fei stel 2007

7/25/2019 Fei Stel 2007

1/25

A GibbsPitzer function for high-salinityseawater thermodynamics

Rainer Feistel a,*, Giles M. Marion b

a Leibniz-Institut fur Ostseeforschung, Warnemunde, Seestrasse 15, 18119 Rostock, Germanyb Desert Research Institute, Reno, NV, USA

Received 8 December 2005; received in revised form 19 March 2007; accepted 14 April 2007Available online 5 May 2007

Abstract

A high-salinity Gibbs function for seawater is derived from Pitzer equations of the sea salt components, in conjunctionwith the 2003 Gibbs function of seawater for low salinities. Various properties, computed from both formulations by ther-modynamic rules, are compared with each other, and with high-salinity measurements. The new GibbsPitzer functionpresented in this paper is valid in the range 0110 g kg1 in absolute salinity, 7 to +25 C in temperature, and 0100 MPa in applied pressure. The formulation is expressed in the International Temperature Scale 1990 (ITS-90), andis consistent with the International Standard for Fluid Water (IAPWS-95), and with the 2005/2006 equations of stateof ice Ih.

2007 Elsevier Ltd. All rights reserved.

Keywords: Seawater; High salinity; Thermodynamic properties; Gibbs function; Pitzer equations

1. Introduction

Concentrated seawater solutions are found in exceptional natural waters like the Western AustralianShark Bay, in salt works and desalination plants, and particularly in the brine pockets of very cold seaice. Only very few experimental investigations of high-salinity seawater are known to the authors (Higashi

et al., 1931; Bromley et al., 1967, 1970; Bromley, 1968; Connors, 1970; Grunberg, 1970) and a comprehen-sive equation of state is not available. There is an apparent description gap between the thermodynamicformulations of seawater and sea ice properties by (i) the Gibbs function of seawater (Feistel and Hagen,1998; Feistel and Wagner, 2005; Feistel et al., 2005) on the one hand, and (ii) the brine chemistry models(Marion and Farren, 1999; Marion et al., 1999; Marion, 2001) which are based on so-called Pitzer equationsfor the osmotic and activity coefficients, on the other hand. The first one is highly accurate for absolutesalinities up to 50 g kg1 at atmospheric pressure, corresponding to brine salinity of sea ice down to about

0079-6611/$ - see front matter 2007 Elsevier Ltd. All rights reserved.

doi:10.1016/j.pocean.2007.04.020

* Corresponding author. Tel.: +49 381 5197 152; fax: +49 381 5197 4818.E-mail addresses:[email protected](R. Feistel),[email protected](G.M. Marion).

Progress in Oceanography 74 (2007) 515539

Progress inOceanography

www.elsevier.com/locate/pocean
mailto:[email protected]:[email protected]:[email protected]:[email protected]

7/25/2019 Fei Stel 2007

2/25

3 C, and is restricted to standard sea salt composition, while the latter one covers much higher salinitiesand even varying salt compositions.

The present work exploits the Pitzer equations of brine chemistry for medium and high salinities, extendingthem into the form of a GibbsPitzer function of seawater (abbreviated furtheron as GP), which can be usedas a thermodynamic potential similar to the Gibbs function of seawater at lower salinities ( Feistel, 2003abbre-

viated as F03 in the following), and studies the thermodynamic properties of the resulting high-salinityformulation.A particular problem of this crossover formulation is a joint measure of the salt concentration. In

oceanography, salinity is usually expressed as a dimensionless practical salinity on the scale of 1978,PSS-78. This scale is undefined for salinities higher than 42. On the other hand, even at high concentra-tions Pitzer models permit the computation of the absolute salinity from the concentrations and atomicweights of the dissolved species, but absolute salinity, in turn, is not well-defined for standard ocean sea-water (Jackett et al., 2006). A quantitative relation between both measures is estimated (Section 2, Eqs.(2.14) and (2.15)) within an uncertainty sufficient for the accuracy requirements of this paper. For thisreason we formally define here the specific unit psu by 1 psu 1.004867 g kg1, approximately expressingpractical salinity in terms of absolute salinity units, or vice versa. This way all salinity values can be writ-ten in units of either psu or of g kg1 over the full range of concentrations considered in this paper. We

have chosen psu for this purpose for strict numerical consistency with the highly accurate oceanographicdata and formulas.

The extension procedure from the Pitzer equations to the Gibbs potential relies on the fact that the lowestorder terms of the power expansion of the Gibbs function g(S, t,p) of seawater with respect to ion concentra-tion, or salinity, S,

gS; t;p g0g1Sln Sg2Sg3S3=2 g4S

2 1:1

have clear physical meanings. The coefficients giare functions of temperature tand pressure p. The interceptg0(t,p) describes pure water properties and will be taken here from F03 in its zero-salinity limit, which is con-sistent with the scientific pure water standard IAPWS-95 (Wagner and Pru, 2002) over the Neptunian rangeof temperatures and pressures. The linear term g2(t,p) is determined by ionsolvent interaction at infinite dilu-

tion, describing, e.g. molar volumes and entropies, and will be combined from both the Pitzer model andexperimental data of seawater at low and high concentrations. The ideal-solution termg1and the DebyeHuc-kel termg3are supposed to be the same for both the F03 Gibbs function and the Pitzer equations, and can inprinciple be taken from either of them, but practically should be chosen consistently with the Pitzer terms. Allhigher-order terms g4 . . . are strongly influenced by ionion interactions, and will be adopted here entirelyfrom the Pitzer equations.

In Section 2, we introduce the fundamental definitions and quantities required for mixed electrolytesand seawater. In Section 3, we define a single activity potential function of seawater, derived from theGibbsDuhem equation, which comprises all information on ionion interaction forces of the electrolyte.In Section 4, the Pitzer equations available for the particular chemical constituents of sea salt are brieflyexplained. At various Stp points, the values of the activity potential are computed from these equationsand an activity potential function is fitted to these points. In Section 5, this seawater activity potential iscombined with Pitzer functions for the molar volumes, the Gibbs function F03 for pure water properties,and heat capacity measurements, to form the new GibbsPitzer function GP for high-salinity seawater.The required mathematical relations are given. In Section 6, various thermodynamic properties derivedfrom GP are compared with F03 for medium salinities, and the behaviour at higher salinities and theagreement with available measurements is discussed. In the Appendix, tables of various constants andcoefficients are given.

The GibbsPitzer function presented in this paper is valid in the range 0110 psu in practical salinity (PSS-78, Unesco, 1981) or its high-salinity equivalent, 7 to +25 C in temperature, and 0100 MPa in appliedpressure, relative to normal pressure. The new function is expressed in the International Temperature Scale1990 (ITS-90, Preston-Thomas, 1990) and is consistent with the international standard for fluid water,IAPWS-95 (Wagner and Pru, 2002) and the Gibbs potential of ice Ih in its first and second, slightly improved

version (Feistel and Wagner, 2005, 2006; IAPWS, 2006).

516 R. Feistel, G.M. Marion / Progress in Oceanography 74 (2007) 515539
http://-/?-http://-/?-

7/25/2019 Fei Stel 2007

3/25

2. Basic terms and definitions

We consider a mixed electrolyte solution consisting ofN0solvent molecules and N1, . . . , NXdissolved par-ticles ofXdifferent chemical species, written in compact form as particle number vector N = {N0, . . . , NX}, atabsolute temperature T and absolute pressure P. The particle numbers N can be expressed by molalities

m= {m0,

. . .

,m

X} (moles of solvent or solute per mass of solvent), asma

Na

N0M0; a 0;. . .;X 2:1

Maare the molecular weights of the species a= 0, . . ., X. The total molality mof the solute in the solution is

mXXa1

ma: 2:2

The concentration, c, is defined as the mass of dissolved salt per mass of solution as

c PX

a1NaMaPXa0NaMa

2:3

the mole fractions x= {x1, . . . , xX} of the solute as

xa NaPX

b1Nb

ma

m ; a 1;. . .;X 2:4

and the mass fractions waof the solute as

wa NaMaPX

b1NbMb; a 1 . . .X: 2:5

The total mass Mof the solution sample is then given by (NA is Avogadros number),

M 1NA

XXa0

NaMa: 2:6

We define the total number Nof dissolved particles in the sample as

NXXa1

Na: 2:7

Inserting Nafrom(2.4)and Mfrom(2.6), and introducing the average molecular weight of the solute

hMi XXa1

xaMa 2:8

we get for Nfrom Eq.(2.3),

NcMNA

hMi 2:9

and the individual particle numbers, from (2.9) and (2.4),

Na xaNxacMNA

hMi; a 1;. . .;X: 2:10

Ifza is the valence of the dissolved ion a, the electro-neutrality of the solution requires

