MARINE PHYSICAL LABORATORY4 of the Scripps Institution of Oceanography

San Diego, California 92132

EQUATION OF STATE OF PURE WATER XND SEA WATER

F.H. Fisher and 0.E. Dial, Jr.

Sponsored byOffice of Naval ResearchN00014-69-A-0200- 6002

NR 260-103

andNational Science Foundation

DES 70-00094A04

Reproduction in whole or in part is permitted 7 , ""

for any purpose of the U.S. Government <2''

Document cleared for public releaseand sale; its distribution is unlimited[\J2 t-

I November 1975

SIO REFERENCE 75-28

'~U.1 ~2~% ~

29

4

UNCLASSIFIEDSECURITY CLASSIFICATION OF THIS PAGE (Ivhen Dot- E~ntered)

I READ 1NSTRVJCT1ONSSIO REERE 'T DOCUMENTATION PAGE BEFORE CO'.~?LETINrj FCRM

REPORT NUMBER j2 GOVT ACCESSION No .RECIPIENT'S CATALOG w'itABER

SORFRNE75-28_______________

-WATE MPL-U-99/671,., -

31 P-t AND ADDRESS10OMSMEErTFeCAK

Uni rsity of Ca: ifornia, San Diego, Main AREA &WORK UNIT NUMCERS

Physical Laboratory of the Scripps Institutionof Oceanography, San Diego, California 92132

1 CON4TROLLING OFFICE NAME AND ADDRESS 12. REPOR? O'ATE~ '

J 1 NovynbS_1975;,Office of Naval Research, Code 210, Department '13. NUMBEI F *OT

of the Navy, Arlington, Virginia 22217 11, i/ -1IA MONITORING AGENCY NAME & ADDRESS(II different from Controling Office) 1.SCUF~-1t*6f*44,4 r

IUnclassified15a. CLASIFICATION DOWNGRADING

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Document cleared for public release and sale; its distribution is unlimited.

17. DISTRIBUTION STATEMENT (of the abstract entered In Block 20, If different from Roport)

18. SUPPLEMENTARtY NOTES

19. KEY WORDS (Continue on reverse side if necessary and Identify by block number)

volume, pressure, salinity, temperature, sea water, pure water.

0 ABSTRACT (Continue on reveres side If necessary and Identify by block number)

For representing the PVT properties of pu're water and solutions, sea waterin particular, we -haxe salacted the Tumlirz equation

where V is the specific volume as a function of pressure P, Salinity S andI- (over-'j 4

I .~" '*.. EDITION OF I 13V.0 IS 0b:.OLETE - UNCI ASSTFTTD ___

SECU. ' .:1 A , C."' 4

Y----

- - - - 7-7."' r'

20. Abstract (Continued)

Aemperature. For pure water S+O and for sea water V ad Po are thepure water values. For pure water we have obtained an analytic functionwhich fits the data of Kell and Whalley t 8 ppm from 0-100' and-15 ppmfrom 100-1501 up to 1000 bars. The calculated density maximum is at

e r The form of the equation appears to be useful not only forrepresenting the properties of sea water but for deriving partial molalproperties of solutions at low concentrations from high concentrationdata.

----------

UNIVERSITY OF CALIFORNIA, SAN DIEGOMARINE PHYSICAL LABORATORY OF THESCRIPPS INSTITUTION OF OCEANOGRAPHY

SAN DIEGO, CALIFORNIA 92132

EQUATION OF STATE OF PURE WATER AND SEA WATER

F.H. Fisher and O.E. Dial, Jr.

Sponsored byOffice of Naval ResearchN00014-69-A-0200-6002

NR 260-103and

National Science FoundationDES 70-00094A04

SIO REFERENCE 75-28

I November 1975

; Reproduction in whole or in part is permittedi

* for any purpose of the U.S. Government!

Document cleared for public releaseand sale; its distribution is unlimited

F. N. SPIESS, DIRECTORM U /MARINE PHYSICAL LABORATORY

MPL-U-99/67

Fisher, Dial

i9D

PREFACE

This work on the equation of state was done initially in1967. It has led to an equation of state 'or pure vater based onthe work of *.ell and Whalley. Subsequent work by Millero,t in-cluding corrections in the data of Kell and Whalley based on hisresults have yielded a more complex equation of state for purewater and sea water. The results in this report are useful notonly in developing an easily differentiated equation of state of seawater suitable for a number of purposes requiring less accuracythan Millero's equation but providing a basis for the ietpresen-tation of the PVT properties of other solvents and solutions con-taining water such as for dioxane -water mixtures.

