activity coefficients of bicarbonate, carbonate and calcium ions in sea water

Ckocltimica et Corlnochimica Acta 1995, Yol. 29, pp. 9Pi to 965. Pergamon Press Ltd. Printed in Northern Ireland

Activity coefficients of bicarbonate, carbonate and calcium ions in sea water

ROBERT A. BERNER*

Department of the Geophysical Sciences, IJniversity of Chicago

(Receiced 15 July 1964)

Abstract-Measurements of pH of sea water samples equilibrated with known partial pressures of CO, and with calcite and aragonite have enabled the determination of molal activity coefficient

for bicarbonate, carbonate, and calcium ions in two sea water samples at 25°C and 1 atm total pressure. Results are:

For standard artificial sea wat,er (chlorinity = 19.0x,, titration alkalinity = 2.36 x 10e3

molal) :

For natural sea alkalinity = 2.09 X

yHCO,- = 0.550 * 0.007 yc(&2- = 0.021 f 0.004 yCaz+ = 0.203 & 0.010

water from Woods Hole, Massachusetts (chlorinity = 17,0x,, titration low3 molal) :

yHCO,- = 0.561 & 0.006 >‘oo,2- = 0.024 & 0.004

yCaz+ = 0.223 & 0.020

These values indicate the essential validity of the chemical model of GARRELS and THOMPSON

(1962).

The normal range in temperature and salinity of the open ocean, away from immediate

sources of fresh water, is sufficiently small so that the values cited above can be used, as a first approximation, to represent the whole ocean. Only relat#ivcly minor corrections for the effect of depth (pressure) on the activity coefficients need be made for depths less than 5000 m.

Values of the activit’y coefficienm when combined with measurements of pH, moaa+ and carbonate alkalinity can be used t,o calculate the ion activity product, IAP = aC,z+ . r_zco,s-, for sea water. Comparison of IAP with the t.hermodynamic solubility product, KS, of calcite and aragonite indicates that low latitude surface sea water is supersaturated with respect to both phases. Similar calculations for sea water at depth, based on an assumed “average” ocean model, indicat,e that a large proportion of sub-surface sea water is undersaturated with respect to calcite

anti, therefore, all ot,her forms of CaCO,.

A Q~ESTIOK of considerable interest which often confronts individuals studying the oceans is whether a given mass of sea water is saturated or not with respect to calcium carbonate ; in other words : can the given sea water dissolve the calcareous remains of organisms which come into contact with it, or is the water supersaturated so that, calcium carbonate can precipitate inorganically? This paper presents one method by which these questions can be answered in a quantitative manner.

A useful way of expressing the state of saturation of a given sea water sample with respect to calcium carbonate is by means of the ion activity product, here abbreviated as IAP: which is defined as:

IAP = c++ . a,,, z- 3 (1) where (1 = activity.

* Present address: Dept. of Geology, Yale University, New Haven, Connecticut.

947

948 R. A. BERNER

If IAP can be determined, then a comparison of IAP with the thermodynamic solu-

bility constant, K,, for the calcium carbonate phase gives a direct indication of the degree of under, or supersaturation, of the water. Specifically:

If IAP > K,, the water is supersaturated. If IAP = K,, the water is saturated. If IAP < K,, the water is undersaturated.

IAP as defined above can be expressed in terms of measurable quantities as follows :

The molal carbonate alkalinity (SVERDRUP et al., 1942) is given by the expression

A = %CO,_ + 2mcosz- (2)

where m = analytically determined or total molality (includes both free ions and ion pairs)

A = carbonate alkalinity Since :

where y = activity coefficient Therefore :

a = ym (3)

IAP = mca2+ . mco,e- . yca+2 . yc,,,2-- For the reaction :

HCO,- + Hf + COs2-

the thermodynamic equilibrium constant is :

(4)

K, = aH+ . aCO,z- _ YCO,z- . uH+ . vnC0,2- (5)

aHCO,- YHcO,- mHCO,-

Setting (Y~o,~-/YH~o,- ) = R, substituting equation (5) in terms of m,,,os_ into equation (a), and rearranging :

Therefore :

A mco,2- =

2 + (R . a,+,lK,) (6)

IAP = mCs2+A . YW+ . YCO~~- (7)

2 + (R.aH+lK)

The variables A, mca2+ and au+ can be measured directly. Values of K, for t’he appropriate temperature and pressure can be estimated from the data of HARNED

and SCHOLES (1941) and OWEN and BRINKLEY (1941). The purpose of this paper is to present experimental data for the remaining parameters, yHCO,- yco,2- and ycn2+ and thereby enable calculation of IAP. The values of these activity coefficients can, also, be used to test the assumptions of theoretical models, such as that of GARRELS

and THOMPSON (1961) from which the same activity coefficients can be calculated.