0 XX

a1

xaza: 2:11

R. Feistel, G.M. Marion / Progress in Oceanography 74 (2007) 515539 517

7/25/2019 Fei Stel 2007

4/25

The valence factor z2 of the electrolyte is given by

z2 XXa1

xaz2a 2:12

and its ionic strength Ias

I12

XXa1

maz2a

12

mz2: 2:13

In the case of standard seawater, the chemical composition of the solute is assumed to be given by fixedvalues of its mole fractions xa, valences za, and molecular weights Ma. Practical salinity S, as introducedby the Practical Salinity Scale 1978 (PSS-78,Lewis and Perkin, 1981; Unesco, 1981) is defined by the conduc-tivity of specific natural ocean water samples in such a way that a linear relation between salt concentration c(i.e. absolute salinity) and S(i.e. practical salinity) can be assumed (Millero and Leung, 1976; Feistel, 2003;Jackett et al., 2006)

c n S: 2:14

There are several proposals for the composition of sea salt in the literature which we refrain from discussinghere; unfortunately, there is no such common standard specified yet. If a composition is chosen as given inTable A.2for the seawater model used by Feistel (2003), and S is expressed in the practical salinity scale,the factor evaluates to

n c=S 1:004867 g kg1 psu1 1:004867103 psu1: 2:15

Introducing the number of dissolved sea salt particles NS per psu and kg seawater

NS nNA

hMi; 2:16

which takes the valueNS= 1.926845 1022 kg1 psu1 for the composition model ofFeistel (2003)we finally

have Naexpressed by S, M, and the composition x as

Na xaNSSM; a 1;. . .;X: 2:17

Summing up these numbers, we find for the total number Nof dissolved particles in the given sample

NXXa1

Na NSSM: 2:18

Now we derive the corresponding expression for the remaining number of water molecules, N0. We write Eq.(2.6)in the form

MNA N0M0NhMi N0M0NSSMhMi 2:19

and get for N0

N0 MNANSShMi

M0

MNA

M01nS: 2:20

Note thatc =nS< 1 is the mass fraction of salt in the seawater sample, while 1 nS= 1000 g kg1 c is thesamples mass fraction of water. For the solute molalities we have

ma Na

N0M0xa

NS

NA

S

1nS; a 1;. . .;X 2:21

for the total molality we get

mNS

NA

S

1nS 2:22

and for the ionic strength


7/25/2019 Fei Stel 2007

5/25

Iz2

2

NS

NA

S

1 nS: 2:23

Quantitatively, we find from Table A.1 for the ratios NS/NA= 31.99601 mmol kg1 psu1 and

12z2NS=NA 9:91980 mmol kg

1 psu1.In principle, the molality of mixed strong electrolytes like seawater can be defined in different ways. For

example, a solution with 12 1023 ions of Na+, 12 1023 ions Cl, 6 1023 ions Mg2+, and 6 1023 ionsSO24 , contains 2 + 2 + 1 + 1 = 6 mol of ions in our notation (2.2). It terms of neutral molecules, however,there is no unique total molality. Depending on which compounds are supposed to be originally dissolved,it contains 2 mol of NaCl and 1 mol of MgSO4, i.e. 3 mol of salt, or 1 mol of MgCl2and 1 mol of Na2SO4,i.e. 2 mol of salt, or any combination in between. This way, computed from Eq. (2.22), standard seawaterwith S= 35 psu has the molality m = 1.16068 mol kg1, in agreement withGrasshoff et al. (1999, p. 568), butcontrary to m = 0.58052 mol kg1 calculated from formula (4.61) of Millero (2001, p. 253), andm= 0.60597 mol kg1 reported byMillero (2006, p. 62).

The various quantities introduced in this section, describing the concentration and the composition of theelectrolyte seawater, are required for the thermodynamic relations considered in the following parts.

3. GibbsDuhem equation

The thermodynamic properties of a given electrolyte can be derived by thermodynamic rules from its GibbsfunctionG, i.e. its Gibbs energy or free enthalpy, expressed in terms of temperatureT, pressureP, and particlenumbers N= {N0, . . . , NX}:

GN; T;P XXa0

NalaN; T;P 3:1

Here,laare the molecular chemical potentials of the components, which can be written by means of practicalactivity coefficients,ca(m, T, P), as (Falkenhagen et al., 1971)

la l0aT;P kTlnmaca; a 0;. . .;X: 3:2

Here,kis the Boltzmann constant (Table A.1). The activity of the solvent is usually expressed by the practicalosmotic coefficient /, in the form (Falkenhagen et al., 1971)

/ lnm0c0

M0m ; 3:3

and the activity of the solute by the mean activity coefficient, cas

ln c 1

m

XXa1

ma ln ca XXa1

xaln ca: 3:4

At constant temperature and pressure, the fundamental differential ofGis

dGN; T;P XXa0

ladNa; 3:5

which, together with(3.1), leads to the GibbsDuhem equation

0XXa0

Na dlaN; T;P 3:6

Next, we reformulate this equation in terms of osmotic and activity coefficients. Inserting (3.2)into(3.6)atconstant Tand P, we obtain

0 N0 d lnm0c0 XX

a1

Na d lnmaca; 3:7


7/25/2019 Fei Stel 2007

6/25

This equation can be rearranged eventually into the well-known Bjerrum relation

dm1 / m d ln c 0 3:8

If the composition of the solute is fixed, as in the case of sea salt, and its total concentration is variable,expressed by salinity S, molality m, or ionic strength I, it is useful to introduce an activity potentialfunction w,

wm; T;P 1/ ln c 3:9

which vanishes for ideal solutions,/= 1,c= 1. We call wa potential function since osmotic and activity coef-ficients can be obtained from it by partial derivatives, as follows from (3.8):

/ 1 m ow

om

x;T;P

I ow

oI

x;T;P

S1nS ow

oS

x;T;P

3:10

ln c o mw

om

x;T;P

oIw

oI

x;T;P

1nS2 oSw=1nS

oS

x;T;P

3:11

The fact that the osmotic and the activity coefficient are not independent functions, but can be derived from asingle function by Eqs.(3.10) and (3.11), is thus a direct consequence of the GibbsDuhem Eq.(3.6). The pres-sure derivative ofw is related to the excess volume, Eq. (5.15), which we shall consider later, and its temper-ature derivative to the excess entropy and the mixing heat, Eq.(6.9).

The non-ideal behavior of the solution seawater is thus entirely described by only its activity potentialfunction w, which we are going to determine quantitatively in the following section.

4. Activity potential of seawater

To estimate the seawater activity potential function, Eq.(3.9), it is necessary to calculate the osmotic coef-ficient (/) and the activity coefficients (c) of individual species, Eq.(3.4). In the Pitzer approach, the osmoticcoefficient is given by

/ 1 2P

mi

A/I3=2

1 bI1=2

XXmcmaB

/caZCca

XXmcmc0 U

/cc0

XmaWcc0a

XX

mama0 U/aa0

XmcWcaa0

XX

mnmckncXX

mnmaknaXXX

mnmcmafn;c;a

4:1

The activity coefficient for a cation M is given by

lncM z2MF

Xma2BMaZCMa

Xmc2UMc

XmaWMca

XXmama0WMaa0

zMXXmcmaCca2XmnknM XXmnmafnMa 4:2Similarly, the activity coefficient for an anion X is given by

lncX z2XF

Xmc2BcXZCcX

Xma2UXa

XmcWcXa

XXmcmc0Wcc0X

jzXjXX

mcmaCca2X

mnknXXX

mnmcfncX 4:3

And finally, the activity coefficient for a neutral species N is given by

lncN X

mc2kNc X

ma2kNa XX

mcmafNca 4:4

In these equations, Fis defined as

F fc XXmcmaB0ca XXmcmc0U0cc0 XXmama0U0aa0 4:5

http://-/?-http://-/?-

7/25/2019 Fei Stel 2007

7/25

where

fc A/I1=2

1bI1=2

2

b ln1bI1=2

4:6

Zis defined as

ZX

mijzij 4:7

and I(the ionic strength) is defined in Eq. (2.13).In these equations,n,c, andarefer to neutral, cation, and anion species, A/is the DebyeHuckel parameter

(Archer and Wang, 1990) andB,C,U,W,k, andfare interaction parameters among the various solutes. For amore complete description of these relationships, see Pitzer (1991, 1995). For the specific interaction param-eters actually used in the implementation of the Pitzer approach (FREZCHEM model), see Spencer et al.(1990); Marion and Farren (1999); Marion (2001, 2002) and Marion et al. (2005).

To fully implement the GibbsPitzer function requires a model that can estimate / and c as functions oftemperature and pressure. The Pitzer interaction parameters (B, C, r, W, k, and f) are written as functionsof temperature (see the above references). The pressure dependence of equilibrium constants, activity coeffi-cients, and the activity of water [aw=f(/)] must be specified to estimate the pressure effect on seawater ther-modynamics. The pressure dependence of equilibrium constants (K) is given by

ln KP

KP0

DV0r PP0

RT

DK0r P P02

2RT ; 4:8

where Ris the gas constant

DV0r X

V0i nV0H2O

V0MXcr; 4:9

and

DK0r X

K0i nK0H2O

K0MXcr: 4:10

In these equations, V

0

i and K

0

i are the molar volumes and compressibilities of chemical species involved in areaction (r).The pressure dependence of the activity coefficient (c) is estimated by

ln cPi

cP0

i

!

