyd (14), Bjerrum and Gjaldbaek (15), and others. McCoy’s results, as recalculated by Stieglitz (16), led to a value of & = 5.4 X lo-” or pK?= 10.27 at 25”; Seyler and Lloyd found KZ = 4.3 X 10-n or pK2 = 10.37; and Bjerrum and Gjaldbaek give & = 6.0 X lo-” or pKZ = 10.22 at 25”. Although it was recognized by McCoy that the concentration of the solution in which the con-

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A. B. Hastings and J. Sendroy, Jr. 451

stant was determined affected the constant obtained, no systema- tic study of the effect of varying ionic strength on the second dis- sociation constant has been made.

As in the case of the first dissociation constant we are here con- cerned with the mass law equation,

(4) aH’ x “‘201’ = KI

aHCOa’

which in the logarithmic form becomes

(5) log cxH. + log ezcoI. - log ctHO03, = log Kz

Substituting -y2[CO/] and YJHCO~‘] for LYCO~. and ffnoo$ respec- tively,

Equation 5 may be written

(6) log aII. f log yz [COf - log yl [HCO,‘] = log Ka

We have shown in the previous section that if [HCOs’] be taken equal to [NaHC03] then - log y1 = 0.5 4; By making the as- sumption that Na2C03 also is a strong electrolyte, so that [COX”] = [Sa&O& Equation 6 may be rewritten

(7) log aH. + log [Na&O,] + log -yz - log [NaHCOJ + 0.5 dp-= log 1G

Transposing log yz and O.SdL to the right-hand side, dividing through by minus one and substituting p%u for - log ffn’, Equa- tion 7 becomes

(8) pin - log [Na&OJ + log [NaHC03] = - log Kz + log y2 + 0.5 4; = PIG

Our experimental work has had for its purpose the determination of pKz’ in solutions of different ionic strengths. From these results we have been able to determine the relation between log yz and the ionic strength.

EXPERIMENTAL.

As in the case of the first dissociation constant, sixteen different solutions were prepared, representing eight different ionic strengths. The CO2 of each solution was determined gasometri- cally and the total alkali (NaOH + Na2C03 + NaHC03) titri-

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452 Dissociation Constants of Carbonic Acid

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A. B. Hastings and J. Sendroy, Jr. 453

metrically. Using Michaelis’ (17) value of 13.475 for pK, at 38”, paon ’ was obtained. From the activity coefficient of OH’ in solutions of different ionic strength, as given in Lewis and Randall, th? NaOH concentration was calculated.

From the ratio of Na&O3 + NaHC03 to total CO, the Na&O:s and NaHC03 were calculated. A ratio of NaGOX to NaHC03 was chosen which gave the maximum accuracy and was sufficiently high so that the H&O, concentration could be neglected. The

9.7

9.8

9.9

g 10.0

10.1

10.2

10.3

10.4’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ J 0 0.05 010 a15 020 LX5 030 a35 040 04.5 050

FIG. 2. Values of pII*’ are plotted as ordinates and of 4; as abscissrp. The line represents values of pKz’ calculated as pKa’ = 10.22 - 1 .l&. The points represent experimental results. .

pan of each solution was determined electrometrically using as ~0 the values found with 0.1 N HCl at the beginning of each experiment.

From these data and Equation 8, we have calculated pK,’ for each solution. Our results are tabulated in Table II and graphi- cally represented in Fig. 2 with pK2’ as ordinates and d; as abscisss. The equation which satisfies this line is

pK,’ = 10.22 - 1.1 4;.

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454 Dissociation Constants of Carbonic Acid

Combining with this Equation 8, and putting pKz = 10.22, we obtain

- log yz = 1.6 &.