THEORETICAL BASIS FOR ACTIVITY COEFFICIENT DETERMINATIONS

Carbonate and bicarbonate

For the reactions: CO, + H,O + H’ + HCO,- CO, + H,O + 2H+ + COs2-

Activity coefficients of calcium ions in sea nat,er 949

the thermodynamic equilibrium constants at 25”C, one atmosphere total pressure are respectively :

rr, = 1~-m = %I+ - uHC%- P

(8) co, * uHzO

2 li’

c = [email protected] = ” =+ * aCOs2-

P (9)

CO, ’ nH,O

Values for the equilibrium constants are based on the data of HARNED and DAVIS (1943), HARNED and BONNER (1945), and HARNED and SCHOLES (1941). Substituting

the activity of water, O-98, in standard sea water of 19-O%, chlorinity and rearra~lging equations (8) and (9):

(10)

10-18.15 . aco,2- =

P,, 2

ad (11)

Carbonate alkalinity can be expressed in the form :

Therefore :

A= aHCO,- 2aC0,2- +

YHCO,- Yco,*-

P A=_!?!!!

[

10-7'82 10-17435

-+

aH+ YHCO,- Yco,z-%I+ 1

(12)

(131

The carbonate aIkalinity (after minor correction for boric acid dissociation) is independent of Pco, as can be seen from the reaction: H,O + CO, + COs2- + 2HCO,-. Therefore, measurement of pH of the same sea water sample at two different fixed values of Pco, at a known value of A allows the calculation of the two remaining ~knowns, yuCo,- and ycO,e-, by simultaneous solution of two equations of the type of equation (13). In this manner the activity coefficients for bicarbonate and carbonate ions are determined. These coefficients are not to be confused with the activity coefficients for free bicarbonate and carbonate ions, but, instead, are empirical parameters representing the ratio of the activity of the ions to their t,otal, or analytically determined, molalities.

The effects of ion pair formation, complexing, and ionic strength are implicitly included in the values for Ynco,- and ycO,a-. Because the cationic composition and ionic strength of sea water are negligibly affected by the changes in pH and Pco, used to obtain the two equations of type (13), these activity coefficients can be con-

sidered as constants for a given sea water sample of fixed temperature, pressure, and salinity. Therefore, the solution of the two equations for assumed constant values of once,- and yco,z- is justified.

Calcium

Determination of the activity of calcium is based on the fact that sea water is, in general, relatively close to saturation with respect to CaCO,. IJnder normally 10%

10

950 R. A. BERNER

partial pressures of carbon dioxide, the addition of CaCO, to sea water results in the solution or precipitation of negligibly low amounts of calcium relative to the total calcium in solution. This is clearly shown by the following.

The precipitation or solution of calcium carbonate can be represented by the reactions :

CO, + Hz0 + CaCO, + Ca2+ + 2HCO,-

CaCO a + Ca2+ + COz2-

In either case the change in carbonate alkalinity is twice the change in calcium concentration, i.e. :

2Amca2+ = AA (14)

The change in carbonate alkalinity upon equilibration of sea water with CaCO, at a fixed partial pressure of CO, can be determined from the data of the present study. The carbonate alkalinity is calculated by means of equation (12) from already determined values of anon,- and yco,z- and values of auto,- and aco,2- calculated on the basis of equations (8) and (9) from the equilibrium pH. The change in alkalinity (after boric acid correction) is simply the difference between this value of A and the original measured value, and the change in calcium concentration is one half the change in carbonate alkalinity.

At the partial pressure of CO, used in this study, 10-2.50 atm, the equilibrium pH (see below) for saturation with calcite or aragonite indicates that the increase in alkalinity and calcium concentration, resulting from the lowering of pH and con- sequent dissolution of CaCO,, amounts to only a few per cent of the dissolved calcium originally present. This small increase in calcium concentration does not change the overall ionic strength, composition, or complexing properties of the sea water, and, therefore, does not alter the value of ycaa+. Also, the molality of total dissolved calcium can be corrected for the small amount added by CaCO, dissolution.

