The apparent molal volume and compressibility of seawater fit to the Pitzer equations

Journal of Solution Chemistry, Vol. 29, No. 8, 2000

The Apparent Molal Volume and Compressibilityof Seawater Fit to the Pitzer Equations

Denis Pierrot and Frank J. Millero*

Received March 31, 1999; revised February 7, 2000

The density and compressibility of seawater salt solutions for ionic strengths 0to 0.8 m, temperatures 0–408C, and applied pressure 0 to 1000 bar are fitted tothe Pitzer equations. The apparent molal volumes and compressibilities (Xf ) arefitted to equations of the form

Xf 5 X 0 1 AXI/(1.2 m) ln(1 1 1.2 I 0.5) 1 2 RT m(b(0)X 1 b(1)Xg( y) 1 m CX)

where I is the ionic strength, m is the molality of sea salt, AX is the Debye–Huckelslope for the volume (X 5 V ) or compressibility (X 5 k) and g( y) 5 (2/y2)[1 2 (1 1 y)exp(x)] where y 5 2 I 0.5. The Pitzer parameters b(0)X, b(1)X, and CX

are fitted to functions of temperature and pressure in the form

Y X 5 oi oj aij(T 2 TR)iP j

where aij are adjustable parameters, Y X is the Pitzer parameter, T is the temperaturein K, TR 5 298.15 K, and P is the applied pressure in bars (P 5 0 at 1 atm or1.013 bar). The standard deviations of the seawater fits are 8.331026 cm3-g21

for the specific volumes, 0.000731026 bar21 for the compressibilities, and0.6331026 K21 for the thermal expansibilities. At 258C, the measured densitiesof seawater are compared to the calculated values using Pitzer coefficients forthe major sea salts. The results agree with the measured values to within 4531026

g-cm23.

KEY WORDS: Apparent molal volumes; apparent molar compressibilities; sea-water; Pitzer equations; sea salts.

Rosenstiel School of Marine and Atmospheric Science, University of Miami, Miami, Florida33149.

7190095-9782/00/0800-0719$18.00/0 q 2000 Plenum Publishing Corporation

720 Pierrot and Millero

1. INTRODUCTION

Precise knowledge of the PVT properties of seawater and other naturalwaters as a function of temperature and pressure is required to examine theirphysical behavior. For example, the density r and thermal expansion a areused in the calculation of the vertical stability of a water column and theadiabatic temperature gradient.(1,2) Since the thermodynamic properties ofseawater diluted with pure water can be used to determine the properties oflakes and rivers,(3–5) it is useful to fit these properties to equations thatapproach pure water as the salinity decreases. In earlier studies,(6–9) thethermochemical properties of seawater were examined using an extendedDebye–Huckel equation, which was an extension of the Guggenheim equa-tion(10) as formulated by Lewis and Randall.(11) Unfortunately the PVT proper-ties(2,12) were not fitted to an equation of the same form, which made itdifficult to integrate the thermochemical properties as a function of pressure.Numerous studies have shown that properties of pure and mixed electrolytesolutions can be successfully modeled by the use of the Pitzer(13) equations,which are readily integrated to yield the derived quantities. We have decidedto refit all of the thermodynamic properties of seawater to these equations.This will provide a thermodynamic representation of seawater diluted withwater that is consistent with that used to examine other electrolytes. Theseequations will yield infinite-dilution values consistent with those determinedfor pure electrolyte mixtures and will be readily applicable to low salinitywaters (estuaries) as well as deep, open ocean waters. The treatment presentedhere is made possible by the fact that the relative composition of the world’soceans is nearly constant, which allows us to treat a complex mixture ofelectrolytes like seawater as a solution of a fictitious “sea salt”(2) dilutedwith water.

The results of this study will give reliable estimates of the propertiesof seawater, estuarine waters, rivers, and lakes. In this paper, we describethe results of fitting the densities and compressibilities of seawater(14,15) tothe Pitzer equations from I 5 0 to 0.8 m ionic strength, 0–408C, and 0–1000bars applied pressure. Similar fits to the thermochemical properties of seawa-ter (heat capacities, enthalpies, and free energies) will be described else-where.(16) Calculated compressibilities and thermal expansibilities will becompared to values calculated using a high-pressure equation of state ofseawater.(17)

2. VOLUME AND COMPRESSIBILITY DATA

Precise volume and compressibility measurements for water and seawa-ter are available(14,15) over a wide range of applied pressure (0–1000 bar),

Molal Volume and Compressibility of Seawater 721

temperature (22–408C), and salinity S (0–40). These measurements havebeen used to determine the equation of state for seawater.(17,18) The data forseawater consist of 467 points at 1 atm (P 5 0) and 1546 points for appliedpressures between 99.83 and 1000 bars, for a total of 2013 measurements.The precision of the densities (r) is 331026 g-cm23 at 1 atm and 531026

g-cm23 over the entire ranges of P, t, and S.(17,18) The specific volumes derivedfrom these data are directly analyzed in this paper.

The most reliable estimates of the isothermal compressibilities (bT 5(1r)(r/P)T) of seawater at 1 atmosphere are those determined from soundvelocities,(17) where the largest source of errors comes from the uncertaintiesin the thermal expansibility, a 5 (1/V )(V/T )P 5 2(1r)(r/T )P . Thecompressibility in bar21 is related to the sound velocity by:

b0T 5 106/r0(U 0)2 1 0.1 T(a0)2/r0c0

p (1)

where the superscript zero indicates that the quantity is at zero applied pressure(1 atm or 1.013 bar). U is the speed of sound in cm-s21; r is the density ing-cm23; T is the temperature in K; a is the expansibility in K21, and cp isthe specific heat in J-g21.

