www.elsevier.com/locate/marchem

Marine Chemistry 10

Dissociation constants of carbonic acid in seawater as a function

of salinity and temperature

Frank J. Millero *, Taylor B. Graham, Fen Huang, Hector Bustos-Serrano, Denis Pierrot

Rosenstiel School of Marine and Atmospheric Science, University of Miami, Miami, FL 33149, United States

Received 17 June 2005; received in revised form 2 December 2005; accepted 2 December 2005

Available online 31 January 2006

Abstract

Potentiometric measurements of the stoichiometric constants on the seawater pH scale for the dissociation of carbonic acid in

seawater (K1*=[H+][HCO3

�]/[CO2] and K2*=[H+][CO3

2�]/[HCO3�]) have been made as a function of salinity (1 to 50) and

temperature (0 to 50 8C). The results have been fitted to the equations (T/K)

pKi � pK0i ¼ Ai þ Bi=T þ CilnT :

The values of pKi0 in pure water are taken from the early work of Harned and Davis (1943) and Harned and Scholes (1941)

pK01 ¼ � 126:34048þ 6320:813=T þ 19:568224lnT

pK02 ¼ � 90:18333þ 5143:692=T þ 14:613358lnT :

The value of the adjustable parameters Ai, Bi and Ci for pK1* are given by (r =0.0054 and N =466)

A1 ¼ 13:4191S0:5 þ 0:0331S � 5:33E � 05S2

B1 ¼ � 530:123S0:5 � 6:103S

C1 ¼ � 2:06950S0:5:

For pK2* the parameters are given by (r =0.011 and N =458)

A2 ¼ 21:0894S0:5 þ 0:1248S � 3:687E � 04S2

B2 ¼ � 772:483S0:5 � 20:051S

C2 ¼ � 3:3336S0:5:

0304-4203/$ - s

doi:10.1016/j.ma

* Correspondin

E-mail addre

0 (2006) 80–94

ee front matter D 2006 Elsevier B.V. All rights reserved.

rchem.2005.12.001

g author.

ss: fmillero@rsmas.miami.edu (F.J. Millero).

F.J. Millero et al. / Marine Chemistry 100 (2006) 80–94 81

The values of pK1* and pK2* determined in this study are in good agreement with the seawater (SW) measurements of Mehrbach

et al. (1973) and Mojica-Prieto and Millero (2002) from S =15 to 45 and 0 to 40 8C. The values of pK1* near S =35 are also in

reasonable agreement with the measurements in artificial seawater (ASW) of Goyet and Poisson (1989) and Roy et al. (1993) from

0 to 35 8C. The values of pK2* in real seawater, however, do not agree with the measurement made in artificial seawater at

temperatures above 5 8C. Calculations of pK1* and pK2* near 25 8C using an ionic interaction model (Millero and Roy, 1997)

suggest that the pK2* results in SW are more reliable than in ASW.

The equations from this study should be valid from S =0 to 50 and t =0 to 50 8C for most estuarine and marine waters (check

values at S =35 and t =25 8C are pK1*=5.8401 and pK2*=8.9636).

D 2006 Elsevier B.V. All rights reserved.

Keywords: Seawater; Carbonic acid; Dissociation constants; pK1; pK2; Titration; Modeling

1. Introduction

To examine the thermodynamics of the carbonic

acid system in seawater from measurements of pH,

total alkalinity (TA), total carbon dioxide (TCO2) and

the partial pressure of carbon dioxide ( pCO2) one

needs reliable constants for the dissociation of carbonic

acid.

CO2 þ H2O X Hþ þ HCO�3 ð1Þ

HCO�3 X Hþ þ CO2�3 : ð2Þ

The stoichiometric constants are given by

K1* ¼ Hþ½ � HCO�3

� �= CO2½ � ð3Þ

K2* ¼ Hþ½ � CO2�

3

� �= HCO�3� �

: ð4Þ

The concentrations (mol kg SW�1) are the total stoi-

chiometric values, [H+]= [H+]F+[HSO4�]+ [HF] (where

the subscript F is used to designate the free proton

concentration). Theoretically the measured stoichiomet-

Table 1

A summary of the measurement made on the dissociation constants of carbon

Reference Media Salinity range

Hansson ASW 5–40

Mehrbach et al. SW 19–43

Goyet and Poisson ASW 10–50

Roy et al. ASW 5–45

Mojica-Prieto and Millero SW 12–45

This study SW 1–50

a The values in parentheses are the number of measurements made and th

functional form (Eqs. (11)–(14)).

ric constants (K1* and K2*) for carbonic acid in sea-

water are related to the thermodynamic constants (Ki0)

by

K1* ¼ K0

1aH2OcCO2=cHcHCO3

ð5Þ

K2* ¼ K0

2cHCO3=cHcCO3

ð6Þ

where ai is the activity and ci is the activity coeffi-

cient of species i.

A number of workers have made measurements of the

stoichiometric constants for the dissociation of carbonic

acid in real seawater (SW) (Mehrbach et al., 1973;

Mojica-Prieto and Millero, 2002) and artificial seawater

(ASW) (Hansson, 1973; Goyet and Poisson, 1989; Roy

et al., 1993). A summary of these studies is given in

Table 1 along with the standard errors of the individual

measurements being fit to the same equation (Eqs. (11)–

(14)). All of the individual studies have similar standard

deviations, r =0.004 to 0.007 for pK1* and r =0.002 to

0.010 for pK2*. The measurements by Mehrbach et al.

(1973) and Mojica-Prieto and Millero (2002) were

made on real seawater (SW) while the measurements

made by Hansson (1973), Goyet and Poisson (1989)

and Roy et al. (1993) were made on artificial seawater

ic acid in real (SW) and artificial (ASW) seawater by various workers

Temperature range (8C) r(pK1*)a r(pK2*)

a

5–35 0.0070 (62) 0.010 (62)

2–35 0.0043 (30) 0.010 (33)

�1–40 0.0057 (84) 0.010 (90)

0–45 0.0044 (80) 0.002 (80)

5–45 0.0040 (80) 0.008 (80)

0–50 0.0054 (466) 0.011 (458)

e standard errors are based upon fitting the measurements to the same

F.J. Millero et al. / Marine Chemistry 100 (2006) 80–9482

(ASW). The values of pK1* determined by Mehrbach et

al. (1973), Hansson (1973), Goyet and Poisson (1989)

and Mojica-Prieto and Millero (2002) were determined

from potentiometric titrations. The values of pK2* de-

termined by Hansson (1973) and Goyet and Poisson

(1989) were also determined by potentiometric meth-

ods. Roy et al. (1993) determined pK1* and pK2* using

the hydrogen electrode method. The pK2* measurements

by Mehrbach et al. (1973) and Mojica-Prieto and Mill-

ero (2002) determined pK1*+pK2* on seawater that had

been stripped of CO2 with HCl. The values of pH=0.5

(pK1*+pK2*) of these solutions were determined after

the addition of HCO3�. If the pH is determined using a

spectroscopic method (Clayton and Byrne, 1993), then

one should be able to determine the pK2* to F0.005

which is better than the precision from most of the

titration studies.