The determination of yca2+ thereby proceeds as follows: from the equilibrium pH at Pco2 = lo- 2.50, the activity of carbonate ion is calculated by means of equation (9). From the equilibrium expression (at 25°C and 1 atm pressure):

for the reactions:

Kcalcite = lo-*.34 = yca2+ . mca2+ . aco,e-

wKaragonite = 1O-s.15 = yca2+ . mca2+ . aco,e-

CaCO 3 calcite * Ca2+ + C032-

(15)

(16)

CaCO 3 aragonite * Ca2+ + C032-

ycaZ+ is then calculated from acoSa- and corrected values of mca2+. Values of Kcalcite

ad Karagonite are based on the data of LATIMER (1952) and JAMIESON (1953)

Marterials METHODOLOGY

Two sources of sea water were studied : standard artificial sea water (LYMAN and FLEMING, 1940) of chlorinity 19*0%, and titration alkalinity 2.36 x 1O-3 and filtered

Activity coefficients of calcium ions in sea water 951

natural sea water from Woods Hole, Massachusetts, of chlorinity 17*0%, and titration alkalinity 2.09 x 10-3.

Sources of CaCO, for ycaa+ determinations were reagent grade calcite (Fisher Scientific Co.) and two natural aragonites from Herrungrund, Hungary, and Cleater Moor, Cumberland, England. Both aragonite samples gave identical results. The reagent grade calcite was in the form of a uniformly fine powder of ~20 ,u particle diameter and contained no detectable intermixed aragonite or other phases as shown by microscopic examination and X-ray powder diffraction. The aragonite samples occurred as large, translucent to transparent, single crystals and contained little Sr2+ ( < 1%) etc., in solid solution and no calcite detectable by X-ray powder diffraction_ Prior to use the aragonite was crushed and ground for a few minutes in a mortar and pestle in order to provide sufficient crystalline surface area for solution equilibration.

As a source of carbon dioxide, gas mixtures of air and CO, were employed. The

gas mixtures were obtained from the Matheson Co., Joliet, Illinois, and made up to order. Analyzed concentrations of CO, used were 290 & 10, 300 f 10, 3150 + 100, and 3200 + 100 parts per million by volume. Because at the total pressure of the experiments, one atmosphere, the gases of the CO,-air mixture behave essentially as ideal gases, the partial pressure, in atmospheres, for each mixture was assumed to be the same as its volume concentration.

Procedure

The washed CO,-air mixture was bubbled against atmospheric pressure through 50-75 ml samples of sea water by means of a fine glass capillary which enabled the rapid formation of small bubbles. During equilibration the sea water was constantly stirred by means of a teflon coated magnet driven by a magnetic stirrer. Glass and saturated calomel electrodes were used to measure pH, and potentials were constantly monitored by means of a Beckman Expanded Scale pH Meter connected to a Varian strip-chart recorder. The electrodes were standardized against fresh Beckman pH 6.86 and 9.18 powder buffers before and after every run. Buffer checks were always. within &O-O2 pH of each indicated buffer value. Temperature was controlled to 25.0 & 0.3”C by means of a shallow stirred water bath held at a constant temperature by a mercury thermoregulator.

In runs at Pco, = 10-2.5 which is about ten times the normal atmospheric value, the solution vessels were covered with a loose fitting transparent plastic film to pre- vent appreciable back diffusion from the air. Without such a cover, the equilibrium pH was found to vary by several hundredths of a pH unit depending upon the rate of bubbling. Rate of stirring and the presence of a CaCO, suspension were found to have negligibly small effects on the electrode potentials.

In runs using CaCO,, the powdered material was added to each sea water sample after saturation with CO,. To minimize equilibration time, relatively large amounts of CaCO, were used ( m3 g per 60 ml sea water). Concordance of results for yCas7 (see Table 1) using unground calcite and two different ground aragonite samples indicates no appreciable effects due to excess surface free energy from grinding. Also, soaking of calcite in sea water for several hours prior to use had no effect on the extrapolatecl equilibrium pH.

As indicated by the chart recorder, the equilibrium pH of runs free of CaCO, was

952 R. A. BERNER

Table 1. Experimental data for activity coefficients, 25-O i 0.3”C, 1 atm total pressure

-1% pco, PH YACOs- Rico,~-

Sta~ndard art$cial sea water: Cl- = 19.0x,, A =L 2.36 x 10-3~~h

2.50 7.40

250 2.50 2.50 7.41 1

7-395 Use average value = 7.403 7.405

3.52 8.26 0.553 0.019 3.54 8.29 0.547 0.022 3.54 8.30 0.543 0.025 3.54 8.27 0.555 0.018 2.50 7.48 - 2.50 7.485 ~- -- 2.50 7.595 - 2.50 7.59 - -

B’oods Hole sea water: Cl- = 17*0%,, A = 2.09 x 10-3~~

2.50 7.36 2.50 Use average value = 7.363 2.50

7-37 I 7.37

3.52 8.245 0.561 0.024 3.52 8.26 0.555 0.029 3.54 8.245 0.566 0.020 2.50 7.47 - 2.50 7.49 - - 2.50 7.60 - -

yca2+” Mineral(“)