As described by Millero et al.,(17) the isothermal compressibilities forseawater at 1 atm were generated using the sound-velocity equations of DelGrosso and Mader,(19) the expansibilities from the equation of state of Millero,Gonzalez, and Ward,(20) and the specific heats from the equations of Millero,Perron, and Desnoyers.(21) The errors in b0

T are estimated to be within60.00431026 bar21.(17)

As mentioned above, we are treating seawater as a solution composedof a sea salt. From the average composition of seawater,(22) the molality mand ionic strength I of sea salt can be calculated as a function of salinity by:

m 5 (1/2) o mi 1 mB 5 16.011 S/(1000 2 1.00488 S) (2)

I 5 (1/2) o miz2i 5 19.9243 S/(1000 2 1.00488 S) (3)

where mi is the molality and zi the charge of ion i, and mB is the molality ofun-ionized boric acid. The molecular weight of sea salt is M 5 62.793 g-mol21. It should be pointed out that the molalities of the components ofseawater can also be defined by equations similar to Eq. (2).(22)

For a kilogram of solution, the apparent molal volume Vf and compress-ibility (kf 5 2Vf/P) of sea salt are defined by:

Vf 5 (V 2 1000/rw)/m (4)

kf 5 (bTV 2 1000 bwT /rw)/m (5)

where V is the total volume of the solution in cm3. The subscript w is used


to denote the properties of pure water. The values of Vf and kf can beexpressed as functions of measurable quantities using

Vf 5 1000(rw 2 r)/rrwm 1 M/r (6)

kf 5 1000(bTrw 2 bwTr)/mrrw 1 bTM/r (7)

By rearranging these equations we can derive expressions for the specificvolume (v 5 1/r) and isothermal compressibility of the solutions:

v 5 (1000 vw 1 mVf)/(mM 1 1000) (8)

bT 5 (r/rw)(kfmrw 1 1000 bwT)/(mM 1 1000) (9)

A similar derivation gives the expression for the isothermal expansibility aas a function of the apparent molal expansibility, Ef 5 (Vf/T )P

a 5 (1000 vwaw 1 mEf[v(mM 1 1000)] (10)

The pure-water quantities were derived from the equations of Kell(23) for thedensity and isothermal compressibility and expansibility. The temperatureswere adjusted to the new ITS-90 scale prior to fitting the data. The adjustmentwas made according to Saunders(24) and is given by:

t68 5 1.00024t90 (11)

where t68 is the temperature in 8C on the IPTS-68 scale and t90 is on the ITS-90 scale.(24)

3. TREATMENT WITH THE PITZER EQUATIONS

3.1. Derivation of the Equations

The volume of a mixture can be expressed as

V 5 n1V 01 1 o niV 0

i 1 V Em (12)

where n1 and ni are moles of water and ions, respectively, V 01 and V 0

i are thepartial molal volumes of water and of sea salt ions at infinite dilution, andVE

m is the excess volume of mixing. Substitution of Eq. (12) into Eq. (4) yields

Vf 5 (1/m) o miV 0i 1 (1/m)V E

m (13)

Differentiation of this equation with respect to pressure gives

kf 5 (1/m) o mi k0i 2 (1/m)(V E

m /P)T (14)

where k0i 5 2(V 0

i /P)T is the partial molal compressibility of ion i. The


excess volume of mixing for a multicomponent electrolyte solution is relatedto the excess Gibbs energy of mixing by

V Em 5 (GE

m /P)T (15)

The expression of the excess Gibbs energy based on a virial expansionadded to a Debye–Huckel term was given by Pitzer(13)

GEm /wwRT 5 2 4IAf /b ln(1 1 bI 0.5) 1 2 oc oa mcma[Bca 1 (oc mczc)Cca]

1 oc oc8 (2Qcc8 1 oa maCcc8a) 1 oa oa8 (2Qaa8 1 oc mcCaa8c)

1 2 oc on mcmnlcn 1 2 oa on mamnlan

1 2 on on8 mnmn8lnn8 1 on m2nlnn 1 . . . (16)

where T is the kelvin temperature, R is the universal gas constant, 83.1441cm3-bar-mol21 2K21, Af is the Debye–Huckel slope (defined for the osmoticcoefficient), b is a parameter set to a constant value of 1.2 kg1/2-mol21/2, Iis the ionic strength of the mixture, ww is the weight of solvent in kg, andmi and zi are the molality and charge of a particular cation (c) or anion (a).B and C are parameters related to short-range interactions of ions of oppositesign and are usually determined from the pure binary electrolyte solutions.Both are function of temperature, but B is also a function of ionic strength.The values of Qij refer to interactions between ions (i-j) of the same signand are zero for pure electrolyte solutions. Their values are determined fromcommon-ion mixtures. The values of Cijk are for triplet interactions (c-a-c,a-c-a) and the lcn parameters account for interactions between neutral solutes(n) and themselves or other neutral solutes (n8), cations (c), or anions (a).

Neglecting the interaction parameters for neutral solutes, the differentia-tion of Eq. (16) with respect to pressure at constant temperature yields theexpression for the excess volume of mixing in the Pitzer formalism.

V Em /ww 5 (AVI/b) ln(1 1 bI 0.5) 1 2RT oc oa mcma[BV

ca 1 (oc mczc)CVca]

1 RT oa oc8 (2QVcc8 1 oa maCV

cc8a)

1 RT oa oa8 (2QVaa8 1 oc mcCV

aa8c) (17)

A second differentiation of Eq. (16) results in an equation of similar formwhere the V subscript or superscript is replaced by k. The value of AV, theDebye–Huckel slope for the apparent molar volume, is given by(25)

AV 5 2 4 R T(Af /P)T (18)

The subscript and superscript V indicate the differentiation of the correspond-


ing ion interaction parameter with respect to pressure at constant temperature[for example, BV

ca 5 (Bca /P)T, where BVca is the osmotic pressure term for

the pure electrolyte ca].The expressions for the pure electrolyte parameters are given by

BVca 5 b(0)V

ca 1 b(1)Vca g(a1I 0.5) 1 b(2)V

ca g(a2I 0.5) (19)

where

b(i)Vca 5 [b(i)

ca /P]T with i 5 0, 1, 2 (20)

g(x) 5 2/x2[1 2 (1 1 x)exp(2x)] (21)

CVca 5 (Cca /P)T 5 (Cf

ca /P)T/(2.zcza.1/2) (22)

For the ionic strength function g, a1 is 2 for 1:1 and 2:1 electrolytes, whilea1 5 1.4 and a2 5 12 for 2:2 electrolytes.

