THE DETERMINATION OF THE EQUILIBRIA INVOLVING CALCIUM, HYDROGEN, CARBONATE, BICARBONATE,

AND PRIMARY, SECONDARY, AND TER- TIARY PHOSPHATE IONS.*

BY I. NEWTON BUGELMRSS AND A. T. SHOHL.

(Prom the Department of Pediatrics, Yale Unicersity, New Haven.)

(Received for publication, October 18, 1923.)

Quantitative equilibrium relations have been developed for the ions, Ca++, H+, HCO;, Cob, HIPOT, H,POI, and PO?-, and expressed in a single equation. The equilibrium constants have been det.ermined experimentally under conditions within physiological limits. This system of calcium salts has been select.ed because (a) t,heir ions are involved in the physiology and pathology of bone calcification and of nervous irrit.abilit,y (e.g. rickets and t,etany); (h) these ions have not been included in other studies of electrolyte equilibria (1) since the calcium and phos- phorus are too small to affect the osmotic and hydrion equilibria of the blood. However, the hydrion concentration directly affects the ionization of the calcium salts of this syst,cm. The general character of this equilibrium is such that a change made in the concentration of one of t,he components produces measurable changes in all the others. With a knowledge of the equilibrium constants of Obese ions one can predict changes in the ionic con- centrations in the blood as well as in other aqueous systems.

THEORETICAL.

Derivation ofEqua~ions.-At the hydrion concentration of blood, COT and PO? are negligible and so the equilibria involve only the ions, Ca++, II+, HCO,, IIPO;, and H2P0,, and the molecules, COz, HsC03, Ca(HCO&, CaHP04, and Ca(ILPO& Such a system may be evolved theoretically by adding to water in

* Aided by a grant from the Loomis Fund.

649

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650 Ca++, H+, HCO;, HPO;, and H,PO;

sequence: (a) carbon dioxide, (b) calcium carbonate, and (c) disodium phosphate.

(CL) Carbon dioxide gas dissolved in water yields carbonic acid and its products of dissociation.

CO,~gas) * CO,(dissolved) +H,O~H,CO,tiHf+HCO;tiH++CO~

Applying the law of mass action and denoting the molecular and ionic concentrations by enclosing their formulas within brackets, the following equations are obtained.

K, [CO,] = [H,CO,] (1)

K, [H,CO,] = [H+].[HCO,] (2)

K, [HCO,] = [H+] . [COT] (3)

X0, K1, and Kz are the equilibrium constants of carbon dioxide in water.

(b) Calcium carbonate introduced into a solution containing carbonic acid causes the following additional equilibria.

CaCO, (solid) SCaCO,(dissolved)+H,CO,SCa(HCO,),~Ca++2HCO; 0

Ca++ + COT

Applying the laws of mass action we obtain

and

[Ca++]e[CO,] = K, (4)

K, [H,CO,] = [Ca++]*[HCO,]’ (5)

Equation (5) may likewise be derived by dividing equation (3) KIKs. into the product of equations (2) and (4) so that & = K

The desired relationship between Ca++, H+, and HCO, folloks at once by dividing either equations (5) by (2) or (4) by (3).

[Ca++].[HCO;] K K LH+l =j$ -< ==K, (6)

or

[H+l .- [Ca++l= K, LHCO,I (6 a)

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I. N. Kugelmass and A. T. Shohl 651

As a matter of fact equation (6) follows at sight by applying the mass action law to the very evident equation:

H+ + CaCO, (solid) $ Ca++ + HCO;

The lengthy development, however, has been presented so that the equilibrium constants could be calculated from others that are defined in the literature.

(c) Disodium phosphate introduced into the aqueous solution of carbonic acid produces additional equilibria.

Na,HPO,(dissolved) e 2Xa+ + HPO, + H2C0, + Na*

f HCO; + Naf f H,PO,

It follows then that

[H,CO,l-[HP071 [HCO;]. [H,PO~~ = K6

(7)

Replacing [HzC03] by its equivalent from equation (2) we get the equilibrium constant for secondary phosphate.