Vi V0iP P

0

RT ; 4:11

where Vi V0iis calculated for cations (M) and anions (X) using the following equations:

VMV0M z

2Mf2RT

Xma B

vMa

Xmczc

CvMa

h iRT

XXmcma z

2MB

vca

0 zMCvca

4:12

VX V0Xz

2Xf2RT

Xmc B

vcX

Xmczc

CvcX

h iRT

XXmcma z

2XB

vca

0 jzXjCvca

4:13

The volumetric interaction parameters in these equations are defined as

Bvca dBca

dP

4:14

and

Cvca dCca

dP

4:15

Finally, the pressure dependence of the activity of water is given by

ln aPw

aP0

w Vw V

0wP P

0

RT

4:16


7/25/2019 Fei Stel 2007

8/25

where

Vw Mw

1000 g kg11000 g kg1

PmiMi

q

XmiVi

Mw1

PmiMi

q XmiVi 4:17

q 1000 g kg1

PmiMi

1000g kg1

qw

PmiV

0i V

ex

1P

miMi

vw P

miV0i V

ex; 4:18

and

Vex

RT

Av

RT

1

bln1bI1=2 2

XXmcma B

vca

Xmczc

Cvca

h i 4:19

In Eqs.(4.16)(4.19),Miis the molecular weight,q is the seawater density, andqw is the density of pure water.

In principle, the mean molar volume of sea salt in (4.18)

V0 1

m XmiV0i 4:20

should be independent of salinity, however, there is a slight salinity dependence in the FREZCHEM model.For example, the FREZCHEM values at S= 35 psu and S= 70 psu (P= 100 MPa and t= 25 C) areV0 10:1761 cm3 kg1 and V0 10:1726 cm3 kg1. This dependence arises because of the way bicarbonateand carbonate species are distributed. At S= 70 psu, the alkalinity (equivalents/kg) increases twofold com-pared to S= 35 psu. The ionic strength is different between 35 and 70 psu and therefore the activity coeffi-cients are different. Consequently, so is the molal distribution of HCO3, CO3, CaCO

03, and MgCO

03 because

this distribution is based on equilibrium constants. In estimating the density, the denominator summationterm of(4.18)does not include CaCO03 and MgCO

03.

Given the activity awand the molecular weight Mwof water (Table A.2), the osmotic coefficient is calcu-lated with the Eq. (3.3):

/ lnaw

Mw m 55:50844 mol kg1lnawP

mi 4:21

For a more complete description of these pressure calculations, see Marion et al. (2005).The implementation of the Pitzer approach is through the FREZCHEM model (Marion and Farren, 1999;

Marion, 2001, 2002; Marion et al., 2003, 2005). The molalities of seawater constituents (mi) were estimated byEq.(2.4)

mi xim 4:22

wherexiis the mole fraction (Table A.2) andmis the seawater molality (m= 1.16072 mol kg1 atS= 35 psu).

These calculated molalities are identical to the seawater molalities given by Millero and Sohn (1992) forS= 35 psu. The seawater molalities at S= 35 psu were multiplied by S/35 psu to estimate molalities at S.

There are several minor seawater species that are not part of the FREZCHEM model including Sr 2+, Br,F, BOH4, and H3BO

03 (Table A.2). In applying the FREZCHEM model to simulate seawater properties,

we lumped Sr2+ with Ca2+, Br and F with Cl, an d BOH4 with alkalinity HCO3 2CO

23

OH BOH4. The neutral H3BO03 species was included with other neutral species such as CO2(aq).

In addition to lumping constituents, there are several other differences in how the Gibbs model (F03)and the Pitzer model (FREZCHEM) handled compositions. The FREZCHEM model explicitly recognizesthe ion-pairs, CaCO03 and MgCO

03, while the F03 model only deals with stoichiometric (total) concentra-

tions. This difference leads to minor differences in calculated ionic strengths. For example, at S= 35 psu,the F03 model gives I= 0.7226 mol kg1, while the FREZCHEM model gives I= 0.7223 mol kg1. Usingthe FREZCHEM seawater model requires a specification of the partial pressure of carbon dioxide (PCO2 ),which in our simulations was assigned a constant value of 36 Pa. Solution equilibrium with CO 2(g) leadsto CO2(aq), which is a minor species that is explicitly included in the FREZCHEM model, but is missing

from the F03 model (Table A.2).

http://-/?-http://-/?-http://-/?-http://-/?-

7/25/2019 Fei Stel 2007

9/25

In none of the simulations reported in this work were solid phases allowed to precipitate. Carbonate min-eral equilibria were excluded from the FREZCHEM simulations to prevent carbonate precipitation from nor-mally supersaturated seawater. The lower temperature limit in our simulations was set by the freezing point ofseawater (ice formation). The upper concentration limit (S= 110 psu) is where gypsum starts to precipitate.

In principle, the activity potential wfor fixed chemical composition x,

wS; t;p 1/mx; T;P XXa1

xa ln camx; T;P 4:23

is given this way as an analytical function of molality, temperature, and pressure, but this function proved tobe too complicated for practical use in oceanography. Therefore, regular grid points (S,T,P) were computednumerically, and a polynomial

wS; t;p Xi;j;k

Qijkxiyjzk 4:24

was fitted to these points, using the dimensionless variablesx ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

S=40 psup

,y =t/40 C, andz =p/100 MPa.Here,t is Celsius temperature, andp is gauge pressure, relative to normal pressure. The coefficients Qijkof thefit are given inTable A.5in Appendix. The rms of the fit was 9 105.

5. Gibbs function of seawater

Now that we have an activity potential wof seawater quantitatively available for salinities up to 110 psu, weneed to build this function into the Gibbs potential of seawater in a suitable way to be able to compute arbi-trary thermodynamic equilibrium properties from it. For this purpose, we start from the specific Gibbs energy,given by Eqs.(3.1) and (3.2)

gS; t;p 1

MGN; T;P

1

M

XXa0

Nal0aT;P kTlnmacam; T;P 5:1

After substituting particle numbers by molar fractions and salinity, and separating the solvent term, this equa-tion becomes

gS; t;p NA

Mw1nSl00T;P NSSkT/m; T;P NSS

XXa1

xal0aT;P kTlnmacam; T;P 5:2

Rearranging the terms, we obtain the final formula

gS; t;p gwt;p SCt;p NSSkT ln S

1nS wS; t;p

; 5:3

where we have introduced the Gibbs function of pure water,

gw

t;p

NA

Mw l0

0T;P; 5:4

and a function C(t,p), collecting all remaining linear salinity terms,

Ct;p NSkT lnNS

NA 1

ngwt;p NS

XXa1

xal0akTlnxa: 5:5

Both these functions describe properties of seawater at infinite dilution, and are independent of the activitypotential function. The Gibbs function of pure water can be obtained from the IAPWS-95 standard ( Wagnerand Pru, 2002), or from the Gibbs function of seawater in the zero-salinity limit, which is consistent withIAPWS-95 in the Neptunian range of t and p, as

gwt;p g0; t;p: 5:6

http://-/?-http://-/?-

7/25/2019 Fei Stel 2007

10/25

The second function,C(t,p), is related to the molar entropy and molar volume of sea salt. It can in principle beobtained from the linear salinity terms of the Gibbs function F03, which has the form,

gF03S; t;p 1 J=kgXj;k

g0jk g1jkx2 lnx

Xi>1

gijkxi

( )yjzk 5:7

expressed in dimensionless variablesx ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

S=40 psup

,y =t/40 C, andz =p/100 MPa. Due to Plancks theoryof ideal solutions, its coefficients g1jkvanish for j> 1 andk> 0, thus being consistent with Eq.(5.3). Then,C(t,p) can be computed from Eq. (5.3)as

Ct;p limS!0

gS; t;p g0; t;p NSkTSln S

S : 5:8

The function C(t,p) we determined from(5.7)and (5.8)showed that the F03 and GP models are fairly con-sistent formulations. In particular for heat capacity and density, however, the GP results could be improvedsignificantly by an alternative derivation ofC(t,p), described below.

We first consider the density formula (4.18),

v

vw PmiV0i Vex1

PmiMi : 5:9

We want to compare this expression, seen as a power series in S, with the density formula which follows fromthe GibbsPitzer equation of state(5.3)

vS; t;p og

op

S;t

vwt;p SoCt;p

op NSSkT

owS; t;p

op : 5:10

We rewrite the denominator of(5.9)as

1X

miMi 1mhMi 1mnNA

NS

1

1nS 5:11

and get from(5.9)using Eq.(4.20)

v 1 nSvw mV0 Vex : 5:12

Assuming thatvw and V0 do not depend on S, we obtain the series expansion

v vw S NS

NAV0 nvw

1nSVex : 5:13

Thus we get, comparing equal powers ofSfrom (5.13)with those from(5.10),

oCt;p

op

NS

NAV0 nvw 5:14

owS; t;p

op

1nS

NSSkT

Vex Vex

mRT; 5:15

with the gas constant R=NAk. We have determined V0 as a tppolynomial, fitted to regular grid points of

V0 1m

PmiV

0i computed with the FREZCHEM model at S= 45 psu. Upon integration of(5.14)over pres-

sure, we find eventually for C(t,p) the expression,

Ct;p Ct; 0 ngwt; 0 NS

NA

Z p0

V0 dpngwt;p 5:16

The remaining unknown function, C(t,0), which is related to the normal-pressure molar entropy of sea salt,can be obtained from experimental data of heat capacity. We have fitted 3 coefficients of its temperaturepolynomial to 93 measurements of Bromley et al. (1967) for La Jolla seawater with salinities up to117.4 psu and temperatures up to 25.5 C (scale 1948), resulting in an rms of 6 J kg1 K1, and to 36 measure-ments ofMillero et al. (1973)with salinities up to 40 psu and temperatures up to 25 C (scale 1968) with an

http://-/?-http://-/?-

7/25/2019 Fei Stel 2007

11/25

rms of 7 J kg1 K1. Millero et al.s (1973) data were converted from chlorinities Cl to salinities by theformula S/psu = 1.80655Cl/(g kg1), taken fromLewis and Perkin (1981).