For seawater solutions, the expression for the excess volume of mixing,neglecting the higher-order terms, is simplified to

V Em 5 (AVI/b) ln(1 1 bI 0.5) 1 2RTm2(BV

ss 1 mCVss) (23)

Substituting Eq. (23) into Eq. (13) gives

Vf 5 V 0ss 1 AvI/(mb) ln(1 1 bI 0.5) 1 2RTm(BV

ss 1 mCVss) (24)

where V 0ss is the infinite-dilution apparent molal volume and the subscript

“ss” refers to sea salt. The sea salt Pitzer parameters are related to theindividual ionic Pitzer parameters by the following equations:

V 0ss 5 (1/m) o miV 0

i (25)

Bvss 5 (1/m2) oc oa (mcma)BV

ca (26)

CVss 5 (1/m3) oc oa (mcma) oc (mczc)CV

ca (27)

Because of the small contribution from 2:2 electrolytes (e.g., MgSO4), theB term can be expressed as a function of a b(0) and b(1) term only, so thatBV

ss 5 b(0)Vss 1 b(1)V

ss g(2=I ). This, and the fact that we neglected the nonelec-trolyte Pitzer terms, imply that there is no direct correspondence between thesea salt and ionic values of b(0) and b(1).

Experimentally, the errors associated with apparent molal volumesincrease as the molality of the solution decreases, and fitting these quantitiescould yield large errors in the derived infinite-dilution parameters unless thedata are weighed by a factor inversely proportional to the molality. This istaken care of in the expression for the specific volume, Eq. (8), where theapparent molal volume is multiplied by such a factor. Therefore, fitting


specific volumes instead of apparent molal volumes will yield more reliableparameters. Combining Eqs. (8) and (24), and grouping all the known termson the left, leads to the form of the equation used to fit the seawater data:

v 2 [1000/(mM 1 1000)]vw 2 [m/(mM 1 1000)]DHv 5 [m/(mM 1 1000)]V 0ss

1 [2RTm2/(mM 1 1000)](b(0)Vss 1 b(1)V

ss g(2=I ) 1 mCVss) (28)

where

DHv 5 AvI/(mb) ln(1 1 b=I ) (29)

Similar expressions for the isothermal apparent molal compressibilities are

kf 5 k0ss 2 AkI/(mb) ln(1 1 b=I )

2 2RTm[b(0)kss 1 b(1)k

ss g(2=I ) 1 mCkss] (30)

and

bT 2 [(1000 r)/(rw(1000 1 mM ))]bwT 2 [(m r)/(1000 1 mM )] DHk

5 [(m r)/(1000 1 mM )]k0ss 2 [2RTm2 r/(mM 1 1000)]

[(b(0)kss 1 b(1)k

ss g(2=I )] 1 mCkss) (31)

where

DHk 5 2AkI/(mb) ln(1 1 b=I ) (32)

and

k0ss 5 2(V 0

ss /P)T (33)

b(i)kss 5 [b(i)v

ss /P]T with i 5 0, 1 (34)

Ckss 5 (CV

ss /P)T (35)

The other terms were defined above.

3.2. Debye–Huckel Limiting Slopes

As mentioned by Krumgalz et al.,(26) the values for the Debye–Huckelslopes used by different authors are not always consistent. The values used


Table I. Pitzer–Debye–Huckel Slopes for the Apparent Molal Compressibilities Usedin this Studya

104 Ak

P ⇒ 1 50 200 400 600 800 1000

0 8C 22.2944 22.2626 22.1696 22.0546 21.9489 21.8514 21.76145 22.6405 22.6018 22.4885 22.3485 22.2197 22.1010 21.9915

10 22.9785 22.9327 22.7988 22.6336 22.4819 22.3425 22.214015 23.3177 23.2645 23.1090 22.9176 22.7425 22.5819 22.434420 23.6655 23.6043 23.4257 23.2066 23.0069 22.8243 22.657125 24.0278 23.9580 23.7545 23.5057 23.2796 23.0737 22.885730 24.4102 24.3308 24.1001 23.8190 23.5645 23.3335 23.123335 24.8174 24.7275 24.4670 24.1505 23.8651 23.6070 23.372840 25.2543 25.1529 24.8593 24.5039 24.1846 23.8969 23.636845 25.7258 25.6114 25.2812 24.8828 24.5263 24.2061 23.917650 26.2369 26.1081 25.7370 25.2909 24.8932 24.5374 24.217855 26.7930 26.6481 26.2313 25.7321 25.2887 24.8935 24.539760 27.3999 27.2368 26.7688 26.2102 25.7162 25.2773 24.885865 28.0639 27.8803 27.3547 26.7296 26.1790 25.6917 25.258670 28.7920 28.5852 27.9945 27.2949 26.6810 26.1398 25.660575 29.5919 29.3589 28.6946 27.9108 27.2260 26.6247 26.094180 210.4725 210.2095 29.4616 28.5828 27.8184 27.1499 26.562285 211.4435 211.1462 210.3031 29.3167 28.4626 27.7188 27.067690 212.5159 212.1793 211.2275 210.1189 29.1636 28.3353 27.613095 213.7023 213.3206 212.2442 210.9965 29.9266 29.0032 28.2014