Comparisons of the measurements made in ASW

and SW by various workers for S =35 seawater as a

function of temperature are shown in Fig. 1. These

comparisons are made relative to the measurements of

Mehrbach et al. (1973) as reformulated by Dickson and

Millero (1987). This was done since laboratory (Lee et

al., 1996, 1997, 2000; Lueker et al., 2000; Mojica-

Prieto and Millero, 2002) and field (Wanninkhof et

al., 1999; Millero et al., 2002) studies on SW indicate

0 10 20 30 40

0 10 20 30 40

-0.03

-0.02

-0.01

0.00

0.01

0.02

0.03

HanssonGoyet & PoissonRoy et al.Mojica & Millero

Δ p

K1

Δ p

K2

Temperature (°C)

-0.06

-0.04

-0.02

0.00

0.02

Fig. 1. A comparison of the values of pK1*and pK2* from 0 to 40 8Cand S =35 determined by various workers with the results of Mehr-

bach et al. (1973) as refit by Dickson and Millero (1987).

that these constants are more reliable than those made

in ASW. The pK1* measurements in SW by Mehrbach

et al. (1973) and in ASW by Goyet and Poisson (1989)

and Roy et al. (1993) are in reasonable agreement

(within 2r =0.014). The pK1* results of Hansson

(1973) from 5 to 25 8C in ASW are slightly higher

than the measurements of Goyet and Poisson (1989)

and Roy et al. (1993). The pK2* results in SW do not

agree with the results in ASW above 5 8C (see Fig. 1).

These results suggest that the values of pK2* in SW and

ASW are different. As discussed above internal consis-

tency studies with field and laboratory measurements of

seawater have shown that the pK2* measurements of

Mehrbach et al. (1973) in SW are more reliable than

the measurements made in ASW. The pK2* SW mea-

surements of Mojica-Prieto and Millero (2002) are in

good agreement with the results of Mehrbach et al.

(1973).

2. Modeling the carbonate system in natural waters

The differences in the values of pKi* near 25 8Cgoing from ASW to SW can be attributed to changes

in the activity of the H2O and CO2 or the activity

coefficients of HCO3� and CO3

2�. For example, the

increase in K1* and decrease in K2* near 25 8C can be

attributed to a decrease in cHCO3and an increase in cCO3

when going from ASW to SW. If these effects are due

to the boric acid in real seawater (Mojica-Prieto and

Millero, 2002), they can be attributed to interactions of

HCO3� with B(OH)3 or B(OH)4

� and interactions of

CO32� with B(OH)3 or B(OH)4

�. It is also possible

that the differences in the pK2* in SW and ASW are

related to an organic acid that is in all seawater (Millero

et al., 2002). Further measurements are needed to de-

termine the cause of these differences.

The measured stoichiometric constants (K1* and

K2*) for carbonic acid in seawater are related to

the thermodynamic constants by Eqs. (5) and (6).

The activity coefficients of H+, HCO3�, CO3

2� and

activities of CO2 and H2O can be determined from

ionic interaction models (Pitzer, 1991). Millero and

Roy (1997) have developed a carbonate model valid

from 0 to 50 8C and I =0 to 6 m. The model considers

the ionic interactions in solutions of the major

components of seawater and other natural waters

(H–Na–K–Mg–Ca–Sr–F–Cl–Br–OH–HCO3–B(OH)4–

HSO4–SO4–CO3–CO2–HF–B(OH)3–H2O). The model

has been used to predict the activity coefficients of major

and minor components of ions required to determine the

dissociation constants of all the acids needed to examine

the carbonate system in natural waters (H2CO3, B(OH)3,

Table 2

The composition of artificial seawater used by various authors

Hansson Roy et al. Goyet and Poisson Seawatera

NaCl 0.422 0.412598 0.409411 0.41040

Na2SO4 0.028 0.02824 0.028217 0.02824

KCl 0.010208 0.00908 0.00937

CaCl2 0.010 0.010372 0.01033 0.01028

MgCl2 0.054 0.052815 0.05327 0.05282

NaF 0.000071 0.00007

KBr 0.000823 0.00084

SrCl2 0.00009 0.00009

Na2CO3 0.001

NaHCO3 0.00205

B(OH)3 0.00042

a In mol (kg soln)�1, Millero (1996).

F.J. Millero et al. / Marine Chemistry 100 (2006) 80–94 83

H2O, HF, HSO4�, H3PO4, H2S, NH4

+). The predicted

dissociation constants for a number of acids in seawater

have been shown to be in good agreement with experi-

mental measurements (Millero and Roy, 1997).

The model can be used to examine the effects of

composition on the carbonate constants in seawater

and to compare the measurements made by various

studies. The compositions of artificial seawater used in

Model Comparisons

0 10 20 30 40 50

Δ p

K1 (M

eas

- Cal

c)Δ

pK

2 (M

eas

- Cal

c)

-0.06

-0.05

-0.04

-0.03

-0.02

-0.01

0.00

0.01

0.02

Mehrbach et al.Goyet & PoissonRoy et al.Mojica and Millero

0 10 20 30 40 50-0.08

-0.06

-0.04

-0.02

0.00

0.02

0.04

Temperature (°C)

Fig. 2. A comparison of the values of pK1* and pK2* from 0 to 40 8Cand S =35 determined by various workers with the model results of

Millero and Roy (1997).

the various studies are given in Table 2. A calculation of

the values of pK1* and pK2* using the compositions

shown in Table 2 indicate that the differences in pK1*

and pK2* are all within F0.002 which is well within the

experimental error of the measurements (see Table 1).

Changes in the values of SO42� and F� show the largest

effects on the values of pKi* (an increase of 0.00001 in F�

increases the pK’s by 0.005 and an increase of 0.0001 in

SO42� increases the pK’s by 0.005). These effects cannot

account for the increase in pK1* and decrease in pK2*

when going from SW to ASW near 25 8C.A comparison of the model calculations of pK1* and

pK2* at S =35 and from 0 to 45 8C with the fitted

measurements of Mehrbach et al. (1973), Goyet and

Poisson (1989), Roy et al. (1993) and Mojica-Prieto

and Millero (2002) are shown in Fig. 2. The model

calculations of pK1* are in reasonable agreement with

all of the measurements from 10 to 45 8C. The model

calculations of pK2* are in agreement with the measure-

ments in seawater by Mehrbach et al. (1973) and

Mojica-Prieto and Millero (2002). Large offsets occur

in the values of pK2* made in ASW by Goyet and

Poisson (1989), and Roy et al. (1993). The model

0 1 2 3 4 5 6 7

0 1 2 3 4 5 6 7

5.7

5.8

5.9

6.0

6.1

6.2

6.3

6.4

ModelMehrbach et al.

Carbonic Acid

S0.5

8.8

9.0

9.2

9.4

9.6

9.8

10.0

10.2

10.4

ModelMehrbach et l.

pK

1 p

K2

Fig. 3. A comparison of the values of pK1* and pK2* at 25 8C as a

function of the square root of salinity (S =0 to 45) of Mehrbach et al

(1973) with the model results from Millero and Roy (1997).

.

F.J. Millero et al. / Marine Chemistry 100 (2006) 80–9484

appears to be in error below 10 8C probably due to the

scarcity of measurements of pK1* and pK2* in NaCl

solutions at low temperatures. In summary, the model

supports the measurements in real seawater by Mehr-

bach et al. (1973) and Mojica-Prieto and Millero (2002).

The measurements made by Mehrbach et al. (1973)

and Mojica-Prieto and Millero (2002) on SW were not

made in dilute solutions so they may not give reliable

constants for estuarine waters. This is shown in Fig. 3

by comparing the measurements of pK1* and pK2* of

Mehrbach et al. (1973) with the model at 25 8C. In this

paper, we present measurements of real seawater over a

wide temperature (1 to 50 8C) and salinity (1 to 50)

range. The seawater results of this study have been

fitted to equations that are valid for all marine waters

over a wide range of salinity and temperature.