[H+] .[HP~;]

[W’O,l = K,.K, = K,

Multiplying equations (5), (6), and (7) we obtain

[ca+f]**[HCO;]* *[HPO,‘]

[H+l-[H,PO,l a.K,=(K,)a.K,=K, (9)

Also this equation follows directly by inspection from a system of reacting components expressed by the equation:

2 CaCO, (solid) + H+ -k H,PO; Z$ Caf+ + 2 HCO; + Ca++ + HPOY

This equation is expressed in terms of the ratio of phosphates. For systems in which the total phosphate concentration varies, further relationships must be derived. From equation (8) it follows that

K, [HPO,‘l [H,PO,] = [H+]

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652 Ca++, H+, HCO;, HPOt, and H,PO;

and

[HPOTl + [HJ’O,] = K, + [H+l [HPO,‘I IG

(11)

Dividing through equation (11) by [‘Ca++] we obtain

[HPOFI + h’O,I = I<, + [H+l --(Ca++] . [HPO,] K, . [Ca++]

(11 a)

Substituting,the soIubility product constant

[ca+f]. [HPO,‘] = K 9 (12)

and solving the equation for total ionic phosphate, we obtain

[HPOTI + [W’O,I = [&$, (131

b+fl = [HPO;] :“[a,PO,]

This equation expresses the equilibrium in terms of the sum of the primary and secondary phosphate ions and serves as an inde- pendent method of arriving at the solubility product constant of CaHP04. However, the ratio of the phosphates is a function of the hydrion concentration, and a known value of one phosphate determines the amount of the other. Therefore, the secondary phosphate, which constitutes about 85 per cent of the total phosphates under physiological conditions, expresses simpler relationship with the ions in t.he system. Multiplying equation (6) by (12) we obt,ain

[c~++]*.[Hco;].[HPoT]

[H+l =,,i-~~~=~)K~=~)K~=K~~ (14)

This relation follows directly by inspection from a system of reacting components expressed by the equation

CaCO, (solid) + CaHPO,(solid) + H+ $ Ca++ + HCO; + Ca+.+ + HPOS

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I. N. Kugelmass and A. T. Shohl 653

Systems studied1 under conditions in which the tertiary phosphate ion concentration is significant, may be expressed by the following relations. Multiplying

[Ca++j’. [POT]’ = K 11 (15)

which represents the solubility product of Ca3(P04) 2, by equation (14) we obtain

[ca++15.[I-IC0,1.[HP0,‘l.[P0~12 = K,eK,eK [H+l

11

Since the solubility product constant is as yet not accurately known, we may eliminate this constant by utilizing the values of the third dissociation constant of HJPOd. This value may be writt.en

(17)

Multiplying the square of equation (14) by equation (17) we obtain

Systems in which the PO? concentration is significant likewise contain CO: in appreciable amounts. The equation expressing all the ions under consideration follows by multiplying equation (4) by (18).

[ca++15.[HC0312~[CO~I.[HPO~I~[P~1= K K = K,, c19j

[H+l 8’ 10

These equilibrium equations for the ionic components of the car- bonates and phosphates of calcium must hold no matter what other coexistent species-aggregated, associated, molecular, or ion&-may be present.

1 At the suggestion of Dr. Holt who is determining the solubility product constant of the tertiary calcium phosphate we have developed the equilib- rium relations for these ions coexisting with PO?.

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654 Ca+‘, H+, HCO;, HPO,‘, and H,PO;

EXPERIMENTAL.

Method of Procedure.-The various equilibrium constants, governing the relations among the coexisting ions in question, were ev.aluated from experimental data. Calibrated tonometers were three times evacuated, filled with hydrogen, and again evacuated. According to the technique of Austin and coworkers (2) definite volumes of the necessary solutions and calculated amounts of CO2 were added. The tonometers were finally filled with hydrogen at atmospheric pressure and rotated at least an hour in the electrically regulated water bat.h at 38 f 0.02” to attain equilibrium. Samples of the gas phase were taken out of some of these tonometers and CO2 determinations made to check the calculations. Samples of the contained liquid phase were trans- ferred under oil into centrifuge tubes, and after centrifugation analyses were made immediately for total COz, total calcium, total phosphate, secondary phosphate, and hydrion concentration.

