the dissociation of phosphoric acid in nacl and namgcl solutions at 25�c

Journal o f Solution Chemistry, Vol. 18, No. 9, 1989

The Dissociat ion of Phosphor ic Acid in NaC! and NaMgCI Solut ions at 25~

J. Peter Hershey, 1 Marino Fernandez, 1 and Frank J. Millero 1 Received January 30, 1989

The pK~, pK~ and pK~ for the dissociation o f HaPO4 have been measured in NaCI solutions from 0.5 to 6m at 25~ The results have been used to evaluate Pitzer interaction parameters ~(NaCIHzP04) = --0.028 + 0.005, ~(NaHaP04) = -0.075 + 0.025, 0 (HP04CI) = 0.105 + 0.009, 0 (P04C1) = --0.59+0.02 and ~(NaCIHPO4) = -0.003+0.004, ~(PO4NaCIH) = 0.110+0.008. These parameters yield values o f pK~, pK~ and pK~ in NaCI that agree with the measured values with average deviations of +0.04, +0.03 and +0.05 in pKg. Measurements o f pK~ and pK~ were also made in NaMgCI solutions. These results have been used to evaluate ~(~ = -355+0.07, ~(1)(MgHzP04) = -16.9+0.03, ~(~ = -17.5+0.03 and ~(1}(MgHPO,) = 27.4+0.8 at 25~ The results for pK~ in NaMg-Cl solutions were also used to calculate log K ~x = 3.2 + 0.1 for the formation of the ion pair MgHPO~.

KEY WORDS: Phosphoric acid; Pitzer equations; ion pairing.

1. Introduction

Phosphoric acid (H3PO4) is a weak acid found in natural waters including seawater and brines. Knowledge of the ionization constants of H3PO4 is necessary to predict the speciation of various cations that in- teract with H3PO4 ionization products and in the study of the precipita- tion and solubility of various phosphate minerals.

The ionization constants of H3PO 4 have been measured in seawater by a number of workers. {1-4) Values of K~ and K~ in NaC1 media with Mg 2+ and Ca 2§ have also been measured up to a total ionic strength of 0.7m3 2~) Ion pairing constants of the various phosphoric acid dissociation products with Mg 2+ and Ca 2+ have been determined from these pK* measurements. However, most of these studies are limited to

1Rosenstiel School of Marine and Atmospheric Science, University of Miami, 4600 Rickenbacker Causeway, Miami, FL 33149.

875 0095-9782/89/0900-0875506.00/0 �9 1989 Plenum Publishing Corporation

876 Hershey, Fernandez, and Millero

ionic strengths less than lm. Pitzer (s) coefficients have also been determined (6) from these studies. Since most of these studies were made at a fixed ionic strength (0.7m), it was not possible to derive reliable parameters valid over a wide range of ionic strengths.

In the present study we have measured the stepwise ionization constants of phosphoric acid in NaCI media up to 6m. Pitzer parameters ~NaHsPO4, lq/NaC1HzPO 4, 0HPO4C1 ~I]NaCIH1K) 4, 0ClPO 4 and l]]NaC1PO 4 were determined from these results. In addition, we have measured the pK~ and pK~ in mixed NaMgC1 solutions and have calculated the Pitzer interaction coefficients for H2P04 and HP042- with Mg z+ and the ion pairing constant of HP04 z- with Mg 2§ These results should be usefi~A for calculating the activity coefficients for phosphate species in mixed electrolyte solutions in high concentrations.

2. Experimental

The stepwise ionization of H3PO 4 is represented by

HAP04 = H + + H2PO2 (1)

H2P04 = H + + HP042- (2)

HPO~- = H + + PO~ (3)

and the stoichiometric ionization constants are given by

K; = [H+][H2POn]/[H3PO4] (4)

K~ = [H+][HPO~-]/[H2PO4 -] (5)

K3 = [H+][PO43-]/[HP042-] (6)

The values of pKi* = -logKi* were determined from potentiometric titration of Na2HPO4 or K2HPO 4 in NaC1 and NaMgC1 solutions with HC1 or NaOH. The titrations were carried out in a 230 cm 3 jacketed cell ther- mostated at 25.00+0.02~ o9) The titrant was added to the cell with a Metrohm Dosimat automatic burette. The automated system was con- trolled with an Apple II computer.