3. Methods

The measurements were made on Gulf Stream sea-

water that was diluted or evaporated from a salinity near

36. The seawater was filtered through a 0.45 Am filter

and stored at room temperature in 50 L P.P. Nalgene

Bottles before use. The low salinity samples were made

Table 3

The effects of TA (Amol kg�1) and TCO2 (Amol kg�1) levels and pH

(A) Effect of fixing or floating TA and TCO2 on the calculated pK1* and pK

Fixed values of TA and TCO2

Temperature (8C) Salinity TA TCO2 pK1*

20 33.758 2233.7 2003.8 5.900

25 33.758 2233.7 2003.8 5.845

30 33.076 2202.0 1998.2 5.805

35 33.076 2202.0 1998.2 5.763

40 33.076 2202.0 1998.2 5.726

Floating values of TA and TCO2

20 33.758 2233.9 2004.3 5.900

25 33.758 2233.7 2011.0 5.849

30 33.076 2202.0 2004.4 5.809

35 33.076 2202.3 2001.8 5.766

40 33.076 2202.1 2000.5 5.728

(B) Effect of initial pH on the values of pK1* and pK2*

Temperature Salinity Without NaOH With Na

pK1* pK2* pK1*

20 36.054 5.879 8.989 5.880

25 36.054 5.829 8.929 5.832

25 33.758b 5.849 8.974 5.851

30 35.885 5.794 8.867 5.796

35 35.885 5.752 8.782 5.756

40 35.885 5.715 8.696 5.723

a Certified reference material.b From Mehrbach data.

by diluting SWwith pure Milli-Q water (18 mV) and the

high salinity samples were obtained by slowly evaporat-

ing the SW. All the salinities below 42 were directly

determined on a Guildline 8410 PortaSal Salinometer.

The samples at higher salinities were determined from

density measurements on a DMA 60 Mettler/Paar Den-

sity Meter. The salinities were determined from the

density measurements using the 1 atm equation of state

of Millero and Poisson (1981). For samples of salinity

less than 8, sodium carbonate (0.002 m) was added to aid

in the determination of pK2*. Since the pH of seawater

(~8) is not high enough to determine an accurate value of

pK2*, small amounts of sodium hydroxide were added to

increase the pH to ~9–10. These small additions of

NaOH did not significantly change the salinity of the

samples. The addition of NaOH was not necessary in

dilute solutions when sodium carbonate was added.

All the samples were equilibrated to the desired tem-

perature in a Neslab RTE-221 constant temperature

water bath to F0.05 8C before addition to the titration

vessel. Flowing water at the desired temperature was

circulated through the titration cell and around the piston

delivering the HCl during an experiment. The tempera-

tures in the constant temperature bath and in the cell were

on the determination of the pK1* and pK2* at various temperatures

2*

Floating values of TA and TCO2

pK2* DpK1* DpK2* DTA DTCO2

9.051 0.000 �0.003 �0.2 �0.58.943 �0.004 �0.027 0.0 7.2

8.868 �0.004 �0.029 0.0 �6.28.792 �0.003 �0.016 �0.3 �3.68.702 �0.002 �0.010 �0.1 �2.3

9.054

8.974

8.897

8.808

8.712

OH Without With

pK2* DpK1*a DpK2*

a DpK1*a DpK2*

a

9.023 �0.001 0.036 �0.002 0.002

8.945 0.004 0.015 �0.002 �0.0008.967 0.007 �0.006 0.009 0.001

8.868 0.000 0.002 �0.002 0.001

8.794 0.005 0.013 0.002 0.001

8.725 0.010 0.026 0.002 �0.003

Table 4

Measured values of pK1* and pK2* for carbonic acid in seawater as a

function of salinity and temperature

Temperature S pK1* pK2*

1.1 3.467 6.362 9.945

1.2 3.467 6.349 9.935

1.0 4.353 6.323 9.923

1.2 4.702 6.336 9.879

1.2 4.929 6.332 9.856

1.0 5.561 6.307 9.859

1.0 6.278 6.295 9.827

1.2 8.154 6.283 9.721

1.2 8.154 6.276 9.721

1.2 9.965 6.256 9.679

1.1 10.349 6.247 9.684

1.3 10.349 6.236 9.685

1.0 12.328 6.219 9.624

1.0 12.328 6.218 9.655

1.2 12.698 6.229 9.642

1.2 12.698 6.227 9.636

1.0 14.626 6.200 9.602

1.0 14.626 6.202 9.604

1.2 17.100 6.193 9.539

1.2 17.100 6.193 9.571

1.0 20.463 6.168 9.514

1.1 20.463 6.170 9.486

1.2 26.089 6.140 9.432

1.3 26.089 6.137 9.458

1.0 27.724 6.117 9.425

1.0 27.724 6.118 9.435

1.0 27.724 6.117 9.425

1.1 28.220 6.127 9.433

1.1 28.220 6.125 9.421

1.1 29.792 6.128 9.410

1.2 29.792 6.132 9.409

1.2 31.459 6.113 9.395

1.0 31.707 6.106 9.384

1.0 31.707 6.107 9.386

1.0 31.707 6.101 9.388

1.0 33.227 6.100 9.361

1.0 33.227 6.100 9.382

1.1 33.251 6.110 9.362

1.2 33.251 6.109 9.364

1.0 35.771 6.104 9.340

1.2 35.771 6.105 9.335

1.0 35.775 6.092 9.356

1.0 35.775 6.087 9.361

1.0 35.775 6.090 9.357

1.2 37.180 6.101 9.329

1.2 37.180 6.095 9.340

1.0 37.474 6.078 9.326

1.0 37.474 6.083 9.330

1.0 37.474 6.080 9.331

1.0 39.819 6.079 9.311

1.0 39.819 6.074 9.315

1.0 41.091 6.073 9.299

3.2 8.150 6.244 9.733

3.0 8.509 6.244 9.692

3.0 8.509 6.244 9.690

3.0 8.509 6.244 9.692(continued on next page)

Table 4 (continued)

Temperature S pK1* pK2*

3.3 9.355 6.237 9.659

3.0 13.045 6.201 9.606

3.3 17.088 6.176 9.535

2.9 17.73 6.161 9.531

3.0 17.814 6.164 9.530

3.0 20.48 6.143 9.497

3.0 22.729 6.132 9.466

3.0 22.729 6.135 9.467

3.3 25.230 6.124 9.401

3.3 25.230 6.123 9.406

3.0 26.091 6.106 9.420

3.0 26.091 6.107 9.428

3.0 26.091 6.111 9.427

2.4 29.939 6.085 9.378

2.5 29.939 6.086 9.381

3.1 29.939 6.097 9.376

3.2 29.939 6.102 9.361

3.0 30.41 6.084 9.377

3.0 30.41 6.084 9.376

3.0 30.41 6.084 9.377

3.0 35.775 6.070 9.315

3.0 35.775 6.065 9.310

3.0 35.775 6.065 9.304

3.0 35.775 6.064 9.306

3.0 35.775 6.065 9.320

3.0 35.775 6.062 9.321

3.0 35.775 6.066 9.320

2.2 36.302 6.077 9.312

2.4 36.302 6.079 9.308

2.4 36.302 6.081 9.313

3.0 37.097 6.062 9.306

3.0 37.097 6.061 9.307

3.0 37.097 6.062 9.307

3.0 39.725 6.056 9.283

3.0 39.725 6.052 9.280

3.0 43.789 6.042 a

5.1 2.569 6.318 9.972

5.2 2.569 6.319 9.963

5.1 5.221 6.267 9.784

5.2 9.104 6.212 9.664

5.2 14.849 6.159 9.539

5.2 14.849 6.161 9.510

5.0 19.902 6.115 9.466

5.0 19.902 6.118 9.463

5.1 20.258 6.117 9.449

5.1 20.258 6.118 9.448

5.0 22.17 6.102 9.430

5.0 22.17 6.102 9.433

5.0 22.17 6.103 9.435

5.0 23.465 6.092 9.421

6.0 24.00 6.093 9.391

5.0 25.473 6.086 9.399

5.0 25.473 6.085 9.399

5.0 25.473 6.086 9.399

5.0 26.347 6.089 9.367

5.0 26.347 6.091 9.370

5.0 26.347 6.086 9.363

F.J. Millero et al. / Marine Chemistry 100 (2006) 80–94 85

Table 4 (continued)