100 215.0170 214.5833 213.3638 211.9571 210.7574 29.7268 28.8358

a Units: P in bars; Ak in cm3-kg1/2-mol23/2-bar21.

in this study were calculated according to the equations given by Ananthas-wamy and Atkinson,(25) which are outlined in the Appendix. At 1 atm, thePVT properties of pure water were calculated using the equations of Kell.(23)

At higher pressures, the water equation of state of Chen et al.(27) wasused. Our calculations of the slopes agree with the values published inAnanthaswamy and Atkinson,(25) except for the values of the slope forthe apparent molal compressibilities Ak, which we believe are in error.Calculations using a simple polynomial fit of their slopes for apparent molalvolumes as a function of pressure show the same discrepancies as ourcalculations using the full equation for Ak. Therefore, we decided to useour values, which are given in Table I. For comparison purposes, Rogersand Pitzer(28) used a value of 23.931024 cm3-kg1/2 mol23/2-bar21 at 258C,which can be compared with our value of 24.031024cm3-kg1/2-mol23/2-bar21 and the Ananthaswamy–Atkinson(25) value of 24.631024 cm3-kg1/2-mol23/2-bar21.


3.3. Fitting Procedure

The specific volume and compressibility data were fitted, respectively,to Eqs. (28) and (31). The resulting Pitzer parameters [b(0)X

ss , b(1)Xss , etc. . . .]

were fitted to an equation of the form:

YXss 5 oi oj aij(T 2 TR)iP j (36)

where X is V or k, i and j take integer values from 0 to 4, TR 5 298.15 K,and P is the applied pressure in bars. The calculations were performed usingMicrosoft Excel 97 macros written in Visual Basic. The fitting program, alsowritten in Visual Basic, used a linear least-squares procedure by invertingmatrixes according to the Gauss–Jordan method. The coefficients aij resultingfrom the fits are given in Table II. The coefficients needed for the fits weredetermined using an F test. Over the entire range of salinity, temperature,and pressure, 32 coefficients were needed to fit the results, which is the samenumber used in the equation of state of seawater.(17)

The 1 atm specific volume and compressibility data as a function ofsalinity and temperature were fitted first, yielding Pitzer parameters valid atP 5 0 (1 atm or 1.013 bar). The coefficients aij for j 5 0, underlined inTable II, were determined from the specific volume data, while the valuesof aij for j 5 1 were determined from the compressibility data. The standarderror of the 1 atm fits was 3.531026 cm3-g21 in v and 0.731029 bar21 in

Table II. Coefficients of Eq. (36) for V 0, b(0)V and b(1)V Obtained from Fit of SpecificVolumesa

aij i 5 0 i 5 1 i 5 2 i 5 3 i 5 4

Vj 5 0 15.177 0.0814 22.1431023 1.5331025 23.3531027

j 5 1b 5.388231023 25.61231025 1.74431026 22.70531028 4.28310210

j 5 2 22.44431026 5.831029 28.65310210 1.66310211

j 5 3 9.02310210

j 5 4 22310213

b(0)V

j 5 0 1.03531025 23.931028 1.8631028

j 5 1b 26.52431029 29.5310211 21.797310211

j 5 2 25.8310212 3310215 1.73310214

j 5 3 21.08310214

b(1)V

j 5 0 1.47131025 22.031026

j 5 1b 22.6531028 1.56731029

j 5 2 9.4310211

a V 0 in cm3-mol21, b(i)V in kg-mol21-bar21.b From fit of compressibilities at 1 atm.


bT. Comparisons of the experimental and measured values of Vf and kf at15, 25, and 358C as a function of ionic strength are shown in Fig. 1. Thefits are quite good over the entire temperature range. As the figure shows,the values of kf are slightly underestimated at high temperatures, but theyare still well within the uncertainties.

The high pressure data were then fitted by subtracting the 1 atm volumeand compressibility parameters from the left side of Eq. (28), yielding Pitzerparameters valid over the whole range of salinity, temperature, and pressure.This was done to ensure that our pressure equations would reproduce the 1atm specific volume and compressibility data. The standard error of specific-volume fits over the entire salinity, temperature, and pressure range was8.331026 cm3-g21.

Fig. 1. Measured (circles) and fitted (line) apparent molal volumes (top) and compressibilities(bottom) of seawater vs. salinity at 15, 25, and 358C and 1 atm.


4. RESULTS OF THE DATA TREATMENT

4.1. Specific Volumes

As mentioned above, the standard error of the 1 atm fits was 3.531026

cm3-g21. A comparison of the 1 atm measured and calculated specific volumesas functions of salinity and temperature is shown in Fig. 2. They are within831026 cm3-g21. A comparison of the measured and calculated high pressurespecific volumes as functions of salinity, temperature, and pressure is shown

Fig. 2. Comparison of measured and calculated specific volumes (top) and compressibilities(bottom) of seawater at 1 atm. Contour maps showing dependence of Dv and DbT on salinityand temperature: [s(v) 5 3.531026 cm3-g21; s(bT) 5 0.731029 cm3-g21].


in Fig. 3. The deviations appear to be randomly distributed with an averageerror of 20.431026 cm3-g21. The standard deviations of the volume fits are8.331026 -g21 over the entire range of salinity, pressure, and temperature.The deviations are larger at temperatures above 308C and at applied pressuresabove 800 bars. Below these values, the calculated specific volumes do notdiffer by more than 62031026 cm3-g21 from the experimental values, witha standard deviation of 6.531026 cm3-g21. At higher temperatures and pres-sures, however, these deviations are as high as 64031026 cm3-g21.

Fig. 3. Comparison of measured and calculated specific volumes of seawater as a function ofsalinity, temperature, and pressure (s 5 8.331026 cm3-g21).