The emf was measured with an Orion 91-01 research grade glass electrode and a Ag,AgC1 reference electrode connected to a Metrohm 605 pH meter. The reference electrode was prepared by electroplating AgC1 onto a silver billet electrode. The reliability of the electrode pair was checked using a 0.01M HC1 solution of known activity

Dissociation of Phosphoric Acid 877

7 . 0 "

5 . 0 "

3 . 0 "

1 I I I ' NaMgCl t

pK. 2 �9 NaCl

w

1.0 L 0.0 0.5

pK* 1

I I ! ! 1.0 1.5 2.0 2.5

Fig. l. Values of pK1 and pK2 for the lomzataon of H3PO4 m NaC1 and NaMgCI media as a function of ~/I at 25~

coefficients. (1~ Before the start of each titration, the concentrated NaC1 solutions were saturated with freshly precipitated AgC1 to prevent dis- solution of the reference electrode. The emf of the electrode pair is related to the proton concentration by

E = E* - ( R T / F ) l n [H +] (7)

where E* is the apparent standard potential in the medium and R, T and F have their usual meanings. The values of Ki*and E*, were determined from the titrations using a nonlinear least squares program. (H) Values of pKi* were converted from molar to molal units using known densities (~2> of NaC1 solutions.

The NaCI and Na-Mg-C1 solutions were prepared by weight from deionized (Millipore 18Mr2) water and Baker Analyzed NaC1 and MgClz-6H20. Concentrations of MgClz stock solutions were determined by density. 02) The phosphate solutions were prepared from the sodium or potassium reagent grade salts of H2PO~ and HPO~ and the total phosphate concentration was 0.003-0.005M. The concentration of Mg 2§ in the titrand was O.05m and the titrant was standardized 1.000M HCI or NaOH.


tO

12.5 , , ,

12.1

11.7-

11.3 t

10"9 t

lo.51 o o o'5 11o 115

,A

i , i �9 NaCI

21o 215

Fig. 2. Values of pK~ for the ionization of H3PO4 in NaC1 and NaMgC1

media as a function of q7 at 25~

3. Results and Discussion

3.1. pKi*in NaCI Solutions

The values of the measured pKi* on the molal scale in NaC1 solutions at concentration to 6m are given in Table I and plotted vs. 11/2 in Figs. 1 and 2. Least square fits of these results from I = 0 to 6m, forcing the fit through the infinite dilution values of pK1 = 2.146, (u) pK2 = 7.198 (14) and pK3 = 12.35 (m are given by

pK~ = 2 .146- 0.5136qi+ 0.16061

pK~ = 7.198 - 1.4040qI+ 0 .6168I - 0 . 0 8 1 1 4 I 3/z

pK; = 12.35 - 2.0992"~I + 0 .9318I - 0.129113/2

(8)

(9)

(10)

The standard errors of the fits are 0.04, 0.02, and 0.03 for pK;, pK~ and pK~, respectively.

The values of the pK~ in NaC1 media at various concentrations calculated from Eqs. (8 - 10) are compared with the values obtained by other workers in Table II. The pKldetermined by both Johansson and Wedborg (a) and Atlas et al. (2) were first converted from the molar to molal scale. In addition, the pK;of Atlas e ta / . (2) which were measured at 20~ were adjusted to 25~ This was done by using the temperature dependencies of K~ and K~ in 0.7m NaC1 solution (a) to calculate the AH~

Dissociation of Phosphoric Acid

Table I. Experimental Values of pK* for the Dissociation of H3PO4 in NaC1 Media at 25~