F.H. Fisher, october 16, 1975

tF.J. Millero, R.A. Fine and D.P. Wang "The _quati,n of Stateof Seawater," J. Mar. Res., 32, pp. 433-456, (1974).

t9

ii ri r

SIO Reference 75-28

EQUATION OF STATE OF PURE WATER AND SEA WATER

aF.H. Fisher and O.E. Dial, Jr.IUniversity of California, San DiegoMarine Physical Laboratory of theScripps Institution of Oceanography

San Diego, California 92132

ABSTRACT

For representing the PVT properties of pure water and solutions, seawater in particular, we have selected the Tumlirz equation

V=Vco-K 1 S+Po + K 2S + P.

where V is the specific volume as a function of pressure P, Salinity S andtemperature. For pure water S + 0 and for sea water Va, X and Po arethe pure water values. For pure water we have obtained an analytic func-

tion which tits the data of Kell and Whalley to-8 ppm from 0 - 1000and-15 ppm from 100 - IM0 up to 1000 bars. The calculated density maximumis at 4.00o. The fnrm of the equation appears to be useful not only for rep-resenting the pr,,erties of sea water but for deriving partial molal prop-erties of solutions at low concentrations from high concentration data.

INTRODUCTION Since Eckart's.review on the equation ofTostate for pure water and sea water, further pea-The goal of an equation of state of a liquid surements Ivive been made on both pure water- and

is to represent PVT data In a simple form and pro- sea water 3,4, 5/. The high precision specific vol-vide insight intothe physical propertiesofthe liquid. ume data Kell and Whalley obtained for pure waterIn particular, an equation of state for a solution such as a function of temperature and pressure havebeenas seawater should reduce tothat of the pure solvent particularly useful.as the solhe concentration goes to zero. In order Two equations have been used for fittingto understand the effects of adding solute, the equa- specific volume data as a fuiction of pressure, thetion of state for the solvent should be established as original Tait equation, /

accurately as possible. In this paper we considertwo equations which have been used before to rep-resent elect rolvte solution and sea water as well as AV Apure water. VP -;+ (1)

-1-

k

____ ___ ____ ___ ___ ____ ___ ____ ___ ___

Fisher, Dial SIO Reference 75-28

t

and the usual Tait equation (UTE) as McDonald 7/ where V'p is the volume predicted by the giveneqra-denotes it in his recent review on equat ons of sta', tlon at pressure P and the indicated summation is at

B+ I each pressure along a single isotherm; this quantityVp= V I + C log t 2) is the standard deviation of observed data points a-3 + Pbout the fitted curve, sometimes called the standard

error of Estimae. -Me details of how these nalcu-The original Tait, equatdon is equivalent to the. "dmli z eq atio -/ lations were carried out m ay be found in Appesdix 1.Ttimlirz equation-ad lTables 1, 11, and III show the results of calculations

fitting these equctions to the data of Amagat- andVP V"" 1 3Po+ P Kell and Whalley/ and the much higher pressure

data of Vedam and Holtozld2/. Figure 1 showsthetemperature dependence of the varionus parame t rs

that Eckart used in hs analysis. Non-linear fitting for the Tumlirz equation as determined from thetechniques for this equation were developed (seeAp- Kel and Whalley data. Figure 2 shows the temper-pendix 1) to avoid ei rors inherent in less direct ature dependence of the parameters for the UTE aiomethods. fit to the Kell and Whalley data.

We find th:at Kell and Whalley data can bedescribed to their stated precision by using the TABLE ITbmilrz equation at low temperatures (0-900) and Fits of Tumlirz and Usual Taitthe UTE at high temperatures (50-1500). Older pure Equations to Amagat Data.water data were not sufficiently accurate to selectone equation in nreference to the other. Since the Tumlirz EquationTmlirz equaton describes the Kell and Whalley Temp., ---------------------------------data within its stated accuracy over the entire tem- egrees P0 (bars) 91har cc/gm V cc/gm o cc/vperature range, we have chosen it as the basis for x 10 x 10representing P-V-Tproperties o!. -ire water and seawater. The Vwo, X, nd Po parameters can be des-cribed by simple polynomials of temperature, corn- 0 5812.6 1.7228 0.70379 11pletirg the equaton of state for pure water. 5 6421.1 2.0164 0. 686C 27