The infinite-dilution apparent molal volume of seawater can be deter-mined in several ways. The extrapolated values of the apparent molal volumesof seawater at I 5 0 yield values for the partial molal volumes of seasalt.These values can be compared with the values determined by additivity, Eq.(25), and those determined from the apparent molal fits of Millero andSchreiber(12) from the high-pressure equation of state.(17) The infinite-dilutionpartial molal volumes and compressibilities of the major components ofseawater(29) at 1 atm from 0 and 508C have been used to determine V o

ss andko

ss using

V o 5 o (mi/m)V oi 1 mBV o

B (37)

ko 5 o (mi/m)koi 1 mBko

B (38)

where the subscript i refers to ions in the solution and the subscript B to un-ionized boric acid. The results from 0 to 508C have been fitted to the equations

V oss 5 15.016 1 0.09027(T 2 TR) 2 0.0021066(T 2 TR)2 (39)

koss 5 25.3331023 1 7.231025(T 2 TR) 2 1.831026(T 2 TR)2 (40)

where TR 5 298.15 K. The values of V oss and ko

ss determined from Eqs. (39and 40) are compared with the extrapolated values in Fig. 4 along with thosedetermined from the equations of Millero and Schreiber.(12) The values ofV o

ss agree to within 60.2 cm3 -mol21 and the values of koss agree to within

60.231023 cm3-mol21-bar21. These differences are quite reasonable consid-ering that the extrapolation equations used in earlier studies were made usingan extended Debye–Huckel equation.(29,30) As we will show below, infinite-dilution values derived from pure salt solutions are also greatly dependenton the experimental data set used and on the equations used to extrapolatethe results to infinite dilution.

4.2. Compressibilities

A comparison of the 1 atm measured and calculated compressibilitiesas a function of salinity and temperature is shown in Fig. 2. The standarderror of the 1 atm fits was 0.731029-bar21. The deviations appear to berandomly distributed as a function of salinity and temperature, with an averagedeviation of 0.1531029 bar21. The apparent molal volume Pitzer coefficientscan be differentiated with respect to pressure to determine the compressibilityof seawater as a function of salinity, temperature, and pressure. A comparisonof the equation of state values(17) of bT with those determined from thedifferentiation of our Vf fit with respect to pressure over the entire range ofsalinity, temperature, and pressure is shown in Fig. 5. Our calculations agreewith the equation of state to within 60.004231026 bar21 at 1 atm and


Fig. 4. Comparison of infinite dilution apparent molal volumes and compressibilities of seasalt at 1 atm from our results with other estimates.

60.01931026 bar21 over the entire range of salinity, temperature, and pres-sure. The results below a salinity of 30 and pressures below 800 bars arewithin 60.0531026 bar21. As for the specific volumes, the deviations increaseabove 30 8C and 800 bars applied pressure. The standard deviation drops to0.01131026 bar21 below those values.

4.3. Expansibilities

From the differentiation of the Vf equation with respect to temperature,one can calculate the thermal expansibilities a 5 (1/v)(v/T )P. A comparisonof the measured(17) and calculated values of a using Eq. (10) are shown inFig. 6. The agreement is very good with a standard deviation of 0.4531026


Fig. 5. Comparison of measured and calculated isothermal compressibilities of seawater as afunction of salinity, temperature, and pressure (s 5 0.01931026 bar21).

K21 at 1 atm and 0.6331026 K21 over the entire range. Here again, thedeviations become larger above 30 8C and 800 bars. Below these values,the standard deviation is 60.4231026 K21, with an average deviation of0.0931026 K21.

4.4. Estimation of the Volume Properties of Seawater from PureSalt Properties

The estimation of the PVT properties of seawater can be made fromapparent molal properties of the components. Estimates using Young’s rule


Fig. 6. Comparison of measured and calculated thermal expansibilities of seawater as a functionof salinity, temperature, and pressure (s 5 0.6331026 K21).

were made in a number of earlier studies.(4,31–36) Although these estimateswere quite good for dilute solutions, excess properties were needed at highionic strengths.(36,37) More recently, Connaughton,(30) Monnin,(37,38) and Krum-galz et al.(39) have used the Pitzer equations to estimate the properties ofmixed electrolyte solutions. For the volume properties, this can be done usingEqs. (8), (13), and (17). The ionic values of V0

i and the Pitzer parameters forthe major sea salts are available.(26,30,37–39) A compilation of Pitzer parameterspublished by various authors is given in Table III. The quality of the datasets used by the different authors as well as the fitting procedure used will


Table III. Compilation of Volumetric Pitzer Parameters of Salts at 258C Obtained byVarious Authorsa

Electrolyte V 0 105 b(0)V 105 b(1)V 102 b(2)V 106 CV m(max) s Ref.

HCl 17.824 0.055039 20.74008 0.023984 17.48 84 27b

NaCl 16.68 1.2414 20.662 6 18 b, c

16.62 1.2335 0.43543 20.6578 6 97 2716.68 1.234 20.645 5.5 120 2916.58 1.14 1.15 20.555 4 .005d 4216.681 1.255 20.688 6 47 30

Na2SO4 11.48 4.9790 16.1 22.308 1.5 9 b, c

11.487 4.466 18.02 21.313 1.47 40 3011.787 5.308 12.34 22.794 2.2 8 3711.776 5.325 12.932 22.914 1.5 90 27

NaHCO3 23.181 21.162 17.8 1 9 37b

Na2CO3 26.48 5.98 8.16 23.25 1.7 19 37b

NaBr 23.479 0.76074 0.95252 20.34908 8 99 27b

NaB(OH)4 20.673 b

NaF 22.34 2.32766 43b

KCl 26.91 1.55 20.11 21.37 4.5 0.02d 4426.87 1.3949 0.235 20.87 4.7 3 3726.93 1.5030 21.023 4.5 121 b, c