The corresponding tppolynomials for gw and Cderived by means of Eqs. (5.6) and (5.16), and the fit toheat capacities, are given inTables A.3 and A.4. This way, the desired Gibbs function, Eq. (5.3), is entirelydetermined. Both functions gw and Care depending on arbitrary linear functions of temperature which rep-

resent the absolute energy and the absolute entropy of water and of salt. These four numbers cannot be deter-mined from measurements and can either be specified at some tpreference state at practical convenience, orderived from additional theorems like the Third Law (Fofonoff, 1962; Feistel, 2003; Feistel and Wagner,2006). The pure water functiongw taken from F03 automatically obeys these IAPWS-95 reference state prop-erties at the triple point. In order to be consistent with F03, the related two coefficients, R00and R10in TableA.4, ofC were computed from the reference state conditions at the standard ocean, which read

g35; 0; 0 0; h35; 0; 0 0: 5:17

Supposed the Gibbs function to be given, we mention that the osmotic coefficient expresses the differencebetween the chemical potentials of pure water, gw, and of water in seawater, lw=gS(og/oS)t,pas

gwt;p lwS; t;p mRT/S; t;p: 5:18

Hence, /can be computed from the Gibbs function gof seawater by means of the formula:

/ 1nS

NSSkT gS; t;p g0; t;p S

o

oSgS; t;p

; 5:19

which follows directly from (5.18), or equivalently, from solving Eq.(5.3)for w and taking the S-derivative(3.10). Because of Eq.(5.18),/is the responsible quantity for the determination of equilibria between seawaterand the different phases of water.

6. High-salinity properties of seawater

With the formulas derived in the previous sections, the quantitative properties of seawater at high salinitiescan now be discussed again here. For this purpose, we will compare in this section the newly derived GibbsPitzer function (GP) of this paper with the 2003 Gibbs function (F03) in the validity range of F03, up to50 psu at normal pressure and 40 psu at higher pressures. To quantify this comparison, we report the rootmean square (rms) deviation between both formulas over an Stp test cube covering 540 psu, 025 C,and 0100 MPa for the parameters selected, separately for the normal pressure slice and the entire cube. Sup-posed F03 being more accurate than GP in the low-salinity range, these rms values can be considered as anuncertainty estimate of GP in this range. In the salinity range up to 40 psu (high pressure) and 50 psu (normalpressure),Feistel (2003)had compared the F03 formulation with 2050 points from 20 different groups of data;this extensive data set will not be discussed repeatedly here. For higher salinities, we will compare both func-tions in order to see how well F03 may be extrapolated beyond its assumed validity limits, which are markedby solid vertical lines in the figures.

Only few experimental data sets for thermodynamic properties of high-salinity seawater are known to the

authors.Grunberg (1970)had measured, among other quantities, vapour pressure of natural seawater, anddensity and heat capacity of artificial, Ca-free seawater. The composition of the latter is given in TableA.2. Grunbergs (1970)data were plotted into the corresponding diagrams below with their original temper-ature and salinity values without conversion to modern scales.

Bromley et al. (1967)had measured heat capacities of La Jolla seawater, Table A.2. His salinity factor isreported as c/S= 34.45 g kg1/(34.3 psu) = 1.004373 g kg1 psu1. We have converted his calories to Joulesby the factor 4.184 J cal1 (Bromley et al., 1970). The same factor was applied to the enthalpy data ofBromleyet al. (1970). Enthalpy of seawater is depending on an arbitrary linear function of salinity ( Feistel, 2003)expressing the absolute enthalpy of water and of salt. Hence, for comparability, Bromley et al.s (1970)dataneed an adjustment to the reference state used for F03 and GP, in the form

h hBromley ABS: 6:1


7/25/2019 Fei Stel 2007

12/25

Since measured enthalpy data are not available for the triple point of pure water, we have computed the offsetA= 61 J kg1 by adjusting(6.1)to the F03 absolute enthalpy at a nearby point, namely at normal pressure,t= 0 C, and S= 0 psu. InterpolatingBromley et al.s (1970)data table for S= 35 psu, we then have deter-mined B= 4.6 J kg1 psu1 to meet the condition(5.17).

The mixing heat measurements reported byBromley (1968) and Connors (1970)do not need adjustments to

any reference state.Density data ofHigashi et al. (1931)were converted from chlorinities Cl to salinities by the formula (Lewisand Perkin, 1981) S/psu = 1.80655Cl/(g kg1), and their vapour pressure values by the factor (ISO, 1993)133.3224 Pa/mmHg.

Density, q, and specific volume, v, are obtained from

1

q v

og

op

S;t

: 6:2

Fig. 1shows good agreement between F03 and GP densities up to very high salinities, in particular at lowpressures. Moreover, both agree well with Grunbergs (1970)data for artificial seawater at 20 C, their rmsis 0.3 kg m3 for F03 and 0.6 kg m3 for GP. With respect to Higashi et al.s (1931) data at 0 C and25 C, their rms is 0.6 kg m3 for F03 and 0.8 kg m3 for GP. Thus, F03 can be applied to salinities even

beyond 40 psu, but with increasing uncertainty. The rms deviation between both density formulas over theStptest cube is 0.06 kg m3, or 60 ppm, at normal pressure, and 0.4 kg m3, or 400 ppm, for all pressures.The uncertainty of F03 density itself is estimated to 4 ppm at normal pressure and 11 ppm for all pressures(Feistel, 2003).

Specific entropy, r, is obtained from

r og

ot

S;p

6:3

and is displayed inFig. 2. The rms deviation between the F03 and GP entropy formulas over the Stpcubeis 0.3 J kg1 K1 at normal pressure, and 0.5 J kg1 K1 for all pressures.

Specific enthalpy, h, is obtained from

h gTr gT og

ot

S;p

6:4

and is shown inFig. 3. The rms deviation between the F03 and GP enthalpy formulas over theStpcube is0.09 kJ kg1 at normal pressure, and 0.15 kJ kg1 for all pressures. The rms deviation ofBromley et al.s

0 10 20 30 40 50 60 70 80 90 100 110990

1000

1010

1020

1030

1040

1050

1060

1070

1080

1090

1100

1110

1120

1130

Density/(kgm-

3)

Salinity S/psu

Density

a A

h

h

h

h

h

h

bB

G

G

GG

G

G

G c C

H

H

H

H

H

H

dD

eE

Fig. 1. Density of seawater computed from Eq.(5.3)(AE) and from F03 (ae) at t = 0 C, p = 0 MPa (A, a), at t = 20 C, p = 0 MPa(B, b), at t= 25 C,p = 0 MPa (C, c), at t= 0 C,p= 40 MPa (D, d) and att = 0 C,p = 80 MPa (E, e). Measured data points at normal

pressure are ofGrunberg (1970)for artificial seawater at 20 C (G), and ofHigashi et al. (1931) at 0C (h) and 25 C (H).


7/25/2019 Fei Stel 2007

13/25

(1970) readjusted natural seawater data at normal pressure, t= 025 C, and S= 0110 psu is 0.09 kJ kg1

from F03, and 0.02 kJ kg1

from GP. The rms deviation of F03 from Bromley et al.s (1970) data up to50 psu is only 0.02 kJ kg1.

The temperature rise Dtdue to adiabatic mixing of two seawater samples with equal volumes, with salinitiesS1andS2at temperaturetand normal pressure, follows from the mass conservation and the enthalpy balance,Eq.(6.4). The final salinity Sis

SS1qS1; t; 0 S2qS2; t; 0

qS1; t; 0 qS2; t; 0 ; 6:5

and the enthalpy conservation yields

qS1; t; 0hS1; t; 0 qS2; t; 0hS2; t; 0

qS1; t; 0 qS2; t; 0 hS; t Dt; 0 hS; t; 0 cp Dt; 6:6

0 10 20 30 40 50 60 70 80 90 100 110-100

-50

0

50

100

150

200

250

300

SpecificEntropy

/(Jkg-1K-1)

Salinity S/psu

Specific Entropy

aA

bB

cC

d D

Fig. 2. Specific entropy of seawater computed from Eq. (5.3) (AD) and from F03 (ad) at t= 0C, p= 0 MPa (A, a), at t= 10 C,p= 0 MPa (B, b), at t = 20C, p = 0 MPa (C, c) and att = 0 C, p= 80 MPa (D, d).