Methods of Determination of the Components.-The hydrion concentration was determined electrometri&lly. The apparatus consisted of a Leeds and Northrup direct reading potentionometer of low resistance; enclosed lamp and scale galvanometer; a Weston standard cell certified by the Bureau of Standards with a voltage of 1.01896 serving as a basis for all the electrical measurements; saturated KCl-calomel and hydr0ge.n cells and *electrodes and platinized platinum electrodes. The measurements were .standardized daily by a 0.05 N KH phthalate solution (3). Hydrogen was passed through the cell previous to the introduction of a sample from the tonometer. The hydrogen cell was rocked until constant equilibrium values were obtained and the molar concentration of hydrion read directly from a plettcd chr e which related the CH to the E.M.F.

Total Calcium. vl cc. of the centrifuged sample under oil was pipetted into a 15 cc. centrifuge tube and determined according to the method of Kramer and Tisdall (4) which one of the authors (5) has shown to be a safe procedure in the presence of phosphate within the range of the hydrion concentration studied.

Total Carbon Dioxide.-1 cc. of the liquid phase kept under albolene was analyzed for total CO2 by the method of Van Slyke and Stadie (6).

Carbon Dioxide of the G%s Phase.-Samples of the gas phase were ana- lyzed for CO2 by the Haldane-Henderson apparatus (7).

Inorganic Phosphates.-These were determined on 1 cc. of the sample kept under oil by Briggs’ modification of the Bell-Doisy method (8).

Secondary Phosphate.-Secondary phosphate was determined on 1 cc. of t:ie sample kept under oil by a method reported in a separate paper (9).

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I. N. Kugelmass and A. T. Shohl

Preparation and Analysis of MateGals.-Conductivity water was ob- tained by redistilling water made alkaline with KMn04 through a block tin condenser and was kept in Pyrex flasks free’from C02.

Sodium bicarbonate was prepared by the method- of Auerbach and Pick (10). Test-91586 gm. of the salt in conductivity water, titrated with methyl orange as indicator, required 19.01 & 0.1 cc. of 0.1 N HCl. This is in good agreement with the theoretically required amount, 18.88 cc.

Disodium phosphate (XasHPOk) was crystallized twice from distilled water by adding an equal volume of 95 per cent alcohol and cooling in ice water. The solution was stirred constantly until crystallization was complete. Two liquid phases appear on addition of alcohol and crystal- lization takes place at the junction of the two liquid phases. As crystalli- zation proceeds the ‘upper phase disappears, leaving but one phase when precipitation is complete. The crystals were filtered on a Buchner funnel with’suction, washed wit.h alcohol, and dried under 20 to 30 mm. of pressure at 196”, thus yielding the pure product NazHPOr.2H20 (11).

Calcium carbonate was prepared by bubbling COZ into a concentrated solution of calcium hydroxide at.room temperature and washing and drying the resulting precipitate. Stable calcite was thus obtained. The carbon- ate was added to conductivity water free from CO2 in a Pyrex flask con- nected to a COz-free burette. A current of COz-free air was then drawn through the solution to remove the last traces of COZ.

Secondary calcium phosphate (CaHP04) was prepared by adding gradually a solution of disodium phosphate to an excess of a solution of calcium chloride and washing the resultant precipitate with calcium chloride and finally with distilled water.. The solubility product constant of CaHP04 at 38°C. will be reported in a separate paper (12).

Mercurous chloride was prepared electrolytically according to the method of Lipscomb and Hulett (13) from redistilled mercury and constant boiling 1.0 N IICI, prepared according to the method of Hulett and Bonner (14). The calomel obtained, heavily laden with finely divided mercury, was washed free from acid with conductivity water and then with saturated KCl. Portions of this calomel were added to solutions of saturated KC1 for mutual saturation and the clear solution was decanted into the calomel vessel.