- -

- 0.213 0.208 0,193 0.198

- - 0.242 R. G. calcite 0.221 R. G. calcite 0.206 C-aragonitc

-

- R. G. calcite R. G. calcite C-aragonitc H-aragonite

- -- - -

flf Corrected for excess mcSe + formed by solution of CaCO, @) C = aragonite from Cleater Moor, Cumbcrland, England;

H = aragonite from Herrungrund, Hungary.

achieved relatively rapidly, and the necessary duration of runs at Pco, = 1O-2.5 averaged about one hour while those at Pco, = 1O-3.5 averaged about 24 hr. Typical equilibration curves are shown in Figs. 1 and 2. The eq~libration with CaCO, was

considerably slower, and final equilibrium was not achieved after four hours. How- ever, a plot of pH ws. exp (O-01 t) with t in minutes proved to be essentially linear and enabled extrapolation to infinite time. Figures 3 and 4 show characteristic extrapolation curves. As can be seen, the error of extrapolation is not great because the last measurements are within 0.01 pH unit of the extrapolation value. equilibrium with CaCO, was approached only from undersaturation to avoid precipatation of metastable forms of CaCO,

RESULTS

Experimental data and derived activity coefficients are summarized in Table 1. Final results for the activity coefficients at 25”C, 1 atm total pressure, are : ( f values refer to experimental reproducibility):

Standard Artificial Sea Water (LYMAN and FLEMING, 1940)

Cl- = 19-O%,, titration alkalinity = 2.36 x 10-Z m

ynoo,- = 0.550 & 0.007 yeo,z- = 0.023. f 0,004 ycaz+ = 0.203 & 0.010


0' I I 1 I I I

7.3 7.4 7.5 7.6 7.7 7.6 7.9

PH

Fig. 1. Equilibration curve, directly taken from chart recorder, for Woods Hole sea water at PC,,, = 10-2.50.

ISO-

160-

140-

izo-

IOO-

.t E so-

; .-

t 60-

40-

20 -

0' ' I 7.6 7.9 8.0 6.1 6.2 6.3

PH

Fig. 2. Equilibration curve, directly taken from chart recorder, for artificial sea water at PCO, = 10-3’52.

954 R’. A. BERNER

T-55 0 0’1 0.2 0.3 0.4 0.3 0.6 0.7 0.8 0.5 1-O

exp c-o*01 t 1

Fig. 3. Ext,rapo~ation curve for e~llilibration with aragonite in Woods Hole sea water at PC0 2 = 10-2.50. t is time in minutes.

Fig. 4. Extrapolation curve for equilibration with calcite in Woods Hole sea water at PC,, = 10-2,50. t is time in minutes.

Woods Hole Sea Water

Cl- = 17-O%,, titration alkalinity = 2.09 x 1O-3 m

yRCO,- = 0.561 f 0.006

yCo,z- = 0,024 -& 0.004

yCaz+ = 0.223 + 0.020

The values listed above for artificial sea water can be compared with those calculated for a standard sea water of 19*0%,, and 25°C from the theoretical model of GARRELS and THOMPSON (1962). This model is based mainly on ion pair formation

Activity coefficients of calcium ions in 98% water 955

to explain values of the activities of several simple ions in sea water which are too low to be attributed simply to ionic strength effects. Total activity coefficients (yT of CARRELS and THOMPSON) based on the model are:

Yn(lo,- = 0.47

yco,_ = 0.018

Yea ff = 0.25

Good agreement is shown between the value for ~oo,~- and that cited above from the present study. The slightly different values found for anon,- and yca2t infer moder- ately small errors in measurement or in the theoretical calculations. This is probably due, for the most part, to the many rough ass~ptions of the model in estimating the activity coefficients for the free ions after correction for ion-pairing. It is, nevertheless, gratifying how well the theoretical values agree with those determined experimentally, and this, in effect, demonstrates the essential validity of CARRELS and THOMPSON’S model.

HARVEY (1955) presents tables of mixed or non-thermodynamic equilibrium eon- stants for the first and second dissociation of carbonic acid in sea water from which values of yHCo, a.- and ycO,+ can be calculated for a sea water of 19-O%, and 25°C. The tables are taken from the studies of BucH et ak. (1932) and BUCH (1951). Results

are : y~co,- = 0.43

y(.o,z- = O-018

The value for yuo g2- agrees reasonably well with the value found in the present study. The value of yHCOS-, which is 25 per cent lower, is probably due to differences in measuring technique. Original determination of the first (mixed) dissociation constant, Ii,’ was based on q~nhydrone electrode me~urements of pH at high Pco, (pH < 7) and short equilibration periods of about 10 min (HARVEY, 1955, p. 178). -41~0, some of the values of K,’ and K,’ were deduced from observations of other workers on sodium bicarbonate solution and not measured.