0 10 20 30 40 50 60 70 80 90 100 110

-10

0

10

20

30

40

50

60

70

80

90

100

110

SpecificEnthalpyh/(kJkg-1)

Salinity S/psu

Specific Enthalpy

a A

b B

c C

d D

e E

f F

g G

h H

1 1 1 1 1 1 1 1 1 1 1 1

2 2 2 2 2 2 2 2 2 2 2 2

3 3 3 3 3 3 3 3 3 3 3 3

4 4 4 4 4 4 4 4 4 4 4 4

5 5 5 5 5 5 5 5 5 5 5 5

6 6 6 6 6 6 6 6 6 6 6 6

Fig. 3. Specific enthalpy of seawater computed from Eq. (5.3) (AH) and from F03 (ah) at t= 0C, p= 0 MPa (A, a), at t= 5 C,p= 0 MPa (B, b), at t = 10 C,p= 0 MPa (C, c) , at t = 15 C,p = 0 MPa (D, d), att = 20 C,p = 0 MPa (E, e), at t= 25 C,p= 0 MPa(F, f), at t = 0 C,p = 50 MPa (G, g) and at t = 0 C, p = 100 MPa (H, h). Data points (16) are fromBromley et al. (1970)for naturalseawater, readjusted to the F03 reference points.


7/25/2019 Fei Stel 2007

14/25

Values computed from Eq.(6.6)with F03 and with GP are reported in Table 1and displayed inFig. 4, incomparison to mixing heat measurements ofConnors (1970). The rms deviation from the experimental valuesis 3.2 mK for F03 and 6.6 mK for GP, the rms deviation between F03 and GP is 6.4 mK.

Mixing heats Q due to adiabatic mixing of two La Jolla seawater samples with the masses m1 and m2 at25 C and normal pressure with initial salinitiesS1andS2, were reported byBromley (1968). The computation

with F03 and GP follows from the mass conservation and the enthalpy balance, in analogy to Eq. (6.5). Theresults are listed inTable 2and shown graphically inFig. 5. The rms deviation from the experimental values is14 J for F03 and 92 J for GP, the rms deviation between F03 and GP is 88 J. The mixing heat

Q m1m2hS m1hS1 m2hS2 6:7

with final salinity S,

Sm1S1m2S2

m1m26:8

depends only on the effective ionion interaction, expressed by the temperaturesalinity properties of the activ-ity potential w, as follows from Eqs. (6.4) and (5.3),

Q NSkT2 o

oTm1S1wS1 m2S2wS2 m1m2SwS; 6:9

and is independent of the way we determined the function C(t,p), Eqs.(5.16) and (5.8).Specific Gibbs energy (free enthalpy) of seawater, g(S, t,p), is shown inFig. 6. The rms deviation between

the F03 and GP free enthalpy formulas over the Stp cube is 0.005 kJ kg1 at normal pressure, and0.021 kJ kg1 for all pressures.

Isobaric specific heat capacity, cP, is obtained from

cP T or

ot

S;p

oh

ot

S;p

T o

2g

ot2

S;p

6:10

and shown in Figs. 7 and 8. The rms deviation between the F03 and GP heat capacity formulas over theStp cube is 9 J kg1 K1 at normal pressure, and 15 J kg1 K1 for all pressures. The rms deviation of

Table 1Temperature increase Dtand final salinity Sdue to seawater mixing of equal sample volumes with initial salinities S1and S2, computedfrom Eqs.(6.5) and (6.6) with F03 and with GP, in comparison to measurements (exp) ofConnors (1970)

S1(psu) S2 (psu) t(C) S(F03) (psu) S(GP) (psu) Dt (F03) (mK) Dt(GP) (mK) Dt(exp) (mK)

10.0 60.0 1.93 35.487 35.489 62.43 69.83 67.010.0 60.0 1.90 35.487 35.489 62.50 69.87 63.715.0 54.6 2.05 35.105 35.106 39.09 43.40 41.115.0 54.6 2.02 35.105 35.106 39.13 43.43 39.915.0 54.6 2.86 35.105 35.106 38.05 42.69 40.620.0 49.6 2.00 34.971 34.971 21.91 24.12 21.320.0 49.6 1.99 34.971 34.971 21.92 24.13 22.124.8 45.4 2.08 35.183 35.183 10.60 11.59 10.924.8 45.4 2.03 35.183 35.183 10.62 11.60 10.510.24 60.5 25.3 35.836 35.836 27.39 30.84 25.510.24 60.5 24.6 35.836 35.836 28.16 32.23 25.710.24 60.5 21.4 35.839 35.839 31.83 38.40 28.810.24 60.5 20.0 35.840 35.841 33.54 41.02 29.310.24 60.5 15.0 35.844 35.846 40.21 49.95 37.510.24 60.5 14.2 35.845 35.847 41.37 51.33 38.610.24 60.5 9.15 35.851 35.853 49.36 59.61 49.110.24 60.5 3.10 35.860 35.862 60.66 68.72 66.610.24 60.5 2.75 35.861 35.863 61.37 69.22 68.5

10.24 60.5 2.00 35.862 35.864 62.93 70.28 67.7


7/25/2019 Fei Stel 2007

15/25

-80 -70 -60 -50 -40 -30 -20 -10 0

-80

-70

-60

-50

-40

-30

-20

-10

0

ComputedTemperatureD

ifferencet/mK

Measured Temperature Difference t/mK

Mixing Heat, Connors (1970)

c

C

c

C

c

C

c

C

c

C

cC

cC

cCcC

c

Cc

Cc

Cc

Cc

C

c

Cc

Cc

C

c

C

c

C

Fig. 4. Temperature increaseDtdue to seawater mixing, computed from Eqs.(6.5) and (6.6)with F03 (c) and with GP (C), in comparisonto measured Dt ofConnors (1970), compareTable 1.

Table 2Mixing heat Q and final salinity Sdue to seawater mixing at 25 C and normal pressure of samples with initial salinities S1and S2, andmassesm1and m2, computed from Eqs.(6.7) and (6.8), with F03 and with GP, in comparison to measurements (exp) ofBromley (1968)

S1(psu) m1(kg) S2(psu) m2(kg) S(psu) Q (F03) (J) Q (GP) (J) Q (exp) (J)

33.4 0.2715 0.00 3.9892 2.128 27.02 36.45 22.5533.4 0.2698 2.12 3.9807 4.105 1.26 49.71 1.6333.4 0.2938 4.11 4.0022 6.113 10.32 52.26 7.7833.4 0.2804 6.11 4.0071 7.895 13.10 45.12 14.5633.4 0.2710 7.90 3.9677 9.530 13.62 38.64 10.29

33.4 0.2920 11.12 4.0170 12.630 14.06 31.62 11.9233.4 0.2734 13.85 4.0112 15.097 11.53 22.60 8.0833.4 0.2943 15.11 4.0335 16.354 11.36 21.02 8.1233.4 0.2455 0.00 4.0027 1.930 26.39 31.81 25.9433.4 0.2512 1.93 4.0055 3.787 0.39 46.06 3.6433.4 0.2560 0.00 3.9890 2.014 26.63 33.70 24.5633.4 0.2610 2.00 4.0163 3.916 0.25 47.96 3.8133.4 0.2756 3.87 4.0163 5.766 8.89 49.60 10.3833.4 0.2850 5.71 4.0279 7.540 12.88 46.94 13.6833.4 0.2790 7.52 4.0226 9.199 13.92 40.86 13.6433.4 0.2706 9.18 4.0321 10.703 13.65 34.93 17.7833.4 0.2772 10.67 4.0001 12.143 13.56 31.39 13.4733.4 0.2836 12.11 4.0145 13.515 13.13 27.99 13.1033.4 0.2832 13.65 4.0275 14.948 12.08 23.87 9.1233.4 0.2640 14.97 4.0181 16.106 10.34 19.30 10.67

3.0 0.2530 0.00 4.0987 0.174 4.26 2.86 6.0710.3 0.3010 0.00 4.0095 0.719 18.39 7.08 18.33

100.2 0.2620 0.00 4.0140 6.139 257.9 480.28 220.6665.9 0.2501 0.00 4.1265 3.766 55.15 184.67 53.7623.9 0.2400 0.00 4.0244 1.345 27.85 7.39 27.7454.8 0.2600 0.00 4.0370 3.316 16.16 127.31 13.56

106.2 0.2680 0.00 4.0473 6.596 301.74 557.93 283.68101.2 0.1715 33.60 4.1423 36.288 132.49 144.72 106.61

0.0 0.2722 99.80 4.0790 93.557 324.71 531.85 351.5875.2 0.2812 0.00 4.0555 4.876 117.87 278.84 96.57

107.5 0.2425 33.60 4.1276 37.701 208.59 244.59 168.9533.6 0.2350 100.38 3.9943 96.669 103.56 215.80 132.47

0.0 0.2776 33.04 4.1126 31.288 29.51 62.15 31.67


7/25/2019 Fei Stel 2007

16/25

Grunbergs (1970) artificial seawater data at 0 C and 20 C is 10 J kg1 K1 from F03, and 11 J kg1 K1

from GP (Fig. 7). The rms deviation of Bromley et al.s (1967) natural high-salinity seawater data atS= 50.1, 66.7, 83.2, 99.6 and 101 psu is 10 J kg1 K1 from F03, and 4 J kg1 K1 from GP (Fig. 8). The

uncertainty of F03 heat capacity is estimated as 12 J kg

1 K

1 below 50 psu and at normal pressure, whilethe rms deviation of F03 fromBromley et al.s (1970)data below 50 psu is only 0.5 J kg1 K1 (Feistel, 2003).