Mercury.-Redistilled mercury was first shaken in a separatory funnel with a 10 per cent solution of mercurous nitrate, acidified with nitric acid, and then sprayed into a long column of the same solution and finally redistilled by Hulett’s method (15).

Experimental and Calculated Data.-The methods of calculation of the factors involved in the equilibrium equations are presented to interpret the tables given below.

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656 Ca++, H+, HCO;, HPO;, and H2PO;

TABLE I.

System: Calcium Carbonate-Calcium Bicarbonate-Carbon Dioxide.

pco, %’ 2.

mdsfl

Free COs T”$y= fHzCoal. Ca(HCOs)m

[Cd+] =

HHCOJ H+ K&.105 KS

0.0263 0.0526 0.0760 0.1050 0.0263 0 :0523 0.0785 0.1005

nwlsll mols/l

0.0053 0.00068 0.0023 0.00200 0.50*10-7 4.71 0.0071 0.00136 0.0029 0.00250 (0.48.10~‘) 4.61 0.0086 0.00208 0.0033 0.00280 1.11*10-’ 4.22 0.0099 0.00278 0.0036 0.00300 1.32.10-’ 3.89 0.0048 0.0006h 0.00180 0.46.10-7 3.55 0.0075 0.00136 0.0031 0.00255 0.71.10-’ 4.88 0.0086 0.00208 0.0033 0.00275 1.14.10-7 4.00 0.0108 0.00278 0.0039 0.00330 1.45.10-7 5.17

-

142 136 141 181 133 150

PCOn is the partial pressure of COz, either measured or calculated, expressed in atmospheres.

Total CO2 is the volume of CO2 measured in the Van Slyke apparatus at the recorded temperature and pressure, reduced to standard conditions.

c+~,, the absorption coefficient of COz, is the ratio of the volume of CO, dissolved (reduced to d”) to the volume of the water. The value of this ratio for 38” used throughout this work is 0.58 (16). This ratio is, in ac- cordance with Henry’s law, independent of the pressure under the con-

ditions of the experiments. n4 acoz is the value in mols per liter of CO*

dissolved in pure water. Salts’cause a change in the solubility of CO* and hence in acot. The coefficients given were read from curves plotted from Bohr’s data (17) for NaCl solutions, with the assumption that the coefficients of CO2 in our systems are the same as in NaCl of the same

equivalent concentrations. The value of 22 at 38’ is 0.,026.

vMm1 = ;3 pcop is the molal concentration of the dissolved CO*.

H&OS calculated in this way involves the assumption that CO% does not exist as such in solution but hydrated as COe*H20 or H&OS, a fact sub- stantiated by the present work.

Total Ca(HCO& expressed in mols per liter, equals (a) total Ca in

mols per liter, or total [CO,] - [H,COs]

2 where calcium bicarbonate is the

calcium component; (b) totai[COt] - ([NaHCOsl +IHzCOxl) where

2 NaHCO

‘

is also present; (c) total [Cal - ([CaHPOr] + [Ca (HzPO&]) where phos- phates are present.

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atm

osph

ere

0.02

63

0.05

26

0.07

90

0.10

50

0.02

63

0.05

26

0.07

90

--

TAB

LE

II.

Sys

tem

: C

alci

um

Car

bon&

e-C

alci

um

Bic

arbo

nate

-Sod

ium

B

icar

bona

te-C

arbo

n D

ioxi

de.

rota

1 C

O%

m&

/1

0.01

60

0.01

69

0.01

80

0.01

90

0.00

77

0.01

11

- .- -

%oz

--

pc

oz

w.4

0.00

068

0.00

136

0.00

208

0.00

278

0.00

066

0.00

136

0.00

208

-

‘1

( .- -

‘ota

l C

a =

tota

l :a

(HC

Oa)

t.