26.848 1.2793 0.89477 20.7131 4.7 64 27K2SO4 32.167 3.348 23.8 0.65 25 37

31.98 2.14 28.85 0.4 17 b, c

32.05 22.3199 36.414 29.11 0.7 68 27KHCO3 33.448 21.9568 21.604 1.0 NR 39b

33.371 20.2705 16.95 1.0 31 3734.34 7.0283 28.4507 216.738 1.0 1 27

K2CO3 14.054 3.4471 18.043 20.824 5.6 NR 39b

12.327 3.0758 33.269 20.6468 7.6 258 27KBr 33.746 1.066 0.8337 20.7017 5.6 NR 39b

33.689 1.0259 1.1021 20.6641 5.6 152 27KB(OH)4 30.94 b

KF 7.76 2.7663 26.065 43b

MgCl2 14.4 1.6992 27.44 20.56748 6 114 b, c

14.395 1.844 28.46 20.6887 5.8 38 3014.16 1.78 26.38 20.643 NR 0.02d 4513.734 1.3833 20.357 NR 40 4614.083 1.6933 25.2068 20.5698 5.8 188 27

MgSO4 27.48 4.323 19.636 1.3405 0.86 2.4 14 b, c

27.485 5.137 13.19 1.495 2.4 45 3027.839 4.9329 14.838 1.679 0.192 2.9 20 3726.551 4.2551 18.439 0.8889 1.3198 2.5 244 27

Mg(HCO3)2 27.402 b

MgCO3 225.44 b

MgBr2 27.998 20.0468 14.569 1.0696 4.4 NR 39b

28.788 0.60798 3.1073 0.5359 4.4 171 27


Table III. Continued

Electrolyte V 0 105 b(0)V 105 b(1)V 102 b(2)V 106 CV m(max) s Ref.

MgB(OH)4 22.387 b

MgF2 223.64 b

CaCl2 17.612 1.3107 22.4575 20.1265 7.7 595 27b

18.53 1.564 210.48 20.882 7.4 0.3d 4217.419 1.3287 20.217 NR 60 46

CaSO4 24.268 b

Ca(HCO3)2 30.614 b

CaCO3 222.228 b

CaBr2 31.21 2.1496 25.2328 20.8976 5.0 NR 39b

32.3 2.6894 216.029 21.2951 5.0 276 27SrCl2 16.29 4.3451 25.404 43b

18.4 3.2044 29.8894 22.1913 3.0 NR 39SrSO4 25.59 b

Sr(HCO3)2 29.292 — — b

SrCO3 223.55 — — b

SrBr2 29.888 1.7931 13.073 21.2458 3.3 NR 39b

SrB(OH)4 24.277 b

SrF2 221.75 b

a Units: V 0 in cm3-mol21; b(i)V in kg-mol21-bar21; C V in kg2-mol22-bar21.b Parameters used in our model. The infinite dilution volumes were determined by the additivityrule using the following sequence: V 0 (Cl2) is determined from V 0 (HCl), assuming V 0 (H+) 50 cm3-mol21. Then the volume of the cations is calculated from the value for their chloridesalt [for example: V 0 (Mg21) 5 V 0 (MgCl2) 2 2 V 0 (Cl2)]. The volumes for the anions arecalculated from the volumes of their sodium salts, using the sodium value previouslydetermined.

c This work.d Error in Vf.

affect the ability of these parameters to predict the densities of electrolytesolutions. Lo Surdo et al.(40) measured the densities of the four major seasalts at different temperatures with a precision of 531026 g-cm23 and com-bined their results with an extensive literature data set. They fitted the datato a polynomial function of molality and temperature. Dedick et al.(41) didthe same for KCl and K2SO4 solutions. Since the fits of Dedick et al. are inerror, we have refitted their data. The standard deviations of the fits are givenin Table III. Figure 7 shows a comparison of the densities calculated usingeither Lo Surdo et al.(40) fitting equations (for NaCl, Na2SO4, MgCl2, andMgSO4) or our parameters (KCl and K2SO4) with the densities calculatedusing different sets of Pitzer parameters. In most cases, the Krumgalz etal.(39) parameters show several hundred ppm differences with the densitiesof Lo Surdo et al.,(40) whereas Monnin’s parameters(37) reproduce the datamuch better. Using the experimental data published by Lo Surdo et al., we


Fig. 7. Comparison of densities of the major sea salts calculated using different sets of Pitzerparameters at 258C and 1 atm. The reference density rref for NaCl, Na2SO4, MgCl2, and MgSO4

is from the smooth equation of Lo Surdo et al.,(40) and for KCl; K2SO4 is from our calculations.Symbols: solid line (Monnin); dashed line (Krumgalz); circles (this study).

redetermined the Pitzer parameters for the four major sea salts (NaCl, MgCl2,Na2SO4, and MgSO4) at 258C. Our results are given in Table III. Theyreproduce the experimental data to within 1831026 g-cm23 for NaCl, 931026

g-cm23 for Na2SO4, 11231026 g-cm23 for MgCl2, and 1431026 g-cm23 forMgSO4. The calculated densities for these salts using our parameters areshown in Fig. 7 and are compared with the values from Monnin(37,38) andKrumgalz et al.(26,39) Our results agree well with the densities calculatedusing the Monnin(37) parameters.