Adiabatic lapse rate, C, is obtained from

C ot

op

S;r

o2g=otopSo2g=ot2S;p

6:11

and displayed inFig. 9. The rms deviation between the F03 and GP lapse rate formulas over the Stpcube is0.6 mK MPa1 at normal pressure, and 0.9 mK MPa1 for all pressures.

The isobaric thermal expansion coefficient, a, is obtained from

a1

v

ov

ot

S;p

o2g=otopS

og=

opS;t

6:12

0 50 100 1 50 200 25 0 3 00 350 40 0 4 50 500 550 600

0

50

100

150

200

250

300

350

400

450

500

550

600

ComputedMixingHeatQ/J

Measured Mixing Heat Q/ J

Mixing Heat, Bromley (1968)

b

B

b

B

b

B

b

B

bBbB

bBbB

b

B

b

B

b

B

b

B

b

B

b

B

bBbB

bBbB

bBbB

bBbB

b

B

b

B

b

B b

B

b

B

bB

b

B

b

B

b

B

b

B

b

B

Fig. 5. Mixing heatQ due to seawater mixing at 25 C and normal pressure, computed with F03 (b) and with GP (B), in comparison tomeasurements ofBromley (1968), compareTable 2.

0 10 20 30 40 50 60 70 80 90 100 110

-10

0

10

20

30

40

50

60

70

80

90

Spe

cificGibbsEnergyg/(kJkg-1)

Salinity S/psu

Specific Gibbs Energy

a A

b B

c C

d D

Fig. 6. Specific Gibbs energy of seawater computed from Eq.(5.3)(AD) and from F03 (ad) at t = 0 C,p = 0 MPa (A, a), at t = 20 C,p= 0 MPa (B, b), at t= 0 C, p = 40 MPa (C, c) and at t = 0 C, p = 80 MPa (D, d).


7/25/2019 Fei Stel 2007

17/25

0 10 20 30 40 50 60 70 80 90 100 110

3400

3500

3600

3700

3800

3900

4000

4100

4200

4300

SpecificHeatCapacity

cp

/(Jkg-1K-1)

Salinity S/psu

Specific Isobaric Heat Capacity

a A

g

g

gg

g

g

g b B

G

G

GG

G

G

G

c C

d D

Fig. 7. Specific isobaric heat capacity of seawater computed from Eq.(5.3)(AD) and from F03 (ad) at t = 0 C,p = 0 MPa (A, a), att= 20C, p = 0 MPa (B, b), at t = 0 C, p = 40 MPa (C, c) and at t = 0 C, p= 80 MPa (D, d) Measured data points are of Grunberg(1970)for artificial seawater at 0 C (g) and 20C (G) at 0 MPa.

0 5 10 15 20 25

3600

3650

3700

3750

3800

3850

3900

3950

SpecificHeatCapacitycp

/(Jkg-1K-1)

Temperature T/K

Isobaric Specific Heat Capacity

aA

1 1 1 1

2 2 2

3 3 3

3

4 4 4 4 4

4

5 5

5 5 5 5 5 5 5

5 5 5

bB

1 1 1 1

2 2 2

3 3 3

3

4 4 4 4 4

4

5 5

5 5 5 5 5 5 5

5 5 5

cC

1 1 1 1

2 2 2

3 3 3

3

4 4 4 4 4

4

5 5

5 5 5 5 5 5 5

5 5 5 d

D

1 1 1 1

2 2 2

3 3 3

3

4 4 4 4 4

4

5 5

5 5 5 5 5 5 5

5 5 5 eE

1 1 1 1

2 2 2

3 3 3

3

4 4 4 4 4

4

5 5

5 5 5 5 5 5 5

5 5 5

Fig. 8. Specific isobaric heat capacity of seawater atp= 0 MPa, computed from (e5.3) (AE), from F03 (ae), and experimental data (15)ofBromley et al. (1967)atS= 50.1 psu (A, a, 1), at S= 66.7 psu (B, b, 2), atS= 83.2 psu (C, c, 3),S= 99.6 psu (D, d, 4), andS= 101 psu(E, e, 5).

0 10 20 30 40 50 60 70 80 90 100 110

-5

0

5

10

15

20

25

LapseRate/(mKMPa

-1)

Salinity S/psu

Adiabatic Lapse Rate

a

A

b B

c

C

d

D

Fig. 9. Adiabatic lapse rate of seawater computed from Eq.(5.3)(AD) and from F03 (ad) at t = 0 C,p = 0 MPa (A, a), at t = 20 C,

p= 0 MPa (B, b), at t = 0C, p = 40 MPa (C, c) and at t= 0 C, p = 80 MPa (D, d).


7/25/2019 Fei Stel 2007

18/25

and is shown inFig. 10. The rms deviation between the F03 and GP thermal expansion formulas over theStpcube is 9 ppm K1 at normal pressure, and 11 ppm K1 for all pressures. The uncertainty of the thermalexpansion coefficient of F03 is estimated as 0.6 ppm K1 (Feistel, 2003).

Isothermal compressibility, jT, is obtained from

jT

1

v

ov

op

S;t

o2g=op2S;t

og=opS;t 6:13

and is shown inFig. 11. The rms deviation between the F03 and GP compressibility formulas over theStpcube is 10 ppm MPa1, or 3%, at normal pressure, and 6 ppm MPa1, or 2%, for all pressures.

Adiabatic compressibility, jS, and sound speed, U, is obtained from

jS 1

v

ov

op

S;r

v

U2

o2g=otop2S o

2g=ot2S;po

2g=op2S;t

og=opS;to2g=ot2S;p

6:14

Sound speed is shown inFig. 12. The rms deviation between the F03 and GP sound speed formulas over theStpcube is 16 m s1, or 1%, at normal pressure, and 10 m s1, or 0.7%, for all pressures. The uncertainty ofF03 sound speed is estimated as 0.05 m s1.

0 10 20 30 40 50 60 70 80 90 100 110

-100

-50

0

50

100

150

200

250

300

350

ExpansionCoefficient

/(ppmK-1)

Salinity S/psu

Isobaric Thermal Expansion Coefficient

a

A

b B

c

C

d

D

Fig. 10. Isobaric thermal expansion coefficient of seawater computed from Eq. (5.3) (AD) and from F03 (ad) at t= 0 C,p= 0 MPa (A, a), at t= 20C, p= 0 MPa (B, b), at t= 0 C, p= 40 MPa (C, c) and at t= 0C, p= 80 MPa (D, d).

0 10 20 30 40 50 60 70 80 90 100 110

300

325

350

375

400

425

450

475

500

525

CompressibilityT/(ppmMPa-1)

Salinity S/psu

Isothermal compressibility

a

A

b

B

c C

d

D

Fig. 11. Isothermal compressibility of seawater computed from Eq.(5.3)(AD) and from F03 (ad) at t= 0C, p= 0 MPa (A, a), at

t= 20C, p = 0 MPa (B, b), at t = 0 C, p = 40 MPa (C, c) and at t = 0 C, p = 80 MPa (D, d).


7/25/2019 Fei Stel 2007

19/25

The isothermal haline contraction coefficient,b, is obtained from

b 1

v

ov

oS

t;p

o2g=opoSt

og=opS;t6:15

and is shown inFig. 13. The rms deviation between both haline contraction formulas over the Stpcube is3 ppm psu1, or 0.5%, at normal pressure, and 21 ppm psu1, or 2.5%, for all pressures.

Potential temperature, h(S, t,p,pr), for reference pressure pr, is implicitly given by the equation

rS; t;p rS; h;pr 6:16

and displayed inFig. 14for normal pressure as reference pressure. The rms deviation between the F03 andGP potential temperature formulas over the Stpcube is 42 mK for all pressures. The uncertainty of poten-

tial temperature computed from F03 is estimated as 3 mK.Freezing temperature, tf(S,p), is implicitly given by the equation

gIhtf;p gS; tf;p S o

oSgS; tf;p; 6:17

in conjunction with the equation of state of ice Ih (Feistel and Wagner, 2006; IAPWS, 2006) and is shown inFig. 15. The rms deviation between the F03 and GP freezing point formulas over the Stpcube is 2 mK at

0 10 20 30 40 50 60 70 80 90 100 110

1400

1425

1450

1475

1500

1525

1550

1575

1600

1625

1650

1675

1700

SoundSpeedU

/(ms-

1)

Salinity S/psu

Sound Speed

a

A

b

B

c C

d

D

Fig. 12. Sound speed of seawater computed from Eq. (5.3) (AD) and from F03 (ad) at t= 0 C, p= 0 MPa (A, a), at t= 20 C,p= 0 MPa (B, b), at t = 0C, p = 40 MPa (C, c) and at t = 0 C, p= 80 MPa (D, d).

0 10 20 30 40 50 60 70 80 90 100 110

620

640

660

680

700

720

740

760

780

800

820

840

ContractionCoefficient/(ppmpsu-1)

Salinity S/psu

Isothermal haline contraction coefficient

aA

b

B

c

Cd

D

Fig. 13. Isothermal haline contraction coefficient computed from Eq.(5.3)(AD) and from F03 (ad) at t = 0 C, p = 0 MPa (A, a), at

t= 20C, p = 0 MPa (B, b), at t = 0 C, p = 40 MPa (C, c) and at t = 0 C, p= 80 MPa (D, d).