The Tumlirz equation was fit to isotherms 10 6228 4 1.8584 0.70193 15of Wilson and Bradley an , Newton and Kennedy sea- 15 6322.9 1.8742 0.70449 19water oata at constant sainity. While the equation 20 6569. 9 1. 9833 0.69989 23described the data te its prision, the parameter8 30 6851.0 2.1026 0.69741 30of the equdtion, 0, X, a,, P0 varied erratically 40 6888.2 2.1027 0. 70244 27with salinit, and temperature. We found that by con- 50 6787. 1 2.0447 0.71069 25straining X t the same value as for pure water at 60 6704.8 2.0126 0.71670 39that temperature, % and Po become linear functions 70 6189.5 1.7609 0. 73802 51,

of salidty. This procedure yields an equation to 80 5493.9 1.4483 0.76529 19describe the P-V-T-S properties of setowater, which 90 5467.4 1.4262 0.77006 21differs in form from that used by E&sart and Wilson 1(9) 5157.9 1.4694 0. 76901 28and Bradley. 198 3328.4 1.0941 0. 82994 91

CALCULATIONS Usual Tait EquationA. Pure Water B (bars) C bar cc/gm V1 cc/gn, r cc/gm

For their maeasurements, Kell and Whalley x 106

(20 temperatures up to 1500 and 26 pressures from5 to 1000 bars) claim accuracies which vary from 0 2697.7 0. 3180 A.00014 135 ppm at low temperatures 0-50° ) and pressures to 5 3002.,Z 0.3392 1.0oooo 2940 ppm at high temperatu. es and pressures. Al-though Kell and Whalley 2 published both their ex- 10 2893.6 0.3204 1.00026 15perimental and smoothed data, we use their exper!- 20 3052.3 0.3242 1.00174 24mental data to avoid any systematic errors Introduced 30 3185.1 0. 3298 1.00428 27by theix smoothing procedure. 40 3205.5 0.3283 1, 00767 26

Acomputer routine was written to fit equa- 0 3155.1 0.3237 1.01192 23tions (2) and (3) to an isothierm of water data. The 60 3109.8 1.01684 3routine calculated parameter values so as to mini- 0 3109.8 0.3219 1.01684 34mize the quantity, '0 2850.8 0.31112 1.02249 44

8, 2509.5 0.2789 1.02889 1490 2453.1 0.2812 I.03574 27

P 2n (4) 100 2449.8 0.2904 1.04322 36198 1425.3 0.3298 1.15868 82

-2-

I - 12 1 , _ 1 --t l

SIO Reference 75-28

TABLE 11

Fit of Tumlirz and Usual I'jit Equations to Kell and Whalley Data.

Tumlirz Equation Usual Tait Equation- - - -Temp.- . . . . ..--------------------------------- ----------------------------------------------

degrees P0 (bars) X bar cc/gm V cc/gm T x 106 B bars C bars cc/gm Vlcc/gtn a x 10 cc/gmX I0 "3 cc/glli

KellandWhalley

0.000 5825.80 1.73619 0.702201 4. 2685.04 0.31773 1.000101 11.910.002 6323.46 1.92008 0.096711 6.2 2932.58 0.32553 1.000315 11.919.997 6644.11 2.03391 0.695724 4.6 3091.72 0.32919 1.001808 9.024.998 6722.75 2.05508 0.697320 6.1 3131.12 0.32899 1.002973 10.324.998 6723.00 2.05585 0.697202 5.1 3131.51 0.32904 1.002971 9.025,005 6729.02 2.05889 0.697041 5.1 3134.15 0.32931 1.002974 9.530.112 6773.51 2.06654 0.699357 3.5 3155.79 0.32845 1.004410 6.539.999 6783.43 2.05450 0.705013 4.3 3160.30 0.32605 1.007847 5.350.007 6772.43 2.04980 0.709488 4.9 3155.27 0.32584 1.012119 7.860. 001 6591.20 1.96582 0.718884 5.0 3064 46 0.32048 1.017097 5.270.003 6433.54 1.90833 0.726164 4.8 2986.09 0.31825 1.022749 5.380.0031 6189.63 1.81601 0.735672 6.0 2864. 11 0.31395 1.029029 4.190.0071 5924.39 1.72028 0.745605 5.8 ?732.84 0.30982 1.035939 5.6