Temperature S pK1* pK2*

5.1 26.510 6.082 9.367

5.1 26.510 6.083 9.370

5.0 27.79 6.073 9.368

5.0 27.79 6.073 9.374

5.0 27.79 6.075 9.372

5.0 30 6.074 9.345

5.0 30 6.068 9.342

5.0 31.4 6.056 9.332

5.0 31.4 6.057 9.332

5.0 31.4 6.058 9.330

5.0 32.346 6.072 9.313

5.1 32.376 6.060 9.299

5.1 32.376 6.058 9.315

5.0 35.775 6.043 9.284

5.0 35.775 6.042 9.280

5.0 35.775 6.043 9.284

5.0 35.775 6.041 9.280

5.0 35.775 6.046 9.266

5.0 35.775 6.049 9.268

5.0 36.094 6.056 9.284

5.0 36.324 6.056 9.279

5.1 37.653 6.044 9.261

5.2 37.653 6.041 9.259

5.0 39.69 6.034 9.245

5.0 40.27 6.028 9.245

5.0 40.27 6.027 9.242

5.0 40.27 6.031 9.241

5.0 42.69 6.023 9.223

5.0 42.69 6.024 9.219

5.0 42.69 6.022 9.220

5.0 43.538 6.020 a

5.2 46.850 6.013 9.177

5.0 50.754 5.982 a

10.0 5.493 6.206 9.706

10.1 5.493 6.203 9.713

10.1 10.934 6.129 9.538

10.2 10.934 6.131 9.527

10.1 12.43 6.112 9.497

9.9 16.072 6.081 9.437

10.2 16.072 6.088 9.428

10.2 20.777 6.052 9.358

10.2 20.777 6.053 9.356

10.4 23.702 6.046 9.325

10.2 24.00 6.043 9.320

10.1 26.388 6.024 9.295

10.1 26.388 6.021 9.293

10.0 30 6.013 9.268

10.0 32.093 5.999 9.229

10.2 32.093 6.001 9.230

9.9 35.740 5.986 9.190

10.2 35.740 5.989 9.189

10.0 36.2 5.990 9.188

10.0 36.3 5.990 9.188

10.2 38.113 5.977 9.164

10.3 38.113 5.978 9.171

15.0 1.924 6.236 9.899

14.9 2.772 6.209 9.790

able 4 (continued)

emperature S pK1* pK2*

5.1 4.933 6.158 9.642

5.1 4.933 6.162 9.657

5.0 6.447 6.133 9.586

5.0 6.579 6.136 9.576

5.0 6.579 6.138 9.574

5.0 6.579 6.138 9.572

5.0 9.164 6.089 9.450

5.0 9.164 6.086 9.456

5.0 10.191 6.084 9.442

3.7 12.40 6.085 9.429

5.0 15.274 6.035 9.359

5.1 15.274 6.036 9.362

5.0 17.358 6.032 9.331

5.0 17.358 6.029 9.330

5.1 19.551 6.006 9.300

5.1 19.551 6.005 9.298

5.0 21.631 5.993 9.259

5.1 21.631 5.992 9.262

5.2 25.670 5.972 9.216

5.2 25.670 5.970 9.220

5.0 30 5.956 9.188

5.0 31.15 5.960 9.158

5.0 35.00 5.947 9.117

5.0 36.2 5.932 9.107

5.0 36.3 5.932 9.107

5.0 37.25 5.942 9.096

5.0 40.52 5.932 9.060

5.0 40.74 5.930 9.061

0.0 1.663 6.204 9.858

0.0 1.954 6.199 9.794

0.2 3.342 6.153 9.668

0.0 4.48 6.134 9.586

0.0 4.48 6.134 9.587

0.0 4.48 6.133 9.585

0.0 7.122 6.082 9.457

0.0 9.198 6.040 9.392

0.0 9.198 6.043 9.395

0.0 9.198 6.047 9.393

0.5 10.461 6.039 9.355

0.0 13.748 5.996 9.297

0.0 13.748 5.998 9.302

0.0 13.748 5.999 9.295

0.0 16.354 5.983 9.265

0.0 16.354 5.983 9.265

0.0 18.53 5.965 9.228

0.2 20.326 5.959 9.210

0.2 20.326 5.962 9.217

0.2 20.326 5.957 9.208

0.2 20.326 5.958 9.208

0.0 22.154 5.944 9.181

0.0 22.154 5.944 9.183

0.0 25.89 5.924 9.133

0.0 25.89 5.922 9.137

0.0 25.89 5.923 9.137

0.2 31.444 5.903 9.072

0.2 31.444 5.905 9.072

0.0 40.134 5.880 8.988

(continued on next page)

F.J. Millero et al. / Marine Chemistry 100 (2006) 80–9486

T

T

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

Table 4 (continued)

Temperature S pK1* pK2*

20.2 43.061 5.856 8.961

20.2 43.061 5.857 8.958

20.0 47.219 5.858 a

25.2 2.217 6.148 9.695

25.3 2.868 6.134 9.671

25.0 4.213 6.086 9.525

25.0 4.213 6.095 9.560

25.0 5.007 6.064 9.493

25.1 5.940 6.050 9.441

25.1 6.217 6.059 9.459

25.1 12.015 5.980 9.250

25.1 12.015 5.980 9.251

25.2 13.847 5.951 9.222

25.2 13.847 5.951 9.215

25.0 14.742 5.950 9.213

25.0 14.742 5.950 9.207

25.1 16.488 5.934 9.184

25.1 16.488 5.934 9.183

25.0 16.996 5.930 9.171

25.0 16.996 5.931 9.175

25.0 16.996 5.939 9.164

25.0 16.996 5.935 9.151

25.0 18.545 5.917 9.154

25.2 18.545 5.918 9.152

25.1 19.079 5.917 9.145

25.1 19.079 5.916 9.146

25.0 21.147 5.905 9.121

25.0 21.147 5.907 9.112

25.0 21.147 5.907 9.118

25.0 21.147 5.906 9.112

25.1 24.097 5.888 9.077

25.2 24.097 5.887 9.080

25.0 27.713 5.869 9.034

25.0 27.713 5.869 9.033

25.0 27.713 5.869 9.034

25.0 27.713 5.869 9.035

25.0 30 5.858 9.007

25.0 30 5.857 9.006

25.0 30 5.858 9.007

25.0 30 5.857 9.006

25.0 30.00 5.857 9.004

25.0 31.847 5.856 8.983

25.0 31.847 5.853 8.988

25.0 31.847 5.852 8.988

25.0 31.847 5.854 8.983

25.0 33.998 5.842 8.965

25.0 33.998 5.842 8.963

25.0 33.998 5.842 8.965

25.0 33.998 5.842 8.963

25.0 36.055 5.840 8.939

25.0 36.055 5.836 8.941

25.0 36.055 5.836 8.944

25.0 36.055 5.837 8.939

25.0 36.055 5.836 8.942

25.0 36.055 5.833 8.943

25.0 36.25 5.843 8.949

25.0 37.51 5.831 8.926

25.0 38.272 5.837 8.936

Table 4 (continued)