7/25/2019 Fei Stel 2007

20/25

normal pressure, and 21 mK for all pressures. The uncertainty of F03 freezing points is estimated as 2 mK atnormal pressure. For pure water, the agreement of F03 with accurate high-pressure measurements is within30 mK, while the uncertainty is estimated by 2% of the melting pressure (Feistel and Wagner, 2006).

Vapour pressure, Pvap

, of seawater under pressure p, is computed from the formula

gvapT;Pvap gS; t;p S o

oSgS; t;p; 6:18

in conjunction with the IAPWS-95 formulation for water vapour, gvap(T, P), Wagner and Pru (2002) andshown inFig. 16for normal pressure as applied pressure. The rms deviation between the F03 and GP vapourpressure formulas over the Stp cube is 0.09 Pa at normal pressure, and 0.15 Pa for all pressures. The rmsdeviation fromGrunbergs (1970)natural seawater data at 20 C is 25 Pa for F03, and 10 Pa for GP. Therms deviation fromHigashi et al.s (1931) data at 25 C is 63 Pa for F03, and 33 Pa for GP.

The osmotic coefficient can be computed from the formula(5.19)

/ 1nS

NSSkT

gS; t;p g0; t;p S o

oS

gS; t;p 6:19

0 10 20 30 40 50 60 70 80 90 100 110

-1.8

-1.6

-1.4

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

PotentialTempera

ture/C

Salinity S/psu

Potential Temperature

aA

b

B

c

C

d

D

e

E

Fig. 14. Potential temperature for reference pressurepr= 0, computed from Eq.(5.3)(AE) and from F03 (ae) at t = 0 C,p = 20 MPa(A, a), at t= 0 C,p= 40 MPa (B, b), at t = 0 C,p = 60 MPa (C, c), att = 0 C,p = 80 MPa (D, d) and at t = 0 C,p = 100 MPa (E, e).

0 10 20 30 40 50 60 70 80 90 100 110

0

-2

-4

-6

-8

-10

-12

-14

-16

-18

Temperaturet/

C

Salinity S/psu

Freezing Point

a

A

b

B

c

C

d

D

e

E

f

F

Fig. 15. Freezing temperature, computed from Eq.(5.3)(AF) and from F03 (af) atp= 0 MPa (A, a), at p= 20 MPa (B, b), p= 40 MPa(C, c),p = 60 MPa (D, d), p = 80 MPa (E, e) and at p = 100 MPa (F, f). Values computed in conjunction with the equation of state of iceIh,Feistel and Wagner (2006).


7/25/2019 Fei Stel 2007

21/25

and is shown inFig. 17. The rms deviation between the F03 and GP formulas for the osmotic coefficient over

the Stp cube is 0.003 at normal pressure, and 0.005 for all pressures.

7. Conclusion

For high-salinity seawater, neither a fundamental equation of state nor a comprehensive set of experimentaldata was available by now. We have used data of the numerical FREZCHEM model, based on Pitzer equa-tions for the constituents of artificial seawater, in combination with pure water properties from the 2003 Gibbsfunction (F03) of seawater, and experimental data of seawater heat capacities, to construct a new high-salinityGibbsPitzer function (GP) of seawater, depending on salinity, temperature, and pressure. This equation isgiven in Eq.(5.3), with coefficients listed inAppendix A,Tables A.3A.5.

We emphasize that the FREZCHEM model was never explicitly designed for, nor adjusted to, any seawater

properties, in contrast to F03, which was almost exclusively determined by regression to a large number of

0 10 20 30 40 50 60 70 80 90 100 110

800

1000

1200

1400

1600

1800

2000

2200

2400

2600

2800

3000

3200

PressureP

/Pa

Salinity S/psu

Vapour Pressure

a A

G G G G GG

G

HH H

H H H

b B

cC

d

D

eE

Fig. 16. Vapour pressure, computed from Eq. (5.3)(AE) and from F03 (ae) at t= 5C, p= 0 MPa (A, a), at t= 10 C, p= 0 MPa(B, b), at t= 15C, p= 0 MPa (C, c), at t= 20 C, p= 0 MPa (D, d) and at t= 25 C, p= 0 MPa (E, e). Values computed inconjunction with the IAPWS-95 formulation for vapour. Experimental data at 20 C (G) fromGrunberg (1970) for natural seawater,

at 25 C (H) from Higashi et al. (1931).

0 10 20 30 40 50 60 70 80 90 100 110

1.02

1.00

0.98

0.96

0.94

0.92

0.90

0.88

OsmoticCoefficient

Salinity S/psu

Osmotic Coefficient

a

A

0

0 0 0

0

0

0

0

b

B

1

1 1 1

1

1

1

1

c

C

22

2 2

2

2

2

2

d

D

33

3 3

3

3

3

3

Fig. 17. Osmotic coefficient, computed from Eq. (5.3) (AD) and from F03 (ad), in comparison to data (03) reported by Bromleyet al. (1974) at t= 0 C, p= 0 MPa (A, a, 0), at t= 10 C, p= 0 MPa (B, b, 1), at t= 20 C, p= 0 MPa (C, c, 2) and at t= 25 C,p= 0 MPa (D, d, 3).

http://-/?-http://-/?-

7/25/2019 Fei Stel 2007

22/25

highly accurate seawater data. Under these circumstances, the good agreement between both approaches,which are so different conceptionally and with respect to their data background, is a surprising result on itsown. This is the first comprehensive intercomparison of F03 with a rather independent formulation, withexcellent results.

Another quite unexpected finding is that the mere extrapolation of F03 to high salinities, which was neither

intended nor systematically studied before, yields reasonable results in most cases, sometimes even more accu-rate than the new GP formulation itself.The extended and systematic comparison between GP, F03, and available experimental data has revealed

that

In colligative properties (osmotic coefficient, freezing point, vapour pressure), F03 is restricted to the low-salinity range up to 4050 psu, and shows significant deviations with increasing salinity, while GP coversthe entire salinity range in a stable and accurate way.

In thermal properties (enthalpy, entropy, Gibbs energy, heat capacity), but also in density, both formula-tions describe the high-salinity region rather well.

In higher derivatives of the Gibbs function (mixing heat, compressibility, thermal expansion, saline contrac-tion, sound speed) both formulations exhibit substantial deviations, and lacking measurements do not

allow decisive final conclusions. Particularly in those cases where GP already at lower salinities signifi-cantly deviates from F03, the latter one appears more reliable. This conclusion is supported by the differentagreement of both with the mixing heat data of Bromley,Fig. 5.

In general, the high-salinity range is described with only lower accuracy than is the low-salinity range byF03.

For many practical applications, either GP or F03 should be sufficiently accurate formulations, both beingin good agreement with the scarce experimental basis. It must be decided on the particular needs which of bothis to be preferred for a particular purpose.

Acknowledgement

The authors are grateful to H.-J. Kretzschmar for providing literature on industrial high-salinity seawatermeasurements. They thank A. Schroder and B. Sievert for making literature sources available. This paper con-tributes to the tasks of the SCOR/IAPSO Working Group 127 on Thermodynamics and Equation of State ofSeawater.

Appendix A

Tables A.1A.5are given for important general constants, the composition of seawater, and the coefficientsof the new GibbsPitzer equation of state, Eq. (5.3).

Table A.1Physical constants and special values (maximum seven digits given)

Quantity Symbol Value Unit Source

Celsius zero point T0 273.15 K Preston-Thomas (1990)Normal pressure P0 101 325 Pa ISO (1993)Boltzmann constant k 1.380651 1023 J K1 Mohr and Taylor (2005)Avogadro number NA 6.022142 10

23 mol1 Mohr and Taylor (2005)Molar gas constant R=kNA 8.314472 J K

1 mol1 Mohr and Taylor (2005)Salinity number NS 1.926845 10

22 kg1 psu1 Feistel (2003)Salinity factor n 1.004867 g kg1 psu1 Feistel (2003)

Valence factor z2

1.245143 Feistel (2003)


7/25/2019 Fei Stel 2007

23/25

Table A.2Seawater composition model F03 (Feistel, 2003),Grunbergs (1970)artificial calcium-free seawater, and major constituents ofBromleyet al.s (1967)La Jolla seawater

Kind (a) za Ma (g mol1) F03 (xa/ppm) F03 (wa/ppm) Grunberg (wa/ppm) Bromley (wa/ppm)

Na +1 22.98977 418791.8 306563.3 320606 306531Mg +2 24.305 47163.7 36499.9 38182 36865

Ca +2 40.078 9181.5 11716.8 11611K +1 39.0983 9115.1 11347.7 10303 11030Sr +2 87.62 81.0 225.9 290Cl 1 35.453 487445.1 550257.7 575455 550943SO4 2 96.0626 25213.1 77120.1 53212 76923HCO3 1 61.016 84 1661.5 3228.0 4064Br 1 79.904 751.3 1 911.5 2121 1742CO3 2 60.0089 173 330.5B(OH)4 1 78.84036 74.6 187.3F 1 18.9984032 54.6 33.1H3BO3 0 61.83302 293.8 578.4 909H2O 18.01528

F03 components and mass fractions fromMillero (1982), mole fractions recomputed with IUPAC 99 mol masses ( Coplen, 2001), with

sodium re-adjusted to obey electro-neutrality(2.11). Molecular weights are computed as sums over their atomic parts. z: ion charge, M:molecular weight, w: mass fraction, x: mole fraction.