100.005j 5683.60 1.64577 0.753932 7.8 2612.81 0. 30803 1.043459 4.7110. O05! 53"#3.53 1.-54927 0.764236 9.1 2467.3.] 0.30455 1.051602 6.3110.003: 5410.85 1.56000 0.763330 8.5 2477.33 0.30559 1.051600 4.6120.007: 5104.37 1.46075 0.774234 11.9 2324.93 0,30195 1.060370 5.4130.010, 4810.89 1.37208 0.784642 12.5 2179.43 0. 29950 1.069803 4.2140 009: 4544.00 1.29984 0.793891 15.9 2047.01 0.29895 1. 079907 5.4150.016,1 4249.49 1.21429 0.805041 19. 3 1901.13 0.29686 1.090750 4.7

TABLE II

Fit of Tumlirz and Usual Tait Equations to Vedam and Holton Data.

Tmlirz Equation Usual Tait EquationTemp., ----------------------------------------------------------------------------------

degrees P0 bars X x 10"3 V. cc/gm fr x 106 B bars C bars cc/gm Vlcc/gm wx 106

bars cc/gm cC/gm cc/gm

VedamandHolton

30 8024 2.7583 0.6600 334 2730 0.2927 1.0046 178.0I 40 8934 3. 2767 0.6398 597, 2824 0.2995 1.0082 161.050 8956 3.2931 0.6431 599 2833 0.3004 1.0124 156.060 9004 3.3443 0.6443 618 2853 0.3037 1.0173 137.070 8936 3.3353 0.6480 657 2822 0.3046 1.0228 109.080 8760 3.2747 0.6535 718 2742 0.3036 1.0290 81.5

_ -3-

.-

Fisher, Dial

4W0 JIMQ

0 to so W PA off W 70 40 60 S30. W

6 .0

0,1

IMO 40 1 t o0 2 40 00 s0 W ;20 40 00

rawwan ro toW. 0

IS

,S

JO -~' -

_Olb

lir euaio. s. aion

SOONC Z__ M 40 6 010#010I,0 20 do 00 W0 #00" 12 #0

From Table I the goodness of fit, €,for The very accurate data of Kell and V/haleythe Amagat data shows a rather pood uniform fit. Is more revealing. Excellent fits are to the lso-Testandard error Is about the same as the quality therms obaned below 900 with the Tumlirz opa-of the data. The two equations fit about equally weln tion and above 400 with the UTE. In either vent,o that a superiority of one equation over the other the standard error Is about 5 ppm. Figure 3 shws

Is not demonstrated. These fits do rem seem to dis- a plot of the standard error at each temperature forplay any Inherent limitations In the applicability of the fit of both the proposed equations to Kel adeither the Tumlirz or UTE to puewtrdata. fleda.

pureste WhlCydt

SKO Reference 75-28

A

20 20

SI t

b b

0

0$o too 150 0 so too - /-so:

TEMPERATURE o'C) TEMPERATURE C

Fig. 3 Coinpari.'on of the standard deviations (rof fits of the Tumlirz equation (left) and theUsual Tait Equation (right) to the Kell and

Whalley data.

TABLE IV

Fit of Tumlirz Equation to Pure Water Data at 100.

Data PO (br.) xbar V cc/g r 106 c/gmco/gi c

,magat (1893) I 6228 1858 0.7019 is

Amagat (1893) 11 6570 2038 0. 6900 82Nwton & Kennedy 6279 1902 0.69,4 22

Kell & Whalley 6323 1920 0.6967 6

Vedam & Holton 6271 1882 0.7002 26

(low pressure)Wilson & Bradley 6215 1853 0.7022 24

-5-

Fisher, Dial

As fits were made to data from different in- and Whalley as a functim of temperature. On thisvestigators, it became apparent that there were basis we select the Tamlirz equation to representlarge differences in optimum parameter values at the PVT properties of pure water. In our final cal-the same temperature as illustrated in Table IV with culations we have included KellsI data for purethe Tumlirz equation for pure water at 100. Exam- water at atmospheric pressureination of the Thmlirzequation shows thateven small The TImlirz equatior and the temperaturechanges in any one parameter would greatly affect dependence of its parameters are shown in Table V.the predicted voldmes and hence the quality of fit. Comparisons of fits obtained with our equation andHowever, as Tait noted, large changes in one par- that observed by Kell and Whalley are shown inameter may be made provided compensating changes Table VI. It is quite reassuring to note that theare made in the other parameters; for example, in- density maximum at atmospheric pressure accord-creasing X and Po while decreasing V® would have ing to our results occurs at 4.00° .little effect upon the resulting curve. This is dius- Although McDonald does not consider thetrated In Figure 4a ( X vs. Po) and Figure 4b (PO Tumlhrz equation as a best choice for representingvs. V. ) where the sloping line is derived from an- the properties of pure water we believe it providesalysis of Kell and Whalley data. The derivation of a good base for an analytic function to representthese compensating relationships is explained in Ap- specific olumes on an absolute basis. In their rep-pendix II. resentations for the PV isotherm data both McDonald