879

mNaCl pK1 PK2 pK3

0.5 1.864,1.890 6.481,6.459 11.237,11.341 11.244,11.336

1 1.783,1.803 6.327,6.337 11.064,11.024 1.785,1.785 6.343,6.330 11.055,11,062

11.036,11.038 2 1.739,1.731 6.223,6.240 10.906,10.891

1.711,1.732 6.213,6.218 1.775 6.215

3 1.704,1.695 6.233,6.176 10.837,10.854 1.716,1.803 6.178,6.177

4 1.759,1.759 6.185,6.239 10.836,10.841 1.801,1.722 6.208,6.210

5 1.766,1.871 6.231,6.218 10.863,i0.860 1.885,1.807 6.223

6 1.816,1.808 6.276,6.280 10.911,10.910

= 1.35 and AH~ = 4.38 kcal-mo1-1 for the ionization of H3PO4. These corrected values are given in Table II along with the pK* values in NaCI calculated at the same concentration from this work.

The values of pK~ and pK~ at I = 0.7 are in good agreement with the results of Atlas et al. (2) and Johansson and Wedborg. o) The agreement of our pKa* at this ionic strength with the results of Atlas et al. (2) is not as good, but within the combined experimental error of the two studies (+0.05). Our results for pK~ and pK~at I = 0.2 and 0.4m are 0.1 to 0.2 pK units higher than the results of Johansson and Wedborg. ~ We have no simple explanation for this discrepancy. Since the interpolated values obtained from Eqs. (9, 10) agree with the values obtained with the Pitzer parameters (both with and without those derived in this paper), we do not feel it is due to an error in interpolation.

The thermodynamic equilibrium constants Ki for the stepwise ionization of phosphoric acid are given by

K1 = K; Yn 'YHzPO 4 / 'YH3PO 4

g 2 = K~ YH 'YHPO 4 / 'YH2PO 4

K3 = K:~ ~z4 "/p04 / "/m'04

(11)

(12)

(13)


Table II. Comparison of the Values ofpKi Obtained in this Study with Literature Data

Media Constant This Study Literature A

0.20m NaC1

0.40m NaCI

0.69m NaCI

0.7 lm NaCI

0.53m NaCI + 0.05m MgC12

pK 2 6.69 6.55 b 0.14 pK 3 11.59 11.39 b 0.20 pK~ 6.54 6.44 b 0.10 pK~ 11.36 11.25 b 0.11

pK 1 1.83 1.71 a 0.12 pK~ 6.41 6.37 a 0.04 pK~ 11.18 11.13 a 0.05 pK~ 6.40 6.33 b 0.07 pK~ 11.17 11.09 b 0.08

pK 2 6.16 6.13 a 0.03 6.11 b 0.05

a Ref. 2. b Ref. 3.

A convenient way to predict the activity coefficients in single and multi- component electrolyte media is through the use of the specific ion interaction model of Pitzer. ts'6'~6'17) A brief summary of the Pitzer equations is given in the Appendix.

The activity coefficients for H § and H2POz are expressed explicitly by

lnyH = f 7 + 2mca(Bno + ECHo) + mN~mcl(B~cl + CN~O)

+ mN~(20~, + movr~ ,o ) (14)

lnTH2B = f 7 + 2mN,(BNdzB + ECN~2B) + mNamca(B~ca + CN, Cl)

+ mo(20u2Bca + mNa~N~2ncl) (15)

In order to simplify the notation, we have designated H3PO4, H2PO4, HPO42- and P O ~ as HaB, H2B, HB and B, respectively. The value of the activity coefficient for the neutral HaPO4 species can be written by oa)

lnTi43 B = 2[mri~,aaBa + mCl'~,/3BCl + mNa~,HaBNa

+ mn2B~k~i3ai_I2 B + mH3B~,H3BH3B ] (16)


Table IlL Pitzer Coefficients for Phosphoric Acid in NaC1 and Na-Mg-C1 Solutions

881

Salt [3 (~ 13 O) C ~ Ref.

HC1 0.1775 0.2945 0.00080 18 NaCI 0.0765 0.2664 0.00127 18 Na3PO 4 0.1781 3.8513 -0.05154 18 Na2HPO 4 -0.05828 1.4655 0.02938 18 NaH2PO 4 -0.0533 0.0396 0.00795 18 MgC12 0.35235 1.6815 0.00519 18

Mg(H2PO4) 2 -3.55- 16.9 a

MgHPO 4 -17.5 27.4 a

i j k 0ij Vijk Ref.