Temperature S pK1* pK2*

25.0 38.272 5.831 8.922

25.0 38.272 5.829 8.918

25.0 39.011 5.828 8.912

25.0 39.011 5.823 8.915

25.0 39.011 5.825 8.913

25.0 39.011 5.827 8.919

25.0 40.39 5.820 8.904

25.0 40.39 5.821 8.904

25.0 40.39 5.819 8.901

25.0 40.39 5.820 8.902

25.0 42.67 5.813 8.883

25.0 42.67 5.811 8.884

25.0 42.67 5.811 8.885

25.0 44.008 5.811 8.872

25.0 44.008 5.811 8.869

25.0 44.008 5.811 8.870

25.0 44.236 5.808 8.870

25.0 44.236 5.807 8.872

25.0 44.236 5.807 8.870

25.0 45.002 5.808 8.863

25.0 45.002 5.805 8.866

25.0 45.002 5.799 8.866

25.0 45.985 5.794 8.856

25.0 45.985 5.796 8.857

25.0 45.985 5.796 8.867

30.0 0.953 6.187 9.855

30.0 2.435 6.116 9.624

30.0 2.435 6.120 9.629

30.0 2.435 6.120 9.622

30.0 3.804 6.073 9.519

30.0 3.804 6.073 9.522

30.0 5.815 6.025 9.382

30.0 8.305 5.986 9.269

30.0 8.305 5.985 9.274

30.0 8.305 5.986 9.275

30.0 10.29 5.953 9.219

30.0 10.29 5.959 9.212

30.0 10.29 5.957 9.219

30.0 13.277 5.925 9.154

30.0 13.277 5.925 9.154

30.0 13.277 5.925 9.154

30.0 15.5 5.908 9.121

30.0 15.5 5.905 9.120

30.0 15.5 5.908 9.121

30.1 17.231 5.888 9.092

30.2 17.231 5.890 9.096

30.0 17.676 5.891 9.082

30.0 17.676 5.888 9.085

30.0 17.676 5.894 9.086

30.0 18.330 5.880 9.080

30.0 18.330 5.881 9.080

30.1 19.216 5.877 9.068

30.1 19.216 5.876 9.065

30.0 20.853 5.867 9.040

30.0 20.853 5.866 9.038

30.0 21.429 5.863 9.037

30.0 21.429 5.866 9.037

(continued on next page)

F.J. Millero et al. / Marine Chemistry 100 (2006) 80–94 87

Table 4 (continued)

Temperature S pK1* pK2*

30.0 24.674 5.846 8.996

30.0 24.674 5.844 8.995

30.0 25.236 5.839 8.986

30.0 25.236 5.839 8.988

30.0 27.281 5.826 8.964

30.0 27.281 5.829 8.962

30.0 30.045 5.814 8.930

30.0 30.045 5.817 8.932

30.0 30.919 5.814 8.917

30.1 30.919 5.813 8.915

30.0 33.133 5.803 8.897

30.0 33.133 5.801 8.897

30.0 33.133 5.801 8.897

30.0 38.789 5.793 8.842

30.0 38.789 5.789 8.842

30.0 38.789 5.791 8.843

30.0 45.344 5.765 8.797

30.0 45.344 5.763 8.786

30.0 45.344 5.764 8.796

35.0 0.823 6.162 9.825

35.0 2.905 6.061 9.529

35.0 4.395 6.042 9.423

35.0 5.476 6.005 9.333

35.0 8.413 5.942 9.224

35.0 11.931 5.904 9.100

35.0 14.502 5.875 9.056

35.0 14.502 5.876 9.047

35.0 14.502 5.878 9.051

35.0 17.876 5.850 9.000

35.0 17.876 5.851 9.007

35.0 21.521 5.821 8.941

35.0 21.521 5.822 8.942

35.0 24.579 5.805 8.922

35.0 24.579 5.805 8.925

35.0 28.39 5.785 8.880

35.0 30.116 5.777 8.856

35.0 30.116 5.776 8.856

35.0 33.062 5.762 8.828

35.0 33.062 5.764 8.825

35.0 34.881 5.754 8.808

35.0 34.881 5.754 8.808

34.9 39.562 5.739 8.761

35.0 39.562 5.740 8.763

35.0 42.509 5.733 8.734

35.0 42.509 5.728 8.735

35.0 48.513 5.710 a

35.0 48.513 5.708 a

40.0 0.967 6.140 9.738

40.0 1.761 6.104 9.612

40.0 2.91 6.057 9.488

40.0 3.932 6.030 9.403

40.0 9.584 5.909 9.099

40.0 9.584 5.906 9.103

40.0 12.625 5.870 9.011

40.0 13.779 5.854 8.989

40.0 13.779 5.856 8.999

40.0 16.462 5.831 8.942

able 4 (continued)

emperature S pK1* pK2*

0.0 16.462 5.835 8.946

0.0 17.864 5.818 8.939

0.0 17.864 5.820 8.938

0.0 20.501 5.806 8.902

0.0 20.501 5.800 8.895

0.0 20.501 5.799 8.888

0.0 22.149 5.791 8.883

0.0 22.149 5.787 8.867

0.0 24.818 5.768 8.833

0.0 24.818 5.773 8.847

0.0 24.818 5.772 8.847

0.0 26.997 5.761 8.822

0.0 26.997 5.760 8.823

9.9 29.736 5.749 8.790

0.0 29.736 5.747 8.790

0.0 31.008 5.739 8.781

0.0 31.008 5.741 8.774

0.0 31.008 5.740 8.774

0.0 32.604 5.733 8.757

0.0 32.604 5.732 8.756

0.0 38.480 5.709 8.700

0.0 38.480 5.708 8.699

0.0 44.793 5.687 8.646

0.0 44.793 5.681 8.643

5.0 14.513 5.817 8.902

5.1 14.513 5.817 8.918

5.0 17.814 5.792 8.872

5.0 17.814 5.793 8.870

5.0 19.519 5.778 8.848

5.0 19.519 5.777 8.847

5.0 21.709 5.766 8.817

5.0 21.709 5.763 8.820

4.9 22.935 5.751 8.806

5.1 22.935 5.751 8.807

5.0 25.66 5.736 8.760

5.0 25.66 5.738 8.761

5.0 25.66 5.734 8.761

4.9 27.133 5.727 8.749

5.0 27.133 5.725 8.755

5.1 30.140 5.713 8.716

5.1 30.140 5.711 8.719

5.0 32.957 5.698 8.684

5.1 32.957 5.694 8.685

5.3 35.00 5.697 8.644

5.0 36.200 5.683 8.649

5.0 36.300 5.683 8.649

4.9 37.51 5.684 8.644

5.0 43.289 5.659 8.586

0.0 9.534 5.869 8.998

0.0 9.534 5.866 9.001

0.4 12.40 5.831 8.894

9.7 14.975 5.792 8.830

0.2 14.975 5.802 8.849

0.2 17.73 5.774 8.780

0.2 19.717 5.766 8.777

0.0 21.163 5.740 8.759

(continued on next page)

F.J. Millero et al. / Marine Chemistry 100 (2006) 80–9488

T

T

4

4

4

4

4

4

4

4

4

4

4

4

4

3

4

4

4

4

4

4

4

4

4

4

4

4

4

4

4

4

4

4

4

4

4

4

4

4

4

4

4

4

4

4

4

4

4

4

5

5

5

4

5

5

5

5

Table 4 (continued)

Temperature S pK1* pK2*

50.1 21.163 5.742 8.762

50.0 21.250 5.745 8.755

50.3 24.00 5.724 8.686

49.7 28.024 5.694 8.671

49.8 28.024 5.690 8.674

50.1 32.660 5.675 8.618

50.5 32.660 5.673 8.618

50.0 36.3 5.650 8.561

50.0 36.3 5.652 8.561

50.2 36.34 5.653 8.559

50.0 46.291 5.613 a

a: The missing values of pK2* are not given since they are unreliable

due to the precipitation of MgOH at high pH.