In comparingthe fits to the Kell and Whalley and Kell and Whalley normalize their equations todata with equations (2), UTE, and (3), Tumlirz, we the atmospheric pressure specific volumes. This,see that even though UTE parameter B (Figure 2a) of course, makes it easier to include later and moredisplays a rather smooth behavior, C (Figure 2b) accurate atmospheric pressure data. It also makesshows an erratic temperature dependence at the calculation of thermodynamic properties, partiou-higher temperatures. This contrasts with the smooth larly the thermal expansion coefficient, more in-behavior of the Tumlirz parameters (Figure 1) over volved.the whole temperature region except at 50D. Uponre-examining the w vs. T dependence for the twoequations in Figure 3, we should also note that the The Tbmlirz equation for pure water alsoprecision of fits of the Tumlirz equation vary in ago- provides a good basis for representing the proper-cord with the errors In accuracy estimated by KeU ties of sea water as we shall see next.

KELL & WHALLEY

1900- NEWTON & CRINNOY10%4 SON 16 RRADL|IT

102 AuAGA?'b* VEDA S 1401,T014

100 VADAM HOLTON

9150 - WILSON BRADLEY

698 I\

6%_

,, I ...o I , ,,i;100 6200 6300 6400 6100 6X ..

P,, lbeal P.. Ihwrsl

Fig. 4 Plot of X and V. vs. P0 values obtained atoptimum fits of Tumlirz equation to purewater at about 100 from various sources.The straight line Is the compensating rela-"tonshlpderved from Kell and Whalley data.

-6-

SIO Reference 75-28

B Table V

(a) Pure Water, 0 - 100 ° , 8 ppm and 100 - 1500, 15 ppm fits toKell and Whalley data ()k in bars cc/gm) P, Po in bars and Vo , V in cc/gm.

1=,788,316 + 21.55053 T - 0. 4695911 T2 + 3.096363 x 103T3 - .7341182 x 10"5 TA

L2 -3 3 -5P5918. 499 +58.05267 T- 1. 1253317 T + 6. 6123869 x 10 T' - 1. 4661625 x 10 T4

Z .!6180547 -. 7435626 10 T +.3704258 x 10' 6315724 x 10- 6 T +. 9829576 x 10T -. 1197269 x 10 - T5 + .1005461 x 10 1 T6

- .5437898 x 1014 T +. 169946 x 10-16j T - .2295063 x 10 19 " 1'9

B. Sea Water Tumlirz equation. The original data occurs at ir-regular temperature Intervals, complicating com-

The most extensive sea water measure- parison at different salinities. Consequently wements have been made by Wilson and Bradley 3,4/ have used polynomial interpolation to provide dataand Newton and Kennedy . at even 50 irtervals; it Is this Interpolated data that

The Wilson and Bradley data is published in will be referred to as the Wilson and Bradley data.two forms: the original experimental data and the The UTE and Tumlirz equation were usedsmoothed data; the isotherm precision to be + 20ppm to fit the seawater isotherm data at each salinity.in density. With a 22 term parameter equation they The fits were on the order of 30 ppm for both theobtained a fit with a standard deviation of 80 ppm and UTE and Tumllrz equation; therefore, the Tunlirzwith the 10 parameter Tumlirz equation, which they equation was selected for further study because the

r finally chose, they obtained a standard deviation of quality of fit does not restrict the selection and pro -130 ppm. Because a Tumllrz equation was used as vides a better fit to pure water.a part of the smoothing procedure, the smoothed Figure 5 shows x , Po and Vc vs. salinitydata cannot be used to test the applicability of the at 100from thefits of the Tumlirz equation to Wilson

and Bradley, and Newton and Kennedy data. Theparameters do not appear to vary in any consistentway with salinity because random parameter varia-tions resulting from imperfect data obscure the sys-tematic variations with salinity.