H Na C1

H2PO 4 C1 Na

HPO 4 CI Na

PO 4 CI Na H Mg C1

H3PO 4 Na 0.075

H3PO 4 CI 0

MgHPO 4 Na -0.124

0.036 -0.004 19 0.10 -0.028 13, a

-0.105 -0.003 a

-0.59 0.110 a 0 0 19

a This study.

Since the total phosphate concemration under the experimental con- ditions was low (m<0.005), we assumed the mrI ~,rI3BH = mrI2~ ~'a3B~2B =

mHaa ~rt~H3B = 0. Furthermore, we set the term due to C1- - H3PO 4 interactions to zero, that is, ~,I~3nc~ = 0. Eq. (16), thus, reduces to

ln)'H3B = 2[mcl~I-t3BCl + mNa~tt3BNa] = 2mNa~,rhBNa (17)

A summary of the known Pitzer coefficients for the phosphate system in NaC1 solutions is given in Table III. The missing coefficients are ~n3BNa and ~NaH2BCa fo r determining pkg .

The values of K~ in NaC1 solution can be used to calculate the Pitzer interaction coefficients ~-H~Na and ~N~2BC ~. Rewriting Eq. (11) and combining with Eqs. (14 - 16) gives


lJ z E

>~ I

0.20

0.10"

0.00

- .10

- . 2 0

, , , , i , , , i | , , , , l , , I ) | , , , , | , ' ' ' l

0

0

, , , , 1 ' ' ' ' 1 ' ' ' ' 1 ' ' ' ' 1 ' ' ' ' 1 ' ' ' ' 1

0.0 1.0 2.0 3.0 r 5.0 6.0

mcI

Fig. 3, Values of -y/2mNa = ~,HBNa + mc1 ~/H2BNaCl derived for pK1 measurements in NaCI solutions.

- 0 . 1 0 - , , , , i . . . . i . . . . i . . . . i . . . . i . . . .

I t -0.20- �9

�9 - ~ e 0 I �9

E -~. -0.30.

-0.40.

- 0 . 5 0 0 . 0 1 . 0 2 . 0 3 . 0 4 . 0 5 . 0 6 . 0

tuNa

Fig. 4. Values of -ylmc} = 2ermcl + mN~ ~BNaa derived for pK2 measurements in NaCI solutions.


Table IV. Values of pK~ and pK~ in Na-Mg-C1 Solutions at 25~

/a K1 pK 2

0.65 1.861,1.863 6.176,6.180 1 . 1 5 1.775,1.759,1.806 6.122,6.121,6.128 2.15 1.695 6.069 3.15 1.596 6.047 4.15 1.720,1.705 6.084,6.075 5.15 1.756 6.165,6.133 6.15 1.679,1.796 6.170,6.214

a NaC1 + 0.05m MgClz.

y = lnK1 - l n K ~ - In N - InNzB(MED)

= --2mNa~HaBNa + mN,mCl~H2nN~Cl (18)

where lnN2B(MED) is the summation of all the known terms in Eq. (15) except ~N~H2BCl" The values of the known Pitzer interaction parameters r for N2 B and ~'H on the left hand side of Eq. (17) are given in Table II. A least squares fit of y vs. m gives values ~r~2BN~Ca =

00.028+0.005 and ~'H3BNa = 0.075+.025. The average deviation between the measured and calculated pK~ was 0.04 pK units. A linear transformation of Eq. (17) gives y/2mNa = ~H3BNa + mCl~H2BN~Ca/2 and a plot of y/2mNa vs. mo is given in Fig. 3. The solid curve determined from the above fit of y is in good agreement at high molality, but ap- pears to deviate at low molalities because of the large inherent uncer- tainty in values of y/2mN,. The coefficients in Eq. (18) were fitted directly to avoid the necessity of fitting weighted values ofy/2mN,.

