Revista Latinoamericana de Metalurgia y Materiales, Vol. 20, N°1, 2000, 79-86

A LATTICE FLUID APPROACH TO LIQUID METALS: EQUATION OFSTATE PARAMETERS AND PERIODIC PROPERTIES.

a Y. von Bergen and b R. von Bergen

aDepartamento de Física Universidad Simón Bolívar Caracas- Venezuela.E-mail aybergen@net-uno.net

b Departamento de Química Universidad Simón Bolívar Caracas-Venezuela.E-mail rbergen@usb.ve

ABSTRACT

The microscopic and macroscopic parameters of an equation of state of a lattice gas approachare calculated for thirty-three liquid metals. These calculated parameters are compared with theparameters of organic, inorganic and surfactant compounds. The macroscopic and microscopicparameters of liquid metal s related to the energy interaction between occupied sites in the lattice arelarger than other cases. This last aspect indicates that the intermolecular interactions are considerable.Various linear correlation functions between experimental properties and the Sanchez-Lacombeparameters are examined. The presence of transition metal s destroys any possibility to find a goodlinear correlation function. The product re'" is the intermolecular interaction potential and it is relatedto the fluid cohesive forces. The product rv* is the molecular volume ofthe atom.

Key Words: liquid metals, lattice gas, liquid-vapor equilibria ,Periodic properties.

RESUMEN

Se determinan los parámetros microscópicos y macroscópicos de una ecuación de estado basadaen un modelo de redes gaseosas para treinta y tres metales fundidos. Los parámetros calculados secomparan con los parámetros de compuestos orgánicos, inorgánicos y surfactantes, encontrándose, queen los metales fundidos la energía de interacción entre sitios ocupados de la red es mucho mayor queen los otros casos. Lo que indica que las interacciones intermoleculares son considerables. Seexaminaron varias correlaciones lineales entre los parámetros de Sanchez - Lacombe de metaleslíquidos y algunas propiedades experimentales. La presencia de metales de transición destruyecualquier posibilidad de hallar buenas correlaciones lineales. El producto rs- es el potencial deinteracción intermolecular y está relacionado a las fuerzas cohesivas del fluido. El producto rv"proporciona el volumen atómico de los metales.

Palabras Clave: metales líquidos, red de gas, equilibrio vapor-líquido, propiedades periódicas

80 Y. von Bergen and R. von Bergen /Revista Latinoamericana de Metalurgia y Materiales.

l. INTRODUCTIONLiquid metal technology has been the subject of an

impetuous development in the recently years, mainly dueto the application of Iiquid metal s in nuclear techniques.Actually, there are many applications of liquid metals.They are important in the preparation of metal alloys andceramics, as well as, their usage as refrigerants in nuclearreactors [1]. Knowledge ofthe thermodynamic propertiesof liquid metal s have taken especial relevance, thus it.would be convenient to identify a simple theory, thatfacilitate the estimation of thermodynamic properties offluid metals,

-c, • In the present article a lattice gas approach is used todescribe the behavior of liquid metal s with the theory ofSanchez-Lacombe [2] to describe a simple fluido Thistheory is constructed on three microscopic parameters: 1.- r, the effective number (not necessarily an integer) ofsites occupied by an atom in the lattice; 2.· v*, theaveraged volume of one site in the lattice; 3. - e*, Themolar interaction energy between two occupiedneighboring nearby, unbound sites. These microscopicparameters are also related to macroscopic parametersT*, p* and P* which are respectively the characteristictemperature, characteristic pressure and characteristicdensity.

Once obtained the the Sanchez Lacombe (SL)parameters of liquid metals it is possible to determineeasily thermodynamic properties such as liquid vaporequilibrium eurve, spinodal region, enthalpies, ete.

In the present article the parameters of a lattice gasmodel for thirty-three metals are first time calculated byusing densities and vapor pressure experimental data[3,4]. These parameters are compared with organic,inorganic and surfactant parameters previously calculated[5-7]. To understand better the obtained values of thedifferent liquid metal parameters, which are classified asin the periodic table in groups and periods, in order, toexamine if the behavior of them are similar to periodicproperties such as atomic radius, vaporization enthalpy,boiling point temperature cte. Fact that has motivated tolook for linear correlation's inside the groups andperiods. Once the Sanchez- Lacombe State Equation(SLSE) parameters have been calculated, it is possibleto determine linear correlation's that may exist betweenthe different members of elements that belong 10 thesame group or period (of the periodic table) withexperimental properties [4] of such elements as forexample atomic radius, vaporization enthalpy, molecularweight and boiling point temperature.

These calculations are made with the idea to obtain abetter molecular interpretation ofthe (SLSE) parameters.

l.MODELThe lattice gas theory [2] describes a fluid (liquid

metal) as a model in which there are empiy and occupiedsites and each atom of the liquid metal occupied a

constant number of sites, with such a model in