The usefulness of the activity coefficients obtained in the present study is only as good as their applicability to the real ocean. Natural sea water is not always at 25”C, I atm total pressure, and 19*0%, chlorinity. However, it can be shown that the natural variations in temperature, pressure and salinity (as it affects ionic strength and ion pair formation) do not appreciably alter the values listed above.

The normal range of chlorinity in the open sea away from immediate sonrces of fresh water (such as large rivers and melting glaciers) is about 2x, (SVERDRUP et al., 1942). As can be seen above, a difference in chlorinity of 2x, between standard artificial sea water and Woods Hole sea water does not result in an appreciably large change in the activity coefficients. As one might expect, the activity coefficients are slightly higher in the Woods Hole water which has the lower ionic strength. There- fore, normal chlorinity differences from 19-O%, do not effectively mar the usefulness of the measured activity coefficients.

The effect of temperature on the activity coefficients can be directly assessed for bicarbonate and carbonate. Three additional runs on Woods Hole sea water were made at 3 + 1°C according to the methods described above. The only mo~fieation was the use of ice as a crude temperature controlling mechanism which accounts for

956 R. A. BERNER

the f 1” fluctuation. Equilibrium constants for 3’C were taken from the data of HARNED and BONNER (1945), HARNED and DAVIS (1943), and HARNED and SCHOLES (1941), and all necessary buffer and electrode corrections were made. Data are summarized in Table 2. The values obtained are :

yHCO,- = 0.54

yco,2- = 0.03

These values are quite similar to those obtained for the same sea water at 25°C. The

temperature effect on Yea 2+, although not measured, should probably be even less than

that for Yuco,- and yco,2- because of a lesser degree of complexing of Ca2+ in sea water (GARRELS and THOMPSON, 1962). The fact that 25-30” temperature changes do not appreciably affect values for ionic activity coefficients in sea water is reason- able in light of data presented by ROBINSON and STOKES, 1959 (p. 212) for the activity coefficient of NaCl in one molal solutions.

ZEN (1957) presents values for the partial molar volume of aqueous ions as a function of concentration and temperature. From this data the effects of total pressure on Yeast . yco,2- can be estimated on the assumptions that the partial molar volumes in sea water are the same as in a hypothetical CaCO, solution of the same

Table 2. Experimental data for activity coefficients 3 & l”C, 1 atm total pressure, Woods Hole sea water

-1% yco, PH YHCOI- Y co3 2-

2.50 7.24

2.50 7.25 0.54 0.03 3.54 8.21

ionic strength, and that the difference, V - V” (where B” refers to infinite dilution), is independent of pressure, as a first order approximation, for sea water of normal depths ( ~5000 m). The appropriate equation is :

yp* = (v* - VO”)(P - 1) log y1” 2.3RT

where y* = yca2t . yco,z- P* = V&2+ + V’co 2-

PO* = V* at infinitl dilution R = gas constant B = partial molar volume T = absolute temperature P = pressure in atmosphere

Values of Yp*/yI* are tabulated in Table 3 as a function of depth and, therefore, pressure for 10°C. If the above assumptions are at all valid, it is apparent that only minor corrections need be made for the deepest waters. Although no data is available on the pressure effect on the ratio R = (yco ,2-/yHCO,-) which appears along with

YWf . yco,2- in the expression for IAP, judging from Table 3 it should not vary more


than about 10 per cent for pressures less than 500 atm and will not appreciably alter calculated values of IAP.

Table 3. Effect of pressure on y* = ycaz+ . yco,s- calculat,ed from the dat,a of ZeN (1957)

Depth (m) 1000 2000 3000 4000 5000

100 Pressure (atm)

200 300 400 500

Y*m 1.04 1.075 1.12 1.16 1.20 Y*,T,

Actual measurement of partial molar volumes and compressibilities in sea water are needed to refine and check the order of magnitude reliability of the data of Table 3. A crude check is provided by OWEN and BRINKLEY (1941) who have calculated the pressure effect on the solubility of CaCO, in pure water and in a NaCl solution of the same ionic strength of sea water. From their data the pressure effect upon y* in a pure sodium chloride sea water can be directly determined and is less than 10 per cent increase at all pressures less than 500 atm. Recent pressure-solubility measurements of aragonite in sea water (PYTKOWICZ, 1964) enable calculation of yp*/yr* = 1.40 at 400 atm. However, this can best be considered a maximum value because of difficulties inherent in PYTKOWICZ’S method.