Table VI

Comparison of Standard Deviations of Fits to Kell The sea water data was normalized to pureatwater by constraining X to be equal to the value forSand Whalley Data of the Equations of State Kell and

Whalley used and the one we ise. pure water at the same temperature. The X valuesfor Kell and Whalley data were polynomial Interpo -

Standard Deviation x 106 (cc/gm) lated to the temperatures of the sea water data andTemp Fisher & Dial Kell & Whallev least-square fits were made for V and Po at eachtemperature and salinity. Figure 6 shows the re-

0.000 9. 6 44 suits at 100 from which a linear salinity dependenceS10.002 8.0 10101.201.is now evident. A linear variation of Po with ealin-

19. 997 7.4 7.9 ity is in accordance with Tammann's hypothesis R./24. 998 7,0 4.9 ~which states that internal pressure, the usual inter-24.998 5,8 4.62.5 3846 pretation for Po, should increase linearly with sal-3001 6.9 1.1 Inity. Also, V® is seen to vary linearly with sa-30.112 6,9 1 8. linity. Using the linear variations suggested by039. 007 6.7 63.8 Figure 6 we may writel/:?'50.007 6.7 6.0

60.001 5.6 4.3 X70.003 11.2 13.6 Vp =Vo -K S+ P- + KS (5)80.003 8.4 4.2 0 290.007 6.7 4.6

100.005 10.2 5.8 Fquation(5)was fit to the Wilson and Bradley

110.005 11.8 10.3 data by setting V. , X and Po equal to the pure120.007 13,2 9.6 water values calculated for the temperatures atV130.010 143 5.8130.010 14.3 5.8 which the data was taken; attempts to determine the

: k 140.009 18,5 9.1I4 temperature dependence of K1 and K2 were not41 1150,016 22.8 9.0150.016 2..8 .9..0 entirely satisfactory partly because the data were

-7'

Fisher, Dial

2300 70~~~~lo • -ii il i

KEY

& NOIWON , KENNEDY 7200200 WILSON & §RAO4EV

I t0 AO A

2M

0 to 20 30 40 1JSO&INY~~l~m S (%) 20 0 to 20 30 4

.7000

AU

Fig. 5 Plots of X., Po and Vo obtained from optimumfits of Tumlirz equation to sea water data atabout 10' .

a 0a

Aw

0DISCUSSION

A. Pure Water.00

Our calculations have led us to use theTumlirz equation to represent the properties of purewater. It is rather interesting to note that the valueof V © is very close to the specific volume calcula-

ted for the close packed form of water in most0 10 20 0 4 theorl 1 Vhich treat water In terms of a two state

Y $ Me model . Along aqtfter line of thought we note thatI David and Litovltz calculate the temperature de-pendence of the relaxing and non-relaxing campres-

obtained at varying temperatures. In addition ic was sibilities for water and find a cross over point in thefound that thermal expansion coefficients calculated temperature region near the compressibility mini-from our parameters for equation 5 did not agree mum at 450 and sound velocity maximum at 720 . Iawiththe recent work of Bradshaw and Schlelcher 14 /. this paper we have seen that a crossover in the pre-

While we are able to obtain with our equa- cislon of fits of the Tumlirz and UTE equations alsotion fits to the Wilson and Bpfadley data with stan- occurs near the same temperature.dard deviations of about 10" cc/gm which Is com- Since the open packed structure (ice-like)parable to the Fit Wilson and Bradley obtained with predominates at lower temperatures, it is temptingtheir equation (1.3 x 10- 4) we feel that is possible to associate its properties with the Tumlirz equationto obtain a better equation of state by using sound and the close packed structure with the UTE. Wevelocity data to determine K2 and thermal expansion have seen that the TE equation provides a moredata to determine K1 . This has been done In another precise fit to Kell and Whalley's data at higher tem-paperI5/, peratures and we also note that the very high pres-

.- 8

S10 Reference 75-28

A produce a breakdown of the open structure of water.In this sense, the transition from the Tumlirz to theUTE equation appears to be consistent. Using theresults of Davis and Litovitz for the fractions ofopen and closed structure we attempted to fit theKell and Whalley data with a linear combination ofthe Tumlirz equation and the UTH. The results sofar are inconclusive and w. hope to pursue the mat-

6900 - ter later based in part on the work of Frank andQuist 171I I Q The parameter B in the UTF, and Po in theTumlirz equation have been referred to as an inter-nal pressure. Since no experimental data as yet