The 258C parameter values for the other major ions found in seawaterused in this study are also given in Table III. A comparison of the calculated


Fig. 8. Comparison between measured and calculated densities of seawater using different setsof Pitzer parameters at 258C and 1 atm.

and measured values of the density of seawater, using the parameters givenin Table III, as a function of salinity, is shown in Fig. 8. Although the resultsare not as good as those obtained by the Young’s rule method,(3,34) theequations give reasonable estimates for the densities. The difference betweenthe calculated and measured values increases with salinity and reaches amaximum of 4531026 g-cm23 at a salinity of 40. The larger deviations athigher salinities are related to the importance of the mixing parameters Qij

and Cijk. The results obtained using Krumgalz’s parameters do not reproducethe experimental data as well as Monnin’s or our parameters, as found forthe pure salt densities. Similar calculations for the compressibilities and soundvelocities in seawater using the Pitzer methods must wait for reliable Pitzerparameters for sea salts.

ACKNOWLEDGMENT

The authors wish to acknowledge the support of the Oceanographicsection of the National Science Foundation.

APPENDIX: PITZER-DEBYE-HUCKEL SLOPES

The Pitzer–Debye–Huckel slopes used in our work were taken fromAnanthaswamy and Atkinson(25) and are given by the following equations:

Af 5 (1/3) (2pNrw/1000)1/2 [e2/(DkT )]3/2 (A1)

Av 5 24RT(Af /P)T 5 6RTAf(lnD/P 2 bwT /3) (A2)


AE 5 6RTAf{(lnD/P 2 bwT /3)[1/T 2 (3/2)(1/T 1 lnD/T 1 aw/3)]

1 2lnD/TP 2 (1/3)bwT /T} (A3)

Ak 5 6RTAf[2lnD/P2 2 (1/3)bwT /P 2 (3/2)(lnD/P 2 bwT /3)2] (A4)

where N 5 6.02204531023 is Avogadro’s number, rw is the density of water,e 5 4.803242310210 esu is the charge of the electron, D is the dielectricconstant of water, k 5 1.38066310216 erg-K21 is the Boltzmann constant,aw 5 1/V(V/T ) is the coefficient of thermal expansion of water, and bwT

5 21/V(V/P)T is the isothermal compressibility of water. P, as definedearlier, is the applied pressure in bars (P 5 0 at 1 atm). The absolute pressurein bars is (P 1 P0), where P0 5 1.01325 bar.

The dielectric constant of water was calculated from the equations ofBradley and Pitzer(47):

D 5 D1000 1 C ln[(B 1 (P 1 P0))/(B 1 1000)] (A5)

where

D1000 5 U1 exp(U2T 1 U3T 2) (A6)

B 5 U7 1 U8/T 1 U9T (A7)

C 5 U4 1 U5/(U6 1 T) (A8)

The temperature coefficients are given by:

U1 5 342.79 U2 5 25.086631023

U3 5 9.469031027 U4 5 22.0525

U5 5 3115.9 U6 5 2182.89

U7 5 28032.5 U8 5 4214200

U9 5 2.1417 (A9)

At 1 atm, the equations for r0w and b0

wT (the superscript denotes the 1-atm value) were taken from Kell(23) and are given by:

1000 r0w 5 (999.83952 1 16.94518 t 2 7.9870431023 t2 2 4.61704631025 t3

1 1.0556331027 t4 2 2.805425310210 t5)/1 1 1.68798531022t)(A10)


and

106b0wT 5 (50.8849610.6163813 t11.45918731023 t2 12.00843831025 t3

25.84772731028 t4 14.10411310210 t5)/(111.96734831022t)(A11)

where t denotes the Celsius temperature.The expansibility is simply:

a0w 5 (1/V )(V/T )P 5 2(1/r0

w)(r0w /T )P (A12)

Above 1 atm rw and bwT were calculated from the equations of Chenet al.(27)

rw 5 r0w /(1 2 P/Kw) (A13)

where Kw is the secant bulk modulus for water given by:

Kw 5 K 0w 1 A8P 1 B8P2 (A14)

K0w 5 1/b0

wT 5 19652.17 1 148.183 t 2 2.29995 t2 1 0.01281 t3

2 4.9156431025 t4 1 1.0355331027 t5 (A15)

A8 5 3.26138 1 5.22331024 t 1 1.32431024 t2 2 7.65531027 t3

1 8.584310210 t4 (A16)

B8 5 7.206131025 2 5.894831026 t 1 8.69931028 t2 2 1.0131029 t3

1 4.322310212 t4 (A17)

and

r0w 5 0.99983952 1 6.7882631025 t 2 9.0865931026t2

1 1.0221331027 t32 1.3543931029 t4 1 1.47115310211 t5

2 1.11663310213 t6 1 5.04407310216 t7 2 1.00659310218 t8 (A18)

The compressibility above 1 atm is given by:

bwT 5 (K 0w 2 B8P2)/[Kw(Kw 2 P)] (A19)

and the expansibility by:

aw 5 (1/r0w)(r0

w /T ) 2 P(Kw/T )/[Kw(Kw 2 P)] (A20)


Pertinent derivatives are:

lnD/P 5 (1/D)(D/P)T 5 (1/D)[C/(B 1 P 1 P0)] (A21)

lnD/T 5 (1/D)(D/T )P 5 (1/D){D1000(U2 1 2 U3T )

1 C(U8/T 2 1 U9)[1/(B 1 P 1 P0) 2 1/(1000 1 B)]

2 [U5/(U6 1 T )2] ln[(B 1 P 1 P0)/(1000 1 B)]} (A22)

2lnD/P2 5 (1/D2)[D (2D/P2) 2 (D/P)2]

5 2{C/[D(B 1 P 1 P0)2]}(1 1 C/D) (A23)

2lnD/TP 5 [D(2D/TP) 2 (D/T )(D/P)]/D2 (A24)

where

2D/TP 5 {[B 1 (P 1 P0)](C/T ) 2 C(B/T )}/(B 1 P 1 P0)2 (A25)

bwT /P 5 2{1/[Kw(Kw 2 P)]}

3 {2B8P 1 bwT[2Kw(Kw/P) 2 Kw 2 P(Kw/P)]} (A26)

bwT /T 5 {1/[Kw(Kw 2 P)]}

3 [(K0w /T ) 2 P2(B8/T ) 2 bwT(2Kw 2 P)(Kw/T )] (A27)

As mentioned previously, our calculations agree with those of Ananthas-wamy and Atkinson up to the last given significant figure for all the slopes,except for Ak. Our calculated values for Ak are given in Table I.