CALCIUM CARBONATE SATURATION IN THE OCEANS

Upon substitution of the measured values of yHCO,-, yco32--, and yca2+ into equation (7), the expression for IAP becomes :

IAP = 0.20./l . m&Z+

95 + 1*82un+,K2 (17)

Shallow, low latitude sea water can be characterized by the values 25”C, 1 atm total pressure, chlorinity 19.0%“) A = 2.25 x 10-3, pH = 8.15, w++ = 1.03 x 10-2, X, = 10-10.33. Under these conditions:

IAP = 12.5 x 10~~

The solubility product constants for calcite and aragonite at 25°C and 1 atm total pressure (see above) are :

Kcalcite = 4.5 x 10-g (15) K aragonite = 7.1 X lo-’ (16)

Therefore, shallow, warm sea water is supersaturated with respect to both calcite and aragonite as already suggested by the work of WATTENBERG and TIMMERMAN (1936),

GARRELS and THOMPSON (1962), WEYL (1961), and others. The degree of supersaturation of low-latitude, shallow sea water, in terms of ppm

dissolved CaCO,, can be calculated directly from IAP and other measurable parameters. In expressing the results, the conditions of supersaturation must be stated unambiguously, i.e. supersaturation with respect to fixed Pco, or fixed total carbon (CaCO, + dissolved carbon). The appropriate equations for transforming IAP to A (the degree of under- or supersaturation in terms of ppm dissolved CaCO,) are derived

958 R. A. BERNER

and discussed in Appendix I. For fixed Pcoz, low latitude shallow sea water is super-

saturated with respect to calcite by:

A = 52 ppm

The value of 52 ppm is approximately twice that obtained on samples of standard artificial sea water at 25”C, by means of the carbonate saturometer (WEYL, 1961). This technique consists of measurement of pH followed by measurement of pH of an unstirred suspension of CaCO, in the same water sample. The change in pH reflects

the degree of under- or supersaturation and is empirically calibrated against ppm CaCO, by means of acid-base titrations.

Due to a lack of stirring, free CO, exchange between the sample and the atmosphere is not achieved in the saturometer, and, therefore, results obtained by this method are not directly comparable to those based on the calculation of A at fixed P. The results for the saturometer much more closely approximate those derived frk2.A calculations based on fixed total carbon, i.e. a closed system (see Appendix I). This provides some justification for the empirical calibration method which treats the saturometer as a closed system (WEYL, 1961). However, the saturometer is not truly

closed off from the atmosphere, and this may help to explain variability of results obtained from the same sample using the same CaCO, phase.

In order to determine the state of saturation with respect to calcite of any given sea water sample in the ocean, consideration must be given to the effects of temperature and pressure on the thermodynamic solubility product, K,?, for the reaction:

CaC03calcite % Ca2+ + COS2-

The appropriate equations are : For temperature :

8 log K, Aa

aT p=- 2.3RT2 (18)

where Aa0 = difference in partial molar enthalpy between calcium and carbonate ions and calcite, all in the standard state.

For pressure : alog K, -AV

Yz- ap T 2.3RT (19)

where AP” = difference in standard partial molar volume between calcium and carbonate ions and calcite.

Under all pressures and temperatures found in the ocean:

Therefore, K, increases with increasing pressure and also increases with decreasing temperature. Since, with depth in the oceans, pressure increases and temperature decreases, K, must increase, and CaCO, must become consistently more soluble with depth.


The expression for AH” in equation (18) for a constant pressure of 1 atm is :

Aa;;, = “Ry2,,, + A&;298,(T - 298)

where ACPy,,,, = the difference in standard partial molar heat capacity between calcium and carbonate ions and calcite

and the subscripts, (T) and (298), refer to any temperature and 298°K. As a first approximation A(?,” is here considered constant for a temperature range of 25”. Upon integration, equation ( 18) becomes :

K log + =

K (20)

SW81

On substitution of the values:

AFW, = -2950 Cal/mole

AQ$,,, = - 100 cal/deg/mole

R = 1.987 cal/deg/mole

log :‘T’ s(es8) = 5860 [& -i]+ 50.21ogF (21)

The value of AR&) is based on National Bureau of Standards values cited by LATIMER (1952). The value for AC:,,,,, was obtained as follows : SEGNIT et al. (1962)

have determined AGgcaB,, for the reaction:

CO,, + H,O, + CaCO, calcite % Ca,$+ + 2HCO<,

AC&,,) = - 82.1 cal/deg/mole

HARNED and SCHOLES (1941), and HARNED and DAVIS (1943) present data from which the following are calculated:

HC0a2- % H+ + COa2- APP(29,, = -65

H+ + HCO,- % CO,,, + H,O, A%W = $89.6

CO%, % co,, ACE&,, = -42.0

Adding these four reactions, we obtain:

CaCO 3 + Ca2+ + COs2- Aq2,,, = -99.5 cal/deg/mole

OWEN and BRINKLEY (1941) derived the equation:

K 2.3RT log 3 = -AP”(P - 1)

K St11

+ AK;?,, (B + l)(P - 1) - (B + 1)2 2.3log (B&T)] (22)

where ARI, = difference in standard partial molar compressibility at 1 atm pressure B = an empirical parameter characteristic of the solvent

OWEN and BRINKLEY give the values,

A%,, = -157 cm3/atm/mole

B’ = 3000 bars (atm)

960 R. A. BERNER

ZEN (1957) gives values for partial molar volumes of ions which when combined with the density of calcite can be used to calculate APO for any temperature between 0”

and 25°C. Substitution of AT, AK;;, and B into equation (22) therefore enables the calculation of the effect of pressure on K, at the appropriate temperature.

Calculation of K, for a given temperature and pressure, therefore, consists of solution of equation (21) for KScTl at 1 atm, followed by solution of equation (22) for K s,T,p,. The reference value for calcite, K, = 4.56 x 1O-g at 298” and 1 atm pressure, is taken from National Bureau of Standards data given by LATIMER (1952).

Using the above equations, the degree of saturation with respect to calcite can be calculated for a model “average” ocean whose temperature-depth profile and other properties are shown in Table 4. Carbonate alkalinity, mCa2+, and chlorinity (salinity)

Table 4. Distribution of various properties with depth for an average ocean model; Cl- = 19.0x,, A = 2.3 x 1O-3 m, mCaz+ = 0.0103.

Subscripts m, h, and c are explained in t,he text

Depth Pressure Temper- in in ature K calcite IAP, IAP, IAP, m atm “C x log pH, x 109 PH, x log pH, x 10s

0 0 20 4.9 8.2 12.8 8.2 12.8 8.2 12-s 500 50 13 6.2 7.8 5.5 7.6 3.6 8.1 9-8

1000 100 6 7.4 7.6 3.2 7.7 4.0 8.1 9.0 2000 200 5 9.7 7.6 3.7 7.9 6.8 8.1 10.0 3000 300 4 12.8 7.6 4.2 7.8 6.3 8.1 11.2 5000 500 3 21.2 7.6 5.3 7.8 7.9 8.1 13.7

are assumed to be constant with depth. Values for these parameters are given in Table 4. Under these conditions:

IAP = 4.75 x IO-6 Yi”P,

95 + (1*82a,+/K,) ’ y;“l, (23)

Three pH-depth distributions are considered. The values marked pH, and pH, refer respectively to measurements cited by MOORE et al. (1962) for the N. Pacific and HARVEY (1955, p. 37, Fig. 17) for the equatorial Atlantic. Values designated as pH, are considered as a maximum pH limiting case. The distribution of pH with depth is not well known, but results of most studies suggest a consistently lower pH at depth, than at the surface, due to increased partial pressures of CO, resulting from bacterial activity, and to the effect of increased total pressure on the dissociation constants of carbonic acid (BUCH and GRIPENBERG, 1932).

Values of K, and IAP for each depth calculated from the appropriate equations are listed in Table 4 and plotted in Fig. 5. As can be seen, IAP is, in general, less than Kcalcite (the curve for IAP, represents maximum values). Only near the surface is the opposite true. Therefore, if this “average” ocean model is more or less representative of the real oceans, it is clear that the bulk of subsurface sea water is undersaturated with respect to coarsely crystalline calcite and, consequently, with respect to all less stable forms of CaCO,, such as aragonite.

The above statement is based mainly on theoretical calculations and a simplified model ocean. In situ measurement of pH, more accurate determinations of the effect


of pressure on activity coefficients, and accurate experimental determinations of the effects of lowered temperature and especially increased pressure on Kcalcite are badly needed. An initial step in this direction has been made recently (PYTKOWICZ and

CONNERS, 1964). Nevertheless, it is felt that the thermodynamic calculations made here are of sufficient accuracy to warrant the conclusion that large volumes of subsurface water are undersaturated with respect to all forms of CaCO,. Any theory which attempts to explain the distribution of pelagic calcium carbonate sediments

IAP, K, X 10’ 0 4 8 12 16 20 24

Fig. 5. Ion activity product, IAP, and thermodynamic solubility product of calcite, K,, with depth in a model “average” ocean. Depth is in meters. Subscripts of

IAP refer to pH data from the following sources: h = HARVEY (1955, p. 37, Fig. 17)

m = MOORE et al. (1962) c = Maximum pH limiting case (see Table 4)

aa a function of depth to the ocean floor must consider that the oceans may be able to dissolve CaCO, up to depths as shallow as a few hundred meters. The sudden change in per cent CaCO, in sediments at 4000-5000 meters depth (BRAMLETTE, 1961) is

most likely due to a change in rate of solution and not to a sudden change downward from a state of supersaturation to one of undersaturation.