60 - exists on the ultimate tensile strength of water, wecannot ascribe an intermolecular significance to theinternal pressure.W. Going back to the two-state

A model we note that the lower internal pressurevalues of B would be attributed to the disorderedI/close packed state and the much higher P0 values to

6700 open-packed state. Again this seems to be reason-1able from a qualitative viewpoint.

t to 20 30 40

,B. Sea Water

695 Although we have not established the tem-perature dependence of the parameters K1 and K2 to

four satisfaction in this paper, we have clearly dem-Ionstrated the linear salinity dependence of the cor-

rections to VO and Po as shown in equation 5. Thisj promises to be a very useful equation in that atmo-

ato spheric pressure data on the density, sound ve-locity, and heat capacity can be used to determinethe high pressure properties of electrolyte solu-tions. Furthermore, it promises to be useful fordetermining partial molal properties of solutions at

.6s 5 infinite dilution from data at high concentrations.

Unfortunately, most data at atmospheric pressureI I... and elevated pressures up to now on solution densi-

o to 20 30 40 ties are not accurate or precise enough to do a de-salwity $ ff") finite study utilizing equation 5 in this matter../

Experiments are in progress at various labora-Fig. 6 Plots of P0 and V obtained from optimum tories which promise to yield data of the necessary

fits of Tumlirz equation to sea water data of accuracy. It may also be possible to calculate theWilson and Bradley at 10 when X is con- properties of solutions with more than one solute by

pure water, additions of their respective values of K1 and K2 .

'The principal difference in the form of equa-tion 5 from that used by Eckart and Wilson and

10/ Bradley is that X is not a function of salinity. Thissure data of Vedam and Holton 1 0 at lower temper- m y is a n a fucionsaint. Thiratures shows a lower standard deviation when the

is fcise solution data will determine the utility of thisUTE is used for fitting. equation. Yayanos 18/ has used another equation

-* Although the accuracy of the data at higherI temperatures and very high pressures Is notas good 4as that of the low temperature Kell and Whalley Adata, the fact that the UTE provides better a fit atthe high temperature and pressures may be signifi- to analyze his PV isotherms for water and NaCI. Incant. Bkth Increasing temperature and pressure further work we expect to test this equation also.

-9-

Fisher, Dial

ACKNOWLEDGMENTS Intervals about the optimum value found by the pre-vious search. This search was coded Into a computer

This work was initiated partly in response routine which terminated when the optimum B wastodiscussions of the equation of state of sea water 2 0 ' localized to + . 001 bar.at the meeting of the Joint Panel on Oceanographic The fitting procedure for the Tumlirz eaqua-Tables and Standards in Berne, 1967. The authors tion Is quite similar. Values of V., and o* areare grateful for the interest and encouragement of calculated for selected values of Po. A search forC. Eckart, W. Wooster and R. Arthur in this work. the optimum is continued until Po is localized± .001We particularly wish to acknowledge the stimulating bar.discussions with Dr. Eckart. By computerizing nonlinear least-square

This work was supported by the Office of calculations, we are able to obtain the optimum fits.Naval Research and the National Science Foundation. In all cases the standard error is computed to pro-One of us, F.H. Fisher, wishes to acknowledge the vide a simple, uniform measure of the quality of fit;support of UNESCO which made it possible for him indeed, the minimization of this quantity Is the es-to attend the meeting in Berne as a member of the sence of least-square fitting.Joint Panel.

APPENDIX I: APPENDIX II:

COMPUTATIONAL PROCEDURE FOR NONLINEAR, CORRELATED PARAMETER ESTIMATESLEAST-SQUARE CURVE FITTING

Parameter estimates obtained from dif-Nonlinear least-square curve fitting re- ferent sets of data show both random and systematic

quires the use ofiterative approximationto calclate variations. Experience with the Tumlirz equatonparameters of the curve. We use the fit of the UTE has shown that there are large variations In pararn -equation (2), to the Kell and Whalley data at 10.0020 eterestimates obtained from all but the most pre-to Illustrate the iteration. cLse data. FIgure 5 shows that random variations of

We define (r, the standard error, by qua- parameters for seawater data are so large that theytion (4). Least square curve fitting implies the se- obscure any systematic variations with salinity.lection of parameters VI, C, and B to minimize cr By considering different sets of data, we

If the value of B In equation (2) were known, correlated as shown in Figure 4. Also, any onewe could use conventional, linear least-square fit- parameter can be changed without significantly de-ting to determine VI and C and calculate the re- grading the quality of fit provided that compensatingsuiting r. Table VII shows the results of these cal - (I. e., the correlated) changes are made in the other

culations for tabulated values of B. It Is readily parameters. Table V may be considered to tabulateseen that a- smoothly comes to a minimum value the values of X and V. necessary to compensatenear B = 2900 bars. The nonlinear, least-squaws fit for variations of Po. Indeed, when the 'k vs. Poanis Implemented by searching successively smaller Vw vs. Po columns from Table V are plottedin

Figure 4 the previously obtained parameterestimatesare seen to agree closely.