The Pitzer equations for calculating values of 7rm and % can be written explicitly as

ln~'~ = 4f 7 + 2mN~(Bs~rm + ECNatm) + mN~mcl(4B~raCl + 2CN~ca)

+ mcx(20rmca + mNa~tlmNaCl)

lny~ = 9f 7 + 2mN,(BN~B + ECN~B) + mNamcl(9B~acl + 3CNac1)

(19)

+ mca(2 0BC1 q- mna~BNaCt) (20)


Rewriting Eqs. (12, 13) and combining with Eqs. (14, 19, 20) gives

y = lnK2 - lnK~+ ln~2 B - lnTH -- lnTrm(MED)

= 2mca(20rmca + mr~,~rmN~c~)

y = lnK3 - lnK~+ lngr m - l nTa - lnTB(MED)

(21)

= 2ma(20aca + mNa~BN, CL) (22)

where 7rm(MED) and ~'a(MED) are defined in a manner analogous to 7,2B(MED).

Using the known Pitzer coefficients tabulated in Table III to calculate y in Eqs, (21, 22), the values of 0 and ~ can be determined from a least squares fit of y vs. m. The values of these parameters are 0rmcl = -0.105+0.009, ~trmNaca = -0.003+0.004, 0BOa = -0.59--+0.02 and ~tBN,Ca = 0.110-+0.008. The average deviations between the calculated and measured values of pK~ and pK~ are 0.03 and 0.05, respectively. The experimental values of (-y/2rnct) are plotted vs. mN~ in Figs. 4 and 5; the solid curve was determined by evaluating the linear transformation of Eqs. (21, 22) with values of 0 and ~ determined from the linear squares fit o fy vs. m.

3.2. pK* in Na-Mg-CI Solution

Calculation o f Pitzer Coefficients. The values of pK; and pK2 measured in mixed Na-Mg-C1 (mug = 0.05) solutions at ionic strengths to 6rn are given in Table IV. The values of pK; and pK~ vs . [1/2 are plotted in Figs. 1 and 2, respectively. The least squares fits (valid over a range of 0.5 < m < 6) are given by

pK~(NaMgC1) = 2.2932 - 0.68676~]7 + 0.18989/ (23)

pK~(NaMgC1) = 6.5583 - 0.61662~I + 0.18963I (24)

and have standard errors of 0.05 and 0.02, respectively. The values of pK; in NaMgC1 mixtures are slightly lower than the values in NaC1, while the values of pK~ are much lower. This is largely due to the strong interactions of Mg 2+ with H2PO~.

Our experimental value of pK~(NaMgC1) at I = 0.7m is in good agreement with the corrected value of pK~(NaMgC1) of Atlas et al. (z) and the value determined by Johansson and Wedborg C3) (see Table II). Due to solubility limitations at the total PO4 a- level necessary for a

Dissoclatio~ of Phosphor~ Acki 885

0 ,00 , , i , I , , t ~ I . . . . I t �9 ' ' I t ' ' ~ ' I . . . . I

E

-,50'

-1,00

-1.50"

-2 ,00 0.0

t

1.0 2.0 3.0 4.0 5.0 6.0

t u n a

Fig, 5, Values of -ylrr~ = 20act + mr~a ~ttNact derived for pK~ rneasuTements in NaC~ solutions,

reasonable signal in our work, we were unable to measure the pK~ in NaMgC1 media.

The values of pKi*in mixed Na-Mg-C1 solutions were used to calculate the Pitzer interaction coefficients for Mg-HzPO 4 and Mg-HPO4. The total activity coefficients for HgPO4 and HPO4 in Na-Mg-C1 media are given explicitly by

lnYa=a = [Eq. (15)] + mMsrnc~(B't~ca + CMaCl)

+2mMg(BM~2B + ECt~g~s) (25)

l n ~ = [Eq. (19)] + mMF, mcl(4Bh~ + 2CM~I)

+2mMg(BMsrm + ECMcm) (26)

Rewriting Eqs. (11, 12) and combining them with Eqs. (25, 26) gives

y = lnK1 - lnKl*- lnTn - lnT~gB(MED) + ln~n3B

= 2mMs(BMSnzB + ECMsu3s) (27)