APPENDIX I

In this section are presented derivations of equations which can be used to con- vert values of IAP and K, to degree of under- or supersaturation in terms of parts per million dissolved CaCO,. Two situations are considered.

962 R. A. BERNER

1. Closed system of constant total carbon (CaCO, + dissolved carbon) 2. Open system of constant Pco2. For either situation :

K, = jjGjiCtLz+ . yco,z- . Sicilw . riico,z-

IAP = yca2+ . yco,z- . mcaai . mco,

where the symbols with bars refer to saturation. For sea water (see main body of text):

, - mca2+ - mCaa+ << 1

mca2+

Therefore :

and

Therefore :

For the reaction:

l&2+ = Ycaz+

%0,2- w Yco,z-

YHCO,- w Ymco,-

mcO,z- KS __ - IAP mcO,e-

CO2 + H,O + COa2- + 2HCO,-

K, = &co,- m&o,- - . yco,a- mco,z-Go,

Therefore :

Closed system

Let :

-2 %c0,- &n&o3

P ('0, = IAP Pco,

xc = mHpCO, + mHCO,- + mC0,2-

(1’)

(2’)

(3’)

(4’)

(5’)

(87

(mH2C0, includes both H,COSa9 and C02a,)

Under a closed system the only change in EC, upon equilibration, is duo to the solution or precipitation of CaCO,. In other words:

E, - C, = 6iCaa+ - mca2+ (9’) At pH 7.6 and above:

Therefore : mEI,CO, + mHCO,- + mC0,2- m mHCO,- + mc'l,3'-

and :

Ix e m mHCO,- + mCO,z- (IO')

z, - xc, m (?EHco$- - mHco,-) + (%20,z- - m,:03z-) ( 1 1 ‘)“,


Also (see text):

Therefore :

A - A = 2(?5ca2,. - mca2+)

A - A = 2(& - Z.,)

From the definition of A :

A - A = (rnHCO,_ - TnHCO,_) + 2(rnCOs3_ - mco,*-)

From (ll’), (12’), and (13’):

Therefore : mlICO,- - mHCO,- -

-0

and :

C, - Xc = rEco,2- - mco,2-

From (6’) and (16’) :

Let :

mcaPf - mca2+ = mco,z- - mco,2-

r?ics2+ - mcazt = mco,2_ (& - 1)

A = 105(~rE,,z+ - mCaa+)

where A refers to ppm dissolved CaCO,. Therefore :

A = 105mco,2- (G-4

(19’)

From text equation (G) the expression for mco,2_ is:

Finally :

A mco,2- =

2 + (yCo,~-aH+)I(YHCO,-K2)

A = _105A(K/IAP - 1)

2 + l/B

where

B= K 2YHCO-

Open system

Under an open system: P co, = i5 co2

Therefore from (8’) :

fiHCO,- = J H IAp mHC0,-

From (S’), (13’), and (22’):

B - A = mHco,- (J&-lj +2m00az-(&-1)

Since :

A - A = 2(~&~+ - mCae+) = A4

(14)

(12’)

(13’)

(14’)

(15’)

(16’)

(17’)

(18’)

(6)

(20’)

(21’)

(22’)

(237

(14)

R. A. BERNER 964

Therefore :

Substituting the expressions for mHCo,_ and mc0,2- from the text equation (6) and an analogous expression for mHco3-, finally:

A = 105_4 (Jd-11 i&--1)

2 + 4B + 2+1/B I (25’)

where B is defined as before. From (18’) and (25’) :

A = Aclosed +

105A i& - 1)

open 2 + 4B (W

Acknowledgements-The writer is indebted to the following individuals who read and criticized

the manuscript: R. M. GARRELS of Harvard University and R. M. PYTKOWICZ of Oregon State

University. Specimens of natural aragonites were furnished by J. V. SMITH of the University of Chicago, and EDWARD OLSEN of the Chicago Museum of Natural History. Research was sup- ported by an American Chemical Society-Petroleum Research Fund Starter Grant for Funda-

mental Research in the Petroleum Field (type G).

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11

activity coefficients of bicarbonate, carbonate and calcium ions in sea water

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