F.J. Millero et al. / Marine Chemistry 100 (2006) 80–94 89

measured with a Guildline 9540 Digital Resistance Ther-

mometer calibrated by the company. The temperature

inside the cell was measured before and after each titra-

tion. The values agreed to F0.1 8C which is equivalent

to an error ofF0.0009 in pK1* andF0.0016 in pK2*. The

recorded temperature of a run is the mean of the initial

and final values.

The titration system (Millero et al., 1993) consists of a

closed water jacketed plexiglass cell with a ROSS 8101

glass pH electrode and an Orion 90-02 double junction

Ag/AgCl reference electrode. Some measurements were

made using an Orion 80-05 ROSS Reference Half-Cell

electrode. The titrant is delivered with a Metrohm 665

Dosimat titrator and the emf is measured with an Orion

720A pH meter. The system is controlled by a personal

computer (Millero et al., 1993) using a National Instru-

ment’s Labwindows/CVI environment. The titration is

made by adding ~0.25 M HCl (in 0.45 M NaCl) to

seawater past the carbonic acid end point. A typical

titration records the emf after the readings become stable

(0.05 mV) and adds enough acid to change the voltage to

a pre-assigned increment (9 mV). This provides more

data points in the range of a rapid increase in the emf near

the endpoints.

The values of pK1* and pK2* [mol (kg soln)�1] were

determined using a non-linear curve-fitting procedure

developed by Johansson and Wedborg (1982) and Dick-

son (1981). This procedure was modified to a more

buser-friendlyQ Excel version by Dr. Pierrot. The pro-

gram determines the E*, pK1*, pK2*, TA and TCO2 of the

sample from the full titration (N50 pts). The electrodes

are calibrated over the entire range of the titration with

the computer code giving a value of E* that is constant at

a given temperature and salinity. The dissociation or

association constants needed in the computer code

were taken from the literature (B(OH)3 from Dickson,

1990a; HF from Dickson and Riley, 1979. HSO4� from

Dickson, 1990b, H2O from Millero, 1995).

The dissociation constants are on the seawater pH

scale (Dickson, 1984)

½Hþ�SWS ¼ ½Hþ� þ ½HSO�4 � þ ½HF� ð7Þ

where the brackets represent concentrations in mol

(kg soln)�1 (Millero, 1995). The total seawater scale

is given by

½Hþ�T ¼ ½Hþ� þ ½HSO�4 �: ð8Þ

The two pH scales are related by (Dickson, 1984)

pHSWS ¼ pHT þ log 1þ bHSO4SO4½ �T

� �

� log 1þ bHSO4SO4½ �T þ bHF F½ �T

� �ð9Þ

where the subscript T represents the total concentration

and the bs are the association constants for the forma-

tion of HSO4� (Dickson, 1990b) and HF (Dickson and

Riley, 1979) at the ionic strength and temperature of the

solution.

The calculations were carried out in a manner similar

to the methods described by Goyet and Poisson (1989).

As pointed out by these authors, care is needed in

fitting titration data with many variables. To test the

reliability of the computer code a number of titrations

as a function of temperature were made on certified

reference material provided by Dickson (2004) with

known values of TA and TCO2. The derived values

of pK1* and pK2* on these samples are given in Table 3.

The values of pK1* determined with and without the

known values of TA and TCO2 (floating) gave similar

results that were in good agreement with the values of

Mehrbach et al. (1973). The floating values of TA and

TCO2 are also in good agreement with the certified

values. In a second series of studies, we examined the

values of pK2* obtained with and without the addition of

NaOH. These results indicate that to obtain reliable

values of pK2* that agreed with the results of Mehrbach

et al. (1973) or Mojica-Prieto and Millero (2002), one

needs to increase the pH and the concentration of the

CO32� ion. In a few of the titrations at high salinity the

precipitation of Mg(OH)2 appeared to occur yielding

unreliable values of pK2*. This was apparently due to

the loss of the MgCO3 complex. Since the interactions

of Mg2+ with HCO3� are small, it did not cause any

significant errors in the determination of pK1* (Millero

and Roy, 1997).

4. Results and calculations

The titrations were made on seawater samples as a

function of temperature (1 to 50) and salinity (1 to 50).

Salinity

0 10 20 30 40 50

0 10 20 30 40 50

-0.020

-0.015

-0.010

-0.005

0.000

0.005

0.010

0.015

0.020

-0.020

-0.015

-0.010

-0.005

0.000

0.005

0.010

0.015

0.020

+2σ

-2σ

+2σ

-2σ

σ = 0.0054

σ = 0.0054

Δ p

K1

Δ p

K1

Temperature (°C)

ig. 5. The differences between the measured and fitted values of pK1*

s a function of salinity and temperature.

5.4

5.6

5.8

6.0

6.2

6.4

6.6

I0.5

0.0 0.2 0.4 0.6 0.8 1.0 1.28.0

8.5

9.0

9.5

10.0

10.5

Model10oC20oC30oC40oC50oC

pK

1 p

K2

Fig. 4. The measured values of pK1* and pK2* from 10 to 50 8C as

a function of the square root of ionic strength (I). The smooth

curves are the values calculated from the model of Millero and Roy

(1997).

F.J. Millero et al. / Marine Chemistry 100 (2006) 80–9490

The measured values of pK1* and pK2* determined from

the titrations are given in Table 4. These results are the

individual titrations that were used in the fitting of the

equations. The values of pK1* and pK2* as a function of

I0.5 from 10 to 50 8C are compared to the model

calculations (Millero and Roy, 1997) in Fig. 4. The

measurements are in good agreement with the model

over this temperature range and approach the pure

water values in dilute solution. As stated earlier this

is not the case at temperatures below 10 8C at high ionic

strengths (see Fig. 2).

Table 5

Coefficients for the fits of the values of pK1* and pK2* in seawater as a func

Salinity

pK1* coeff. pK2* coeff.

S0.5 a0 13.4191 21.0894

S a1 0.0331 0.1248

S2 a2 �5.33E�05 �0.000368S0.5/T a3 �530.1228 �772.483S/T a4 �6.103 �20.051S0.5 lnT a5 �2.06950 �3.32254Std. error 0.0054 0.011

Number 466 458

F

a

To examine internally the consistency of the mea-

surements made at each temperature the results were

first fitted to equations of the form

pKi*� pK0

i ¼ AS0:5 þ BS þ CS2 ð10Þ

pKi*� pK0

i ¼ AI0:5 þ BI þ CI2 ð11Þ

where the ionic strength I=19.92S / (1000�1.0049S),

and the values in pure pKi0 are determined from Harned

tion of temperature, salinity and ionic strength

Ionic strength

pK1* coeff. pK2* coeff.

I0.5 93.9053 147.2748

I 1.6549 6.0876

7 I2 �0.130 �0.869I0.5/T �3706.9 �5400.9I/T �303.7 �968.4I0.5 lnT �14.4858 �23.2804

0.0053 0.0114

466 458

-0.03

-0.02

-0.01

0.00

0.01

0.02

0.03

0.04

HanssonMehrbach et al.Goyet & PoissonRoy et al.Mojica & Millero

Δ p

K1 (M

eas

- Cal

c)

-0.03

-0.02

-0.01

0.00

0.01

0.02

0.03

0.04

Δ p

K1 (M

eas

- Cal

c)

0 10 20 30 40 50

Temperature (°C)

Salinity0 10 20 30 40 50

Fig. 7. A comparison of our values of pK1* as a function of temper

ature and salinity with literature values.

Salinity

-0.04

-0.03

-0.02

-0.01

0.00

0.01

0.02

0.03

-0.04

-0.03

-0.02

-0.01

0.00

0.01

0.02

0.03

σ = 0.011

σ = 0.011

+2σ

-2σ

+2σ

-2σ

Δ p

K2

Δ p

K2

0 10 20 30 40 50

Temperature (°C)

0 10 20 30 40 50

Fig. 6. The differences between the measured and fitted values of pK2*

as a function of salinity and temperature.