REFERENCES

1. C. Chen and F. J. Millero, Deep-Sea. Res. 23, 595 (1976).2. F. J. Millero, Ocean Sci. Eng. 7, 403 (1982).3. F. J Millero, in Marine Chemistry in the Coastal Environment, T. M. Church, ed., Chap.

2, ACS Symp. Ser. 18 (American Chem. Soc., Washington D.C., 1975).4. F. J. Millero, D. Lawson, and A. Gonzalez, J. Geophys. Res. 81, 1177 (1976).5. F. J. Millero, Aquatic Geochem. 6, (2000).6. F. J. Millero, L. D. Hansen, and E. V. Hoff, J. Marine. Res. 31, 21 (1973).7. W. C. Duer, W. H. Leung, G. B. Oglesby, and F. J. Millero, J. Solution Chem. 5, 509 (1976).8. F. J. Millero and W. H. Leung, Amer. J. Sci. 276, 1035 (1976).9. F. J. Millero, Ocean Sci. Eng. 8, 1 (1983).

10. E. A. Guggenheim, Phil. Mag. 19, 588 (1935).11. G. N. Lewis and M. Randall, Thermodynamics, 2nd edn., K.S. Pitzer and L. Brewer, eds.

(McGraw Hill, New York, 1961).12. F. J. Millero and D. R. Schreiber, J. Marine Res. 41, 323 (1983).13. K. S. Pitzer, in Activity Coefficients in Electrolyte Solutions Vol. 1, R. M. Pytkowicz, ed.,

(CRC Press, Boca Raton, 1979).


14. F. J. Millero, C. Chen, A. Bradshaw, and K. Schleicher, UNESCO Tech. Paper MarineSci. 38, 99 (1981).

15. F. J. Millero and A. Poisson, UNESCO Tech. Paper Marine Sci. 38, 19 (1981).16. D. Pierrot et al., in preparation.17. F. J. Millero, C. Chen, A. Bradshaw, and K. Schleicher, Deep-Sea Res. 27, 255 (1980).18. F. J. Millero and A. Poisson, Deep-Sea Res. 28, 625 (1981).19. V. A. Del Grosso and C. W. Mader, J. Acoust. Soc. Amer. 52, 961 (1972).20. F. J. Millero, A. Gonzalez, and G. K. Ward, J. Maine Res. 34, 61 (1976).21. F. J. Millero, G. Perron, and J. E. Desnoyers, J. Geophys. Res. 78, 4499 (1973).22. F. J. Millero, Chemical Oceanography, 2nd edn. (CRC Press, Boca Raton, 1996).23. G. S. Kell, J. Chem. Eng. Data 20, 97 (1975).24. P. Saunders, WOCE Newsl. p. 10 (1990).25. J. Ananthaswamy and G. Atkinson, J. Chem. Eng. Data 29, 81, (1984).26. B. S. Krumgalz, R. Pogorelsky, Ya. A. Losilevskii, A. Weiser, and K. S. Pitzer, J. Solution

Chem. 23, 849 (1994).27. C. Chen, R. A. Fine, and F. J. Millero, J. Chem. Phys. 66, 2142 (1977).28. P. S. Z. Rogers and K. S. Pitzer. J. Phys. Chem. Ref. Data 11, 15 (1982).29. F. J. Millero, Geochim. Cosmochim. Acta 46, 11 (1982).30. L. M. Connaughton, P. J. Hershey, and F. J. Millero, J. Solution. Chem. 15, 989 (1986).31. F. J. Millero, Deep-Sea Res. 20, 101 (1973).32. F. J. Millero, J. Solution Chem. 2, 1 (1973).33. F. J. Millero and F. K. Lepple, Marine Chem. 1, 89 (1973).34. F. J. Millero, in The Sea, Ideas and Observations, Vol. 5, E. D. Goldberg, ed. (Wiley-

Interscience, New York, 1974), pp. 3–80.35. F. J. Millero and P. V. Chetirkin, Deep-Sea Res. 27, 265 (1980).36. F. J. Millero, A. Mucci, J. Zullig, and P. Chetirkin, Marine Chem. 11, 463 (1982).37. C. Monnin, Geochim. Cosmochim. Acta. 53, 1177 (1989).38. C. Monnin, Geochim. Cosmochim. Acta 54, 3265 (1990).39. B. S. Krumgalz, R. Pogorelsky, and K. S. Pitzer, J. Solution Chem. 24, 1025 (1995).40. A. Lo Surdo, E. M. Alzola, and F. J. Millero J. Chem. Thermodyn. 14, 649 (1982).41. E. A. Dedick, J. P. Hershey, S. Sotolongo, D. J. Stade, and F. J. Millero, J. Solution Chem.

19, 353 (1990).42. A. Kumar and G. Atkinson, J. Phys. Chem. 87, 5504 (1983).43. F. J Millero, The Physical Chemistry of Natural Waters (Wiley, New York, 2000).44. A. Kumar, J. Chem. Eng. Data 31, 19 (1986a).45. A. Kumar, J. Chem. Eng. Data 31, 21 (1986b).46. C. Monnin, J. Solution Chem. 16, 1035 (1987).47. D.J. Bradley and K. S. Pitzer, J. Phys. Chem. 83, 1599 (1979).

The apparent molal volume and compressibility of seawater fit to the Pitzer equations

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Transcript of The apparent molal volume and compressibility of seawater fit to the Pitzer equations