9nO

l8/1

0.00

024

0.00

035

0.00

060

o.oo

o69

0.00

059

0.00

114

0.00

170

C

-

-

ca++

fro

m,

la(H

CO

&

= ,(H

co;).

n&3/

1

0.00

019

0.00

030

0.00

051

0.00

058

0.00

050

0.00

101

0.00

148

- --

CC

XM

X!Il

- tra

tion

of

NaH

CO

s.

m&

?/1

0.01

50

0.01

50

0.01

50

0.01

50

0.00

75

0.00

75

0.00

75

- 1

--

FIC

O;

from

N

aHC

Oa.

mol

s/l

0.01

170

0.01

170

0.01

170

0.01

170

0.00

608

0.00

608

0.00

608

‘I - -

btal

H

CO

,

mol

s/l

0.01

208

0.01

201

0.01

272

0.01

286

0.00

708

0.00

810

0.00

904

-

H+

?WlS

/l

0.17

.10-

7 0.

31*1

0-7

0.53

.10-

7 0.

73.1

0-7

0.23

.10-

1

0.99

.10-

7

- -

K4'

lOJ

4.09

3.

25

3.96

3.

46

3.80

4.

87

5.82

- -

KS

136

116

122

103

154

135

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658 Ca++, H+, HCO;, HPO;, and H,PO;

-rr~Ca(BCO& is the degree of dissociation obtained from the experi- mentally determined values for the corresponding solutions of Ca(H&O& (18). The validity of such procedure follows from Noyes’ rule “that the degree of ionization of a salt in a mixture is practically equal to that of i salt of the same valence type at the same total ion concentration.” The values of y ranged from 0.82 to 0.92 as read from curves plotted for the systems under consideration.

r2NaBCOa is 0.78 for 0.015 M and 0.81 for 0.0075 M NaHCOs. y&aHPO,is 0.62 to 0.66 for the given concentrations as calculated

from conductivity data (19). &[H#OJ in mols per liter is the difference between the total measured

phosphorus and the measured molal HPOI.

TABLE III-S&em: Phosphates and Bicarbonat

PC02 Total CO%

atmo- 8phere

0.0230 0.0526 0.0725 0.0790 0.0945 0.0989 0.1121 0.1410

nzOl8/1

0.00450 0.00565 0.00630 0.00699 0.01278 0.01305 0.01385 0.01520

Free COn [HeCOz].

cYco2 w4 PCO?

0.00068 0.00136 0.00191 0.00205 0.00246 0.00264 0.00303 0.. 00381

T

Tots.1 Cs. Total P. [Ca++] = t IHCO;] Ca(HCOs)

molsll mols/l m&l1

0.00235 0.00064 0.00160 0.00174 0.00262 0.00050 0.00195 0.00214 0.00270 0.00046 0.00200 0.00220 0.00285 0.00044 0.00220 0.00244 0.00200 0.00069 0.00126 0.00137 0.00215 0.00081 0.00132 0.00143 0.00232 0.00078 0.00151 0.00164 0.00250 0.00064 0.00176 0.00193

Bicarbonates.

%:z?

mobjl

0.0075 0.0075 0.0075 0.0075

HCO; TOtal from

NaHCOa. HdO,

--

mols/l mds/l

0.0032 O.OOis O.C@40 0.0044

0.00608 0.0086 0.00608 0.0087 0.00608 0.0091 0.00608 0.0096

W’O;l - “determined,” is obtained directly from the given measured [H,PO;)

values and this ratio calculated follows from equation (9) by substituting appropriate values for the hydrion concentration.

Estimation of the Average Error.-The average errors in analysis of the components of the liquid phase were approximately within the following ranges : 1 per cent for the total CO*; 1 per cent for the hydrion concen- tration; 4 per cent for HP04; 4 per cent for total P; and 4 per cent for total Ca. Analysis of the gas phase at the end of saturation by the Haldane- Henderson apparatus is accurate to about 0.1 vol. per cent. The corre-

sponding tension at 38’ and 760 mm. is ‘$ (760-49) = 0.7mm. of Hg. The

average difference of tensions calculated and analyzed was within 1 mm,

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I. N. Kugelmass and A. T. Shohl 659

We may calculate the degree of uncertainty of the equilibrium constant for the complete equation (14). If d(Ca) is an error in [Ca++], then the error

in K is & .d(Ca), and similarly for each of the other factors. The

general equation from which the average range of error may be calculated is derived by (a) converting the equation into logarithmic form since dK - = d(log K); (b) partially differentiating with respect to each of the K

components; and (c) summating the differentials, squaring, and solving

for $ which on substitution yields an average range of error of i 10

per cent.

of Calcium at Various Carbon Dioxide Tensions.