F.J. Millero et al. / Marine Chemistry 100 (2006) 80–94 91

and Bonner (1945), Harned and Davis (1943), and

Harned and Scholes (1941) as refit by Millero (1979)

pK01 ¼ � 126:34048þ 6320:813=T þ 19:568224lnT

ð12Þ

pK02 ¼ � 90:18333þ 5143:692=T þ 14:613358lnT

ð13ÞThe average standard deviations for the individual tem-

peratures varied from 0.0029 to 0.0078 for pK1*

(weighted average 0.0048, N =466) and 0.0049 to

0.0130 for pK2* (weighted average of 0.010, N =458).

All of the measurements as a function of temper-

ature and salinity have been fitted to equations of the

form

pKi*� pK0

i ¼ Ai þ Bi=T þ CilnT ð14Þ

The adjustable parameters have been fitted to func-

tions of salinity using equations

Ai ¼ a0S0:5 þ a1S þ a2S

2 ð15Þ

Bi ¼ a3S0:5 þ a4S ð16Þ

Ci ¼ a5S0:5 þ a6S: ð17Þ

Similar equations as a function of ionic strength can

be formulated by replacing S with I.

The coefficients used were arrived at by using an F-

test and are shown in Table 5 along with the standard

errors of the fits (r =0.0054 for pK1* and r =0.011 for

pK2*). The differences between the measured and cal-

culated values of pK1* and pK2* are shown in Figs. 5

and 6). Most of the deviations for pK1* and pK2* are

within 2r (0.012 and 0.022, respectively).

5. Discussion

Comparisons of the results of pK1* and pK2* calcu-

lated from Eqs. (14)–(17) with earlier workers shown

in Figs. 7 and 8) are summarized in Table 6. Our

calculated results of pK1* from 0 to 40 8C and S =12

to 45 are in good agreement with the measurements of

Mehrbach et al. (r =F0.0066, N =30) and Mojica-

Prieto and Millero (r =F0.0086, N =59). The pK1*

results of Roy et al. (r =F0.098, N =56) and Goyet

and Poisson (r =F0.0120, N =93) show larger offsets.

It should be pointed out that the Roy et al. results from

-

able 6

omparisons of the standard deviations of the differences between our

easurements and other authors

uthor r(pK1*) No. r(pK2*) No.

ehrbach et al. 0.0066 30 0.013 33

ansson 0.021 70

oyet and Poisson 0.012 93

0.0084a 84

oy et al. 0.0098b 56

ojica and Millero 0.0086c 59 0.014 140

his study 0.0054 466 0.011 458

a Minus the measurements at 10 8C.b Minus the measurements at S =5.c Minus the measurements at S =5.

-0.03

-0.02

-0.01

0.00

0.01

0.02

0.03

-0.03

-0.02

-0.01

0.00

0.01

0.02

0.03

Mehrbach et al.Mojica & Millero

10 20 30 40 50

Δ p

K2 (M

eas

- Cal

c)Δ

pK

2 (M

eas

- Cal

c)

0 10 20 30 40 50

Temperature (°C)

Salinity

Fig. 8. A comparison of our values of pK2* as a function of temper-

ature and salinity with literature values.

F.J. Millero et al. / Marine Chemistry 100 (2006) 80–9492

S =5 are not included in this comparison because they

appear to be in error. The 10 8C results of Goyet and

Poisson also appear to be too low compared to our

work. If this data is eliminated their results are in

good agreement with our results (r =F0.008, N =84).

The pK1* measurements of Hansson show much larger

differences than the other studies (r =F0.021, N =70)

with our work. Our new equations are in agreement

with most of the earlier measurement of pK1* on SW

and ASW within their experimental errors of the pre-

vious studies.

Our calculated results for pK2* are compared to the

measurements of Mehrbach et al. (1973) and Mojica-

Prieto and Millero (2002) in Fig. 8 from S=12 to 45

and 0 to 45 8C. Our calculated results (Table 6) are in

good agreement with the measurements of Mehrbach et

al. (r =F0.013, N =33) and Mojica-Prieto and Millero

(r =F0.014, N =140). Comparisons with the pK2* stud-

ies by other workers using ASW are not shown since

there are large offsets above 10 8C (see Fig. 1). Our

equations fits all of seawater experimental measure-

ments of pK2* within the experimental measurements

of the earlier studies.

Our equations for the dissociation constants of car-

bonic acid are valid over a wide range of salinity and

temperature. The pK1* equations are in agreement with

most of the earlier measurements in SW and ASW

within the standard error of their measurements. The

pK2* equations are in agreement with earlier measure-

ments in SW.

Since the dissociation constants are frequently used

to calculate the parameters (pH, TA, TCO2 and fCO2)

controlling the CO2 system in natural waters, it is

important to examine the errors involved in these cal-

culations using various inputs. This was done by ex-

amining the calculations of two unknown parameters

with an input of pH–TA, pH–TCO2, fCO2–TA, fCO2–

TCO2 and TA–TCO2. The calculations were made at

S =35 and t =25 8C, TA=2400 Amol kg�1 and two

levels of pH (8.094 and 7.576), fCO2 (350 and 1400

Aatm) and TCO2 (2052.3 and 2308.9 Amol kg�1). The

results are tabulated in Table 7 for errors of 0.006 in

pK1* and 0.011 in pK2*. These uncertainties in the

constants do not cause significant errors in the calcu-

lated values of TA and TCO2. The errors due to uncer-

tainties in pK1* range from 0.0002 to 0.006 for pH and

4.7 to 19.5 Aatm in fCO2. The errors due to uncertain-

ties in pK2* range from 0.0004 to 0.0077 in pH and

�17.9 to 2.7 Aatm in fCO2. The uncertainties in pH and

fCO2 are higher at higher levels of TCO2.

On can estimate the probable errors due to uncer-

tainties in pK1* and pK2* from the square root of the sum

of the individual errors squared. It should be pointed

out the errors will be larger when one accounts for the

errors in the experimental parameters (F3 Amol kg�1

in TA, F2 Amol kg�1 in TCO2, F0.002 in pH and F2

Aatm in fCO2). It is clear from these calculations that

one should make direct measurements of fCO2 rather

than calculating it from the other parameters if one

requires high precision or accurate values of fCO2 for

surface waters.

T

C

m

A

M

H

G

R

M

T

Table 7

Uncertainties in the determination of CO2 parameters at S =35 and t =25 8C due to errors of 0.006 in pK1* and 0.011 in pK2*a,b

Uncertainties due to error in pK1*

Input variables fCO2 (Aatm) DTCO2 (Amol kg�1) TA (Amol kg�1) pH fCO2 (Aatm)

pH–TA 350 0.1 – – 4.9

1400 0.6 – – 19.5

pH–TCO2 350 – �0.2 – 4.8

1400 – �0.6 – 19.1

fCO2–TA 350 �3.0 – 0.005 –

1400 �1.5 – 0.005 –

fCO2–TCO2 350 – 3.7 0.005 –

1400 – 1.6 0.006 –

TA–TCO2 350 – – 0.000 4.7

1400 – – 0.001 14.2

Uncertainties due to errors in pK2*

Input variables fCO2 (Aatm) DTCO2 (Amol kg�1) DTA (Amol kg�1) DpH DfCO2 (Aatm)

pH–TA 350 4.9 – – 1.9

1400 2.2 – – 2.7

pH–TCO2 350 – �5.5 – 1.1

1400 – �2.2 – 1.4

fCO2–TA 350 3.7 – 0.002 –

1400 1.9 – 0.001 –

fCO2–TCO2 350 – �4.6 0.001 –

1400 – �2.1 0.000 –

TA–TCO2 350 – – 0.008 �5.81400 – – 0.006 �17.9

a Initial inputs of TA=2400 Amol kg�1, pH=8.094 and 7.576, TCO2=2052.3 and 2308.9 Amol kg�1, respectively for fCO2=350 and 1400 Aatmat 25 8C. The calculations were made using pK1*=5.8372 and pK2*=8.9553 at 25 8C and S =35 from Mehrbach et al. (1973).b The total probable error (pe) can be estimated from the square root of the sum of the errors due to pK1* and pK2* squared,

pe=[(DpK1*)2 + (DpK2*)

2 ]0.5.