Table VII Similarly correlated parameter estimatesoccur with the UTE. These correlations explain the

Least-Square X, Vo, and o* from Fit of Tumlirz success of previous work which constrained the CEquation to Kell and Whalley Data at 100C. parameter because B values could then be selected

which largely compensated for the non-optimum CPn(bar) lambda(bar-cc/gm) V,(cc/gm) a (ppm) values chosen21 / .

The existence of the compensating relations5600 1,530.894 0.727055 60.4 also provides a rationalization for constraining a5800 1,64.056 0. 716 2.5 parameter of the seawater equation. We know such6000 1,840.6 0.716 11260 constraints won't degrade the fit and that the com-6200 1,850.564 0.701888 11.26323. 46* 1,920.082 0. 696711 6.2 pensating relations will reduce the erratic variations

6400 1,963.813 0673502 8.4 of the remaining parameters. In particular, con-6600 2,080.406 0.685117 20.7 straining the k parameter to its value for pure

water at the same temperature produced the linear6800 2,200.346 0.676733 33.7 variations of Va and P0 wlthsallnitywhichTamman*optimum value found after continued search had suggested.

-10-

SIO Reference 75-28

REFERENCES

L .1. C. Eckart, Am. 1. Scl. 256, 225-240 (1958). 12. G. Tamman, (See R. E. Gibson, J. Am. Chem.2. G. S. Kell and E. Whalley, Proc. Roy. Soc. A. Soc. 56, 4 (1934), or H. S. Harned and B. B.

258, 565-614 (1965). Owen, The Physical Chemistry of Electrolytic3. W. Wilson and D. Bradley, U. S. Naval Ord - Solutions, Third Edition, p. 381, Reinhold Pub-

nance Laboratory, NOLTR 66-103 (1966). lishing Corporation, New York, (1958).4. W. Wilson and D. Bradley, Deep Sea Res. 15,

355-363 (1968). 13. F. H. Fisher and 0. E. Dial, Jr., J. Acoust.5. M. S. Newton and G. C. Kennedy, J. Mar. Res. Soc. Am. 45, 325 (1969).

23, 88-103 (1965).6. P. G. Tait, Rept. Sci. Results, Voy. H.M.S. 14. A. Bradshaw and K. Schleicher, Deep Sea Res.,

Challenger Phys. Chem. 2, 1-76 (1888). 17, 691-706, 1970.7. J. R. McDonald, Rev. Mod. Phys. 41, 316-349 15. F. H. Fisher, R. B. Williams, 0. E. Dial, SIO

(1969). Reference 75-30, 1 December 1975.8. 0. Tumllrz, Wien, Akad. Wissensch., Sitzung- 16. C. M. Davis and T. A. Litovitz, J. Chem. Phys.

sberichte, Math-Naturwiss. KI. 118A, 203, Phys. 42, 2563-2576 (1965).(1909). 17. H., S. Frank and A. S. Quist 34, 604-611 (1961).

9. E. H. Amagat, Ann. de Chim. et Phys. 29, 18. A. A. Yayanos, J. Appl. Phys. 41, 2259-226068-136, 505-574 (1893). See N. E. Dorsey, (1970)."Properties of Ordinary Water Substance, 1. 19. Based on preliminary results obtained by Prof.pp. 207-209, II, pp 210-211, Hafner Publish- B. B. Owen.ing Co., New York, (1968). 20. UNESCO Technical Papers in Marine Science,

10. R. Vedam and G. Holton, J. Acoust. Soc. Am,, No. 8, Third report of the joint Panel on Ocean-43, 108-116 (1968). ographic Tables and Standards. UNESCO, 1968.

11. G. S. Kel, J. Chem. Eng. Data 12, 66-69 21. R. E. Gibson and 0. H. Loeffler, J. Am. Chem.(1967). Soc. 63, 898-906 (1941).

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