Table A.3Coefficients of the polynomial gwt;p 1 J=kg

Pj;kg0jk

t40 C

j p100 MPa

k, adopted fromFeistel (2003)

j k g0jk j k g0jk j k g0jk

0 0 101.342743139672a 2 1 1455.0364540468 4 3 152.1963717338410 1 100015.695367145 2 2 756.558385769359 4 4 26.37483772328020 2 2544.5765420363 2 3 273.479662323528 5 0 58.02591258425710 3 284.517778446287 2 4 55.5604063817218 5 1 194.6183106175950 4 33.3146754253611 2 5 4.34420671917197 5 2 120.5206549020250 5 4.20263108803084 3 0 736.741204151612 5 3 55.27230523401520 6 0.546428511471039 3 1 672.50778314507 5 4 6.48190668077221

1 0 5.90578348518236a 3 2 499.360390819152 6 0 18.98438465141721 1 270.983805184062 3 3 239.545330654412 6 1 63.51139366417851 2 776.153611613101 3 4 48.8012518593872 6 2 22.28973171404591 3 196.51255088122 3 5 1.66307106208905 6 3 8.170605418181121 4 28.9796526294175 4 0 148.185936433658 7 0 3.050816464879671 5 2.13290083518327 4 1 397.968445406972 7 1 9.631081193930622 0 12357.785933039 4 2 301.815380621876

a Coefficients subject to the IAPWS-95 reference state.

Table A.4Coefficients of the polynomial Ct;p 1

40 J=kg psu

Pj;kRjk

t40 C

j p100 MPa

kj k Rjk j k Rjk j k Rjk

0 0 9188.35355278883a 2 0 782.557453641887b 4 1 15.99621431323070 1 3297.02334038811 2 1 290.637800851481 4 2 12.13137264317450 2 364.670193748859 2 2 122.903854303638 4 3 6.117484459002780 3 9.26449928889693 2 3 10.9924275136023 4 4 1.060128162339180 4 0.542487256726294 2 4 2.23323275518327 5 1 7.822620717414830 5 0.168923411741451 2 5 0.174613998930967 5 2 4.844289157177330 6 0.0219635191614547 3 0 77.6208956815076b 5 3 2.221652621743571 0 1455.8851627253a 3 1 27.0312351410255 5 4 0.2605381648235011 1 537.251238518538 3 2 20.0716311136508 6 1 2.552820144685681 2 238.818607352217 3 3 9.62844791114828 6 2 0.8959286335319261 3 7.89875909865436 3 4 1.96155070208747 6 3 0.3284148701900561 4 1.1648278639506 3 5 0.0668466091579295 7 1 0.3871182266440591 5 0.0857312665419243 4 0 22.5314914279383b

a Coefficients subject to the seawater reference state, Eq. (5.17).b Determined by fitting heat capacity data ofBromley et al. (1967) and Millero et al. (1973).


7/25/2019 Fei Stel 2007

24/25

References

Archer, D.G., Wang, P., 1990. The dielectric constant of water and DebyeHuckel limiting law slopes. Journal of Physical and ChemicalReference Data 19, 371411.

Bromley, L.A., 1968. Relative enthalpies of sea salt solutions at 25 C. Journal of Chemical and Engineering Data 13, 399402.

Bromley, L.A., Desaussure, V.A., Clipp, J.C., Wright, J.S., 1967. Heat capacities of sea water solutions at salinities of 1 to 12% and

temperatures of 2C to 80C. Journal of Chemical and Engineering Data 12, 202206.

Bromley, L.A., Diamond, A.E., Salami, E., Wilkins, D.G., 1970. Heat capacities and enthalpies of sea salt solutions to 200 C. Journal of

Chemical and Engineering Data 15, 246253.

Bromley, L.A., Singh, D., Ray, P., Sridhar, S., Read, S.M., 1974. Thermodynamic properties of sea salt solutions. AIChE Journal 20, 326

335.

Connors, D.N., 1970. On the enthalpy of seawater. Limnology and Oceanography 15, 587594.

Coplen, T.B., 2001. Atomic weights of the elements 1999. Pure and Applied Chemistry 73, 667683.

Falkenhagen, H., Ebeling, W., Hertz, H.G., 1971. Theorie der Elektrolyte. S. Hirzel, Leipzig.

Feistel, R., 2003. A new extended Gibbs thermodynamic potential of seawater. Progress in Oceanography 58, 43114.

Feistel, R., Hagen, E., 1998. A Gibbs thermodynamic potential of sea ice. Cold Regions Science and Technology 28, 83142.

Feistel, R., Wagner, W., 2005. High-pressure thermodynamic Gibbs functions of ice and sea ice. Journal of Marine Research 63, 95139.Feistel, R., Wagner, W., 2006. A new equation of state of H2O ice Ih. Journal of Physical and Chemical Reference Data 35, 127.

Feistel, R., Wagner, W., Tchijov, V., Guder, C., 2005. Numerical implementation and oceanographic application of the Gibbs potential of

ice. Ocean Science 1, 2938.

Fofonoff, N.P., 1962. Physical properties of sea-water. In: Hill, M.N. (Ed.), The Sea. Wiley, New York, pp. 330.

Grasshoff, K., Kremling, K., Ehrhardt, M., 1999. Methods of Seawater Analysis. Wiley-VCH, Weinheim.

Grunberg, L., 1970. Properties of seawater concentrates. In: Third International Symposium on Fresh Water from the Sea, vol. 1, pp. 31-

39.

Higashi, K., Nakamura, K., Hara, R., 1931. The specific gravities and the vapour pressures of the concentrated sea water at 0175 C.

Journal of the Society of Chemical Industry Japan 34, 166172 (in Japanese).

IAPWS, 2006. Release on an equation of state for H2O ice Ih. The International Association for the Properties of Water and Steam,

Witney, UK, September 2006. http://www.iapws.org/relguide/Ice.pdf.

ISO, 1993. ISO STANDARDS HANDBOOK: Quantities and Units. Geneva: International Organization for Standardization.

Jackett, D.R., McDougall, T.J., Feistel, R., Wright, D.G., Griffies, S., 2006. Algorithms for density, potential temperature, conservativetemperature, and the freezing temperature of seawater. Journal of Atmospheric and Oceanic Technology 23, 17091728.

Lewis, E.L., Perkin, R.G., 1981. The practical salinity scale 1978: conversion of existing data. Deep-Sea Research 28A, 307328.

Marion, G.M., 2001. Carbonate mineral solubilities at low temperatures in the NaKMgCaHClSO4OHHCO3CO3CO2H2O

system. Geochimica et Cosmochimica Acta 65, 18831896.

Marion, G.M., 2002. A molal-based model for strong acid chemistry at low temperatures (

7/25/2019 Fei Stel 2007

25/25

Millero, F.J., 1982. The thermodynamics of seawater. Part I. The PVT properties. Ocean Science and Engineering 7, 403460.

Millero, F.J., 2001. Physical Chemistry of Natural Waters. Wiley Interscience, New York.

Millero, F.J., 2006. Chemical Oceanography, third ed. CRC Press, Taylor & Francis, Boca Raton.

Millero, F.J., Leung, W.H., 1976. The thermodynamics of seawater at one atmosphere. American Journal of Science 276, 10351077.

Millero, F.J., Perron, G., Desnoyers, J.E., 1973. Heat capacity of seawater solutions from 5to 35 and 0.5 to 22 chlorinity. Journal of

Geophysical Research 78, 44994507.

Millero, F.J., Sohn, M.L., 1992. Chemical Oceanography. CRC Press, Boca Raton.

Mohr, P.J., Taylor, B.N., 2005. CODATA recommended values of the fundamental physical constants: 2002. Review of Modern Physics

77, 1107.

Pitzer, 1991. Ion interaction approach: theory and data correlation. In: Pitzer, K.S. (Ed.), Activity Coefficients in Electrolyte Solutions,

second ed. CRC Press, Boca Raton, pp. 75153.

Pitzer, 1995. Thermodynamics, third ed. McGraw-Hill, New York.

Preston-Thomas, H., 1990. The international temperature scale of 1990 (ITS-90). Metrologia 27, 310.

Spencer, R.J., Mller, N., Weare, J.H., 1990. The prediction of mineral solubilities in natural waters: A chemical equilibrium model for the

NaKCaMgSO4H2O system at temperatures below 25C. Geochimica et Cosmochimica Acta 54, 575590.

UNESCO, 1981. Background papers and supporting data on the Practical Salinity Scale 1978. Unesco technical papers in marine science,

37.

Wagner, W., Pru, A., 2002. The IAPWS Formulation 1995 for the Thermodynamic Properties of Ordinary Water Substance for General

and Scientific Use. Journal of Physical and Chemical Reference Data 31, 387535.


fei stel 2007

Documents