O~

E t'N

>~--1

- 2

4 ' ' I ' ' ' I - ' ~ ' I

3 -

2 - 0

I

0 o o

0 - - 4 I I I '

0.00 0.10 0.20 0.50

fO

Fig. 6. Values ofy/2mMg = [3MgrI2B ~ +f013~gH2B derived from pK~ measurements in NaMgC1 solutions.

y = lnK2 - l n K ~ - lnyH - lnyrm(MED) + ln~'n3B

0.40

= 2mMg(BMgaB + ECM~)

Combining Eqs. (28, 29) with Eqs. (A.4 - A.6) gives

rcOa (1) y = 2mM~[~5~ZB +., VMgn2B]

(1) ] y = 2mM tl3 +

where f0 = [1 - (1 + 2~I) exp ( -2~ i ) ] /2 I

fa = [1 - (1 + 1.4~I) exp(-1.4"~I)]/0.98-~?

(29)

(30)

(31)

(32)

It was found that the C ~ terms for the Mg 2+ interaction with H2PO4 and HPO42- and the ~(2) for the Mg2+-HPO4 z- interaction were not statistically necessary. The values of the parameters in Eqs. (29, 30) are ~ z a = -3.6+0.06, 13~gn2B = 16.9+3.0, 13~g~ =-17.5+0.3 and ~ g ~ = 27.4+0.8. It should be noted that the activity coefficient for H3PO4 in NaMgC1 media is represented rigorously in the Pitzer formalism by


0 I ~ I ~ I ~ I

- 4 .

- - 8 '

E t"q

>, - 1 2 �84

- 1 6 -

- 2 0 I j I I |

0.00 0 .10 0 .20 0 .30 0 .40 0 .50

fl

Fig. 7. Values ofy/2mMg = ~MgtIB ~ + f l ~ l~gftB derived from pK~ measurements in NaMgC1 solutions.

adding a Mg 2§ - H3PO4 interaction term to Eq. (17)

lnyHaB = 2msa~,NarlaB + 2mMg~,M~3B (33)

Since we worked at only one molality of Mg 2§ it is not possible to calculate the XMgH3B interaction coefficient from our ternary medium data. Therefore, we assume that ~-mgr~ = ~NaH3B and that the Setchenow relationship as a function of ionic strength is given by

lnTHaa = 0.15I (34)

The [3 (~ and 13 (1) parameters yield calculated values of pK~ and pK~ that agree within 0.03 and 0.01 pK units, respectively, with the measured values. Values of (-y/2mMg) VS. fo and f l from Eqs. (29, 30) are plotted in Figs. 6 and 7 with the solid curve being evaluated using values of 13 (~ and ~(1) determined from the least squares fit ofy.

Ion Pairing Constant for MgZ*-HPO~ -. The values of pK~ in Na- Mg-C1 media can also be used to determine the ion pairing constant of Mg z+ with HPO4 z-. The formation of the ion pair can be represented by


Mg 2§ + HPO~- = MgHPO4

The stoichiometric ion pairing constant is given by

K~om = [MgHB]/[Mg][HB]

(35)

(36)

The value of K~om can be evaluated from (7)

T~m3 = Fvrm(1 + K~grm[Mg]F) "1 (37)

where Tyro3 and Fvrm are the values of the total and free ionic activity coefficients of HPO 2- and can be calculated from Eqs. (26, 29), respectively, using the Pitzer coefficients from Table II. The values of K ~ m are related to the thermodynamic ion pairing constant by

KMe,~m = KMe, m37~Xm/YMg "YFm (38)

The experimental values of Kr~cm and the values of 7Mg and Ym3 from Pitzer's equations can be used to determine the values of Kuom and Yuom. The Pitzer expression for the activity coefficient of the neutral ion pair can be written as

In 7Me, rl B = 2[mclXtuecmlca + mNa~,[MgnB]Na] (39)

Assigning a conventional value of ~,tigrmla = 0 and substituting into Eq. (38) gives

lnK~ejm = ln~tom - lnyMg - lnn~ = lnKMgn~ - 2mNa~,tMgrmlNa (40)

the values of lnK~dm vs. mNa are plotted in Fig. 8. The value of ~'tMOmlNa = -0.124 was calculated from the slope and the value of lOgKMgrm = 2.70 derived from the intercept. The value of K~grm is in good agreement with logKMom = 2.9 (z~ and logKiom = 2.68 (zl) from literature data.