“;E? [Cd-+] = mined. [I-PO;]

--

mols/l m&/l

0.000550.0003f 0.00043 0.00021 0.000380.0002! 0.000350.0002: 0.000530.0003‘ 0.000610.0004~ 0.00058 O.OOQ31 0.000460.0003~

Phosphates. T

[Ca++] = ~~~po;I

mol.9~1

3.00004: I. 00003: 3.00004( 3.oooo4i 3.00008( 3 .OOOlO( 3.00010( 3.00009(

[=‘O;l/IH,PO;I Total Cd+.

-I -I-

4.7 4.5 0.002000.53~10- 4.2 4.1 0.002260.58~10- 2.8 2.6 0.002290.90~10- 2.7 2.5 0.002480.96.10- 2.1 2.0 0.001681.20~10- 2.0 1.9 0.001821.25~10- 1.8 1.7 0.001991.40~10- 1.7 1.5 !0.00215 1.60.10-

H+

- .

7

.,

.7

.7

.,

7

.7

.I

Kc106

3.01 120 2.53 152 2.00 102 2.35 114 5.05 121 5.28 128‘ 5.44 130 5.20 129

KS

I Kb.108 - -

72 70 58 90 61 80 65 75 - -

T

73 63 55 57 70 72 75 62 -

Ks.10’ KwlO~

3.63 4.45 5.62 9.63 2.92 6.83 3.35 6.48 3.66 6.88 4.06 9.78 4.22 9.23 4.53 8.32

Results.-Three series of experiments were done on systems equilibrated in tonometers at 38” at CO2 tensions varying from 17 t.o 110 mm., according to the procedure outlined above. The first series of eight experiments was made on the system, CaC03-Ca(HCO&-CO2 (Table I). The second series of seven experiments was done as above with the addition of 0.0075 and 0.015 M NaHC03 (Table II). The third series of eight experiments was carried out on systems which contained, in addition to CaCOS, WHCQ&, NaHC03, and COz, the components CaHPO and Ca(HzPO& (Table III). The values of the equilibrium con- stants K4 and Ks are derived for each experiment in Tables I, II,

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660 Gaff, H+, HCO;, HPO,‘, and H,PO,

and III; Ks, KS, and-K10 are obtained from the data in Table III. The average values. of the equilibrium constants are summarized in Table IV.

TABLE IV.

Average values of the Equilibrium Constants from the Experimental Data.

Ka.10” K5 Kg.108 Ks.lC= Km.105

Values............. 4.14*0.14 133+ 3 67=+= 7 4=!=0.4 7.6*0.6 No. of determina-

tions. . . . _ . . . . . . . . 23 21 20 8 8

DISCUSSION OF EQUILIBRIU~M CONSTANTS.

Some of the equilibrium constants involved are known accu- rately and others may be interpolated. These may be compared with the values experimentally obtained.

K1, the first dissociation constant of carbonic acid, is the true dissociation constant of carbonic acid and is to be distinguished from the apparent constant, for the concentration of carbonic acid is really the sum of carbonic acid and its anhydride and the apparent dissociation constant K1 is given by the following equa- tion.

K, ([IWO,] + [CO,] > = tH+]*[HCO,l

Walker and Cormack (20) found K1 to be 3.04 X lo-’ at 18” and calculat.ed it to be 3.4 X lo-’ at 25”. L. J. Henderson found it to be 4.2 X 10-T at 38” (21). The true dissociation constant is K’l - where n is the fractional amount of the tot,al CO2 in solution

ezisting in the form of H&OS. Walker and Cormack (20) main- tain that n is always greater than 0.5. We have found n t.o ap- proach unity and so have used the value of Kfl for K, in our cal- culations.