If the simple relationship - log y = 0.5 v2 d; held for the carbonate ion one might expect the relation between - log y2 and t,he 4; to be

- log y* = 0.5 x 22 4; = 2 4;

It is seen that the slope over the range of concentrations with which we worked is somewhat less than that expected from the theory. In view of the fact that no account has been taken of the dimensions of the ions, or of the change in dielectric constant of the solution at the higher concentrations, it is felt that the cor- respondence of our results with what might have been anticipated from theoretical considerations is satisfactory. As in the case of yl, the linear relationship between - log yz and d; appears to hold to a concentration of P = 0.16.

CONCLUSIONS.

The first and second dissociation constants of carbonic acid have been determined at 38” in solutions of varying ionic strength.

When extrapolated to p = 0.0, the first dissociation constant was found to be 4.68 X lOA7 or pK1 = 6.33. The activity coetli- cient of the bicarbonate ion, yl, was found to be related to the ionic strength of the solution from EL = 0.01 to P = 0.18 according to the equation

- log -/I = 0.5 4;

The apparent first dissociation constant is related to the ionic strength of the solution according to the equation

pK,’ = 6.33 - 0.5 4;

The second dissociation constant of carbonic acid was found to be 6.03 X lo-l1 or pK2 = 10.22 at 38”. The activity coefficient of the carbonate ion, yz, was found to be related to the ionic strength of the solution from CL = 0.02 to CL = 0.16 according to the equation

- log 7% = 1.6 4;

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A. B. Hastings and J. Sendroy, Jr. 455

The apparent second dissociation constant is related to the ionic strength of the solution according to the equation

pKz’ = 10.22 - 1.1 4;

These results are in harmony with the theory of the behavior of strong electrolytes in dilute solutions as elaborated by G. N. Lewis, Bronsted, and Debye and Hiickel.

BIBLIOGRAPHY.

1. Milner, S. R., Phil. Mug., 1912, xxiii, series 6, 551. 2. Debye, P., and Htickel, E., Physik. Z., 1923, xxiv, 185. 3. Lewis, G. N., and Randall, M., Thermodynamics, New York and

London, 1923. 4. Bronsted, J. N.,and La Mer, V. K., J. Am. Chem. Sot., 1924, xlvi, 555. 5. Walker, J., and Cormack, W., J. Chem. Sot., 1900, Ixxvii, 5. 6. Kendall, J., J. Am. Chem. Xoc., 1916, xxxviii, 1480. 7. Warburg, E. J., Biochem. J., 1922, xvi, 153. 3. Hasselbaleh, K. A., Biochem. Z., 1916-17, lxxviii, 119. 9. Sorensen, S. P. L., and Linderstrom-Lang, K., Compt. rend. Lab. Carls-

berg, 1924, xv, No. 6. 10. Austin, J. H., Cullen, G. E., Hastings, A. B., McLean, F. C., Peters, J.

P., andVanSlyke, D. D., J. Biol. Chem., 1922, liv, 121. 11. Van Slyke, D. D., and Neill, J. M., J. Biol. Chem., 1924, lxi, 523. 12. Cullen, G. E., J. Biol. Chem., 1922, lii, 501. 13. McCoy, H. N., Am. Chem. J., 1903, xxix, 437. 14. Seyler, C. A., andLloyd, P. V., J. Chem. Xoc., 1917, cxi, 994. 15. Bjerrum, N., and Gjaldbaek, J. K., Kong. Vet.-og Land.-bohojskole

Aarskr., 1919, xlviii, 91. 16. Stieglitz, J., in Chamberlain, T. C., et al., Carnegie Instikution of

Washington, Pub. No. 107, 1909, 234. 17. Michaelis, L., Die Wasserstoffionenkonzentration, Berlin, 2nd edition

revised, 1922.

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A. Baird Hastings and Julius Sendroy, Jr.CONSTANTS OF CARBONIC ACID

AND SECOND DISSOCIATIONSTRENGTH ON THE APPARENT FIRST THE EFFECT OF VARIATION IN IONIC

1925, 65:445-455.J. Biol. Chem. 

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