The values of K ~ 2 n were very small and negative (at m~aca> 2); consequently, we did not analyze the data in terms of the ion-pairing model, but used only the specific ion interaction approach.

The Pitzer coefficients derived in this study can be used to make reasonable estimates of pK~ for H3PO 4 in natural waters. At present it is not possible to account for the interactions of Mg 2+ and Ca 2§ with PO42- over a wide range of ionic strengths. Reasonable estimates can be made, however, to lm using the Pitzer coefficients at 0.7m estimated in earlier studies. (~,17)


8.00 , , , ,

889

| ' I ' I ' I '

r,,

c-

7.00.

6 . 0 0

0

o o o u

C

0

5 . 0 0 , , , . , , , , 0 . 0 0 . 8 1 .6 2 . 4 3 . 2 4 . 0 4 . 8 5 . 6

mNa

Fig. 8. Values of lnKMgHB' = lrlKMgHB - 2tuNa ~,MgliBNa derived from pK~ measurements in NaMgC1 solutions.

4. Appendix The conventional single ion activity coefficients for cations and

anions where M and c are cations and X and a are anions, are given by

in ~/M = ZMZf Y + 2Eama(BMa + EC~) + Z~ZcEamemaBca

+ ZraEcEamcmaCca + Ecme(20Me + Eama~Mca) + EEa< gmama'~aa'M

+ 2EcmcEOMe + Z~(EEe<c'mcrnc'EOce ") + ZM2(EEa<a'mama'EO'aa ") (A.1)

lnyx = Zxf T + 2Ecmc(Bcx + ECcx) + Zx2EcEamcmaBca

+ ZxEcY'amcmaCca + Eama(20xa + Ecmc~xac) + EEc < c'mcmc'~cc'x

+ 2EamaE0xa + Zx2(EEc<c'memc'~O'cc ,) + Zx2(EEa<emama~Oae) (A.2)

whe re f 7 is the Debye-Hi]ckel term given at 25~ by

f r = -0 .392 [~ / (1 + 1.2ql) + (3/1.2)1n(1 + 1.2qI)] (A.3)

Zi is the charge of the ion and E = EilmiZil/2. The parameters, 0 and ~,

890 Hershey, Fernandez, and MUlero

represent higher order terms for the like charged ions and for the triple ion interactions. E0 and Eo' represent higher order electrostatic terms which are functions of ionic strength and of the electrolyte pair type. Their functions have been defined. <19) The second and third virial coefficients for various electrolytes are given by

BMx = ~~ + (213~x/0~/)[1 - a1~/7) exp (-a~',/7)] (A.4)

BMX, = (2~x/O~12)[-l +(l +~1~l+o~l)exp(-al"~I)] (A.5)

CMx = C~r~x/(21ZMZxl l/z) (m.6)

where aa = 2 and for a 2-2 electrolyte by

B~tx = 13~x + (213~/o~/)[1 - (1 + oq~/7) exp (-a147)1

+ ( 2 ~ x + ~z/)[1 - (1 + ~z~/I) exp (-o~1~7)] (A.7)

B~tx, = + (213~x + ~1zI2)[-1 + (1 + ~1"~I + ~12/) exp ( - a l ~ l ) ]

+ ( 2 ~ + a z 2 1 z ) [ - 1 +(1 + o~47 + a~/) exp (-c~z4I)] (A.8)

where a~ = 1.4 and ~ = 12.

Acknowledgment The authors wish to acknowledge the support of the Office of

Naval Research (N00014-86-G0116) and the oceanographic section of the National Science Foundation (OCE86-00284) for support of this study.

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the dissociation of phosphoric acid in nacl and namgcl solutions at 25�c

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