K,, the second dissociation constant of carbonic acid, was determined by Seyler and Lloyd (22) as 4.91 X 1OF’ at 25”, a better result than had been previously obtained (23,24). John- ston (25) recalculated Kz at 25’ from Walker and Cormack’s data and found it to be 6 X lo-‘1 and 6.4 X lo-l1 from Shields’ data (26) on the hydrolysis of sodium carbonate, results identical with those computed by McCoy (24), and those obtained by Auerbach and

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I. N. Kugelmass and A. T. Shohl 661

Pick (10). We have calculated this value by means of the van’t Hoff formula as 7.2 X lo-l1 at 38”.

Kl --,the ratio of the first and second dissociation constants of K*

carbonic acid, Johnston (25) calculated as 5,600 from McCoy’s (24) data on the carbonate-bicarbonate equilibrium at 25”. Using the data given above for 38” the value is 5,800.

KS, the solubility product constant of calcium carbonate, Leather and Sen (27) found to vary inversely with the temperature, but obtained rather irregular results, the value at 40” approximating 0.5 x 10-s. Seyler and Lloyd (28) made studies at “laboratory temperature” (20’) and obtained 0.86 X lOme. They also ob- served that the constant was not affected by the presence in solu- tion of small amounts of added Ca++from other salts, but that the presence of small amounts of neutral salts with no ion in common increased the total calcium in solution. Johnston recalculated the data of Schlmsing (29) and Engel (30) and obtained a con- cordant value of 0.98 X lo-* at 16’. Wells (31) determined the solubility product constant at several temperatures with air containing 3.2 parts COz per 10,000 as the gas phase:

0.93 X lo-* (209 ; 0.87 X lo-* (25”) ; 0.81 X lo-’ (30”)

Osaka (32) recalculated the data of McCoy and Smith (33) and obtained the constant 0.57 X lo-*. The solubilityproduct constant may be calculated from our experimental results at 38” for

4.14 x 10-s Kp$+= 5800 = 0.71 x 10-s

A ,

KZ

a value which compares favorably with the reliable constants presented above.

K 1

= IC~++l.[~CO$ _ K,K, Pf,CO,I K,

Rona and Takahashi (34) found .experimentally at “laboratory temperature” (18”) a value of 11.6 X lo+. Johnston’s calcula- tions from Schlmsing’s data show a general average of 5.3 X lo+ (16’) from Engel’s data 5.47 X 10m5 (ISo) and from Seyler and Lloyd’s data 4.80 X 1O-5 (20”). The value of Kq from our experi-

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662 Ca++, H+, HCO;, HPO;, and H,PO;

mental data at 38” shows a general average, from twenty-two determinations, of 4.14 f 0.14 X 10-5.

Rona and Takahashi (34), a decade ago, found experimentally that the value for this constant was 350 at 18”. In their work the COZ phase was not adequately controlled and so affected the hy- drion concentration. The analyzed Ca(HCOJ 2 was assumed com- pletely ionized and some of the equilibrium constants used in their calculations have since been revised. Calculation of KS from the ration I&/K1 gives 111 and from KS/K2 also 1’11. From our experimental results at 38” the general average of twenty-two determinations for this equilibrium constant is 133 f 3. The deviation of the calculated from the experimental values is 20 per cent, a value greater than the calculated percentage error and hence must be attributed to unknown experimental error or to the value of the constants used though they are the most accurate known.

K = [~,c0,1*[~~O;l = 3 6 [HCO,]. [II,PO,] K,

Since Abbott and Bray (35) found K,, the second dissociation constant of phosphoric acid to be 2 X lo-’ at 25” and 2.4 X lo-’ at 38”, Ks becomes 0.57.

K 8 = [cu++]* . [HCO,]a * [mo;]

[~+l*[Hm,I

The calculated value of KS is 3.0 X lo+. The general average of KS calculated from the experimental data given above is 4.0 f 0.4 x 10-S.