F.J. Millero et al. / Marine Chemistry 100 (2006) 80–94 93

The equations can be used to examine the thermody-

namics of the carbonate system in most estuarine and

marine waters. It should be pointed out that our equa-

tions assume that seawater is diluted with pure water.

This may not be the case for some estuarine systems. If

the composition is known one can use Pitzer models

(Millero and Roy, 1997; Millero and Pierrot, 1998) to

account for the difference in the composition of estua-

rine waters that differ from seawater diluted with pure

water.

Acknowledgements

This work was supported by the Oceanographic

Section of National Science Foundation and the Na-

tional Oceanic and Atmospheric Administration.

References

Clayton, T., Byrne, R.H., 1993. Spectrophotometric seawater pH

measurements: total hydrogen ion concentration scale calibra-

tion of m-cresol purple and at-sea results. Deep-Sea Res. 40,

2115–2129.

Dickson, A.G., 1981. An exact definition of total alkalinity and a

procedure for the estimation of alkalinity and total inorganic

carbon from titration data. Deep-Sea Res. 28A, 609–623.

Dickson, A.G., 1984. pH scales and proton-transfer reactions in

saline media such as seawater. Geochim. Cosmochim. Acta 48,

2299–2308.

Dickson, A.G., 1990a. Thermodynamics of the dissociation of boric

acid in synthetic seawater from 273.15 to 318.15K. Deep-Sea Res.

37, 755–766.

Dickson, A.G., 1990b. Standard potential of the reaction:

AgCl(s)+1.2H2(g)=Ag(s)+HCl (aq), and the standard acidity

constant of the ion HSO4� in synthetic sea water from 273.15 to

318.15. J. Chem. Thermodyn. 22, 113–127.

Dickson, A.G., 2004. Reference material batch information (http://

www-mpl.ucsd.edu/people/adickson/CO2_QC/Level1/Batches.

html).

Dickson, A.G., Millero, F.J., 1987. A comparison of the equilibrium

constants for the dissociation of carbonic acid in seawater media.

Deep-Sea Res. 34, 1733–1743.

Dickson, A.G., Riley, J.P., 1979. The estimation of acid dissociation

constants in seawater from potentiometric titrations with strong

base: I. The ion product of water—Kw. Mar. Chem. 7, 89–99.

Goyet, C., Poisson, A., 1989. New determination of carbonic acid

dissociation constants in seawater as a function of temperature and

salinity. Deep-Sea Res. 36, 1635–1654.

Hansson, I., 1973. A new set of acidity constants for carbonic acid

and boric acid in seawater. Deep-Sea Res. 20, 461–478.

F.J. Millero et al. / Marine Chemistry 100 (2006) 80–9494

Harned, H.S., Bonner, F.T., 1945. The first ionization constant of

carbonic acid in aqueous solutions of sodium chloride. J. Am.

Chem. Soc. 67, 1026–1031.

Harned, H.S., Davis Jr., R.D., 1943. The ionization constant of

carbonic acid in water and the solubility of carbon dioxide in

water and aqueous salt solutions from 0 to 50 8C. J. Am. Chem.

Soc. 65, 2030–2037.

Harned, H.S., Scholes, S.R., 1941. The ionization constant of HCO3�

from 0 to 50 8C. J. Am. Chem. Soc. 63, 1706–1709.

Johansson, O., Wedborg, M., 1982. On the evaluation of potentio-

metric titrations of seawater with hydrochloric acid. Oceanol. Acta

5, 209–218.

Lee, K., Millero, F.J., Campbell, D.M., 1996. The reliability of the

thermodynamic constants for the dissociation of carbonic acid in

seawater. Mar. Chem. 55, 233–245.

Lee, K., Millero, F.J., Wanninkhof, R., 1997. The carbon dioxide

system in the Atlantic Ocean. J. Geophys. Res. 102, 15696–15707.

Lee, K., Millero, F.J., Byrne, R.H., Feely, R.A., Wanninkhof, R.,

2000. The recommended dissociation constants for carbonic

acid in seawater. Geophys. Res. Lett. 27, 229–232.

Lueker, T.J., Dickson, A.G., Keeling, C.D., 2000. Ocean pCO2

calculated from dissolved inorganic carbon, alkalinity and equa-

tions for K1 and K2: validation based on laboratory measure-

ments of CO2 in gas and seawater at equilibrium. Mar. Chem.

70, 105–119.

Mehrbach, C., Culberson, C.H., Hawley, J.E., Pytkowicz, R.M., 1973.

Measurement of the apparent dissociation constants of carbonic

acid in seawater at atmospheric pressure. Limnol. Oceanogr. 18,

897–907.

Millero, F.J., 1979. The thermodynamics of the carbonate system in

seawater. Geochim. Cosmochim. Acta 43, 1651–1661.

Millero, F.J., 1995. Thermodynamics of the carbon dioxide system in

the oceans. Geochim. Cosmochim. Acta 59, 661–667.

Millero, F.J., 1996. Chemical Oceanography. CRC Press, Boca Raton,

FL.

Millero, F.J., Pierrot, D., 1998. A chemical equilibrium model for

natural waters. Aquat. Geochem. 4, 153–199.

Millero, F.J., Poisson, A., 1981. International one-atmosphere equa-

tion of state of seawater. Deep-Sea Res. 28, 625–629.

Millero, F.J., Roy, R.N., 1997. A chemical equilibrium model for the

carbonate system, in natural waters. Croat. Chem. Acta 70, 1–38.

Millero, F.J., Zhang, J.Z., Lee, K., Campbell, D.M., 1993. Titration

alkalinity of seawater. Mar. Chem. 44, 153–165.

Millero, F.J., Pierrot, D., Lee, K., Wanninkhof, R., Feely, R., Sabine,

C.L., Key, R.M., Takahashi, T., 2002. Dissociation constants for

carbonic acid determined from field measurements. Deep-Sea

Res. 49, 1705–1723.

Mojica-Prieto, F.J., Millero, F.J., 2002. The values of pK1+pK2 for

the dissociation of carbonic acid in seawater. Geochim. Cosmo-

chim. Acta 66, 2529–2540.

Pitzer, K.S., 1991. Theory: ion interaction approach: theory and

data collection. In: Pitzer, K.S. (Ed.), Activity Coefficients in

Electrolyte Solutions, vol. I, 2nd ed. CRC Press, Boca Raton,

FL, pp. 75–153.

Roy, R.N., Roy, L.N., Lawson, M., Vogel, K.M., Porter-Moore, C.,

Davis, W., Millero, F.J., Campbell, D.M., 1993. The dissociation

constants of carbonic acid in seawater at salinities 5 to 45 and

temperatures 0 to 45 8C. Mar. Chem. 44, 249–259.

Wanninkhof, R., Lewis, E., Feely, R.A., Millero, F.J., 1999. The

optimal carbonate dissociation constants for determining surface

water pCO2 from alkalinity and total inorganic carbon. Mar.

Chem. 65, 291–301.