K, = [Ca++]. [IWO;]

Values of this solubility product constant found in the literature are conflicting. The concentration of Ca++ and HPOT for the aqueous systems in this work, in equilibrium, with solid CaHPO4 among other components.at various COz tensions, yields 71 X 10e8 as the average value for this constant at 38”. Calculation of the constant from equation (12) gives a general average of 66 X 10m8.

K = [Ca++]2. [Iwo;]. [mql ~- 10 [If+1 =KS.K9=@)Ko=&r

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I. N. Kugelmass and A. T. Shohl 663

Freudenberg and Gyiirgy (36) modified the equation of Rona and Takahashi (34)

[Ca+f].[HCO;] Dfl

= K,

into the empirical form

[Caf+] . [HCO,] . [HPO,‘]

W+l =K,

This equation has neither theoretical nor experiment.al basis, and the values of Ca++ calculated from it are not concordant with our experimental data. The value of Klo calculated from the product KS-K9 or its equivalents above is 8.8 X 10-5. The general average of this constant calculated from the experimental data is 7.6 f 0.6 X 1O-6.

The application of this equation to determine the calcion concentration in normal and pathological conditions will be re- ported in a subsequent paper.

K,, = [CU++]".[PO~]~

This solubility product constant has as yet not been’ accurately determined. Calculated from solubility values at ‘laboratory temperature” obtained in the literature one of the authors found Kll to be 2.8 X 10-zs at 18” (5).

Values for this tertiary dissociation constant range from 3.6 to 5.6 X lo-l3 at 18” (3). The dissociation curve for the phosphates plotted as a function of the hydrion concentration, however, indicates this constant to be of an order of 10-12.

The value of this constant depends on the known values of I&, and K9 and the undetermined value of Klz. Until accurate data at 38” are available the equation cannot be solved. It seems best not to offer an approximation.

K =[Ca++]".[HCO,la.[C~~]~[HPO,'].[PO~]=K,.K 14 PI+1 18

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664 Ca++, H+, HCO;, HPOT, and H,PO;

The value of I(3 is 0.71 X 10m8 at 38”. The value of K14 also depends on the determination of the tertiary dissociation constant at 38”. Because this value is not definitely determined it seems best for the present to use the simpler relationship in which CO; and PO, are not involved, namely

[Ca++]‘- [HCO,]. [HPO;]

[H+l = K,,

SUMMARY.

The equilibrium relations for Ca++, H+, HCO,, COY, H2P0,, HPOT, and PO7 at 38°C. have been studied and their constants: obtained experimentally are :

(1)

(2)

(3)

(4)

(5)

or

(6)

(7)

Or

(8)

(9)

[Ca++]. [HCO,]*

[H&Q] = K, = 4.14 * 0.14 x 1O-6

[Ca++]. [HCO,]

[H+l = K, = 133 * 3 or [Ca++] = 133 $;;]

[Ca++].- [COT] = K, = 0.71* 0.05 x lo-’

[Ca++]. [HPO,“] = K, = 67 * 7 X lo-*

[HPOF] + [H,PO;] = ‘g+%* (l+$g)

Fa++I = ~Hpo~~~~~~po~l (I+%)

[Ca++]‘- [HCO;]‘. [HPOf]

IH+l.bWO,I = K, = 4.0 f 0.4 X 10-O

[Ca+f]’ . [HCO;] * [HPO,‘]

[H+l =KIO= 7.6* 0.6 X 1O-5

[Ca++] = d

(7.6 X 10-5) [H+] [HCO,] . [HPO;]

[Ca++]4.[HCO;]2.[HPO~]*[PO~] [H+j

= K,, = (0.8 X 10-‘).KI,

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I. N. Kugelmass and A. T. Shohl

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I. Newton Kugelmass and A. T. ShohlPHOSPHATE IONS

SECONDARY, AND TERTIARYBICARBONATE, AND PRIMARY,

HYDROGEN, CARBONATE,EQUILIBRIA INVOLVING CALCIUM,

THE DETERMINATION OF THE

1924, 58:649-666.J. Biol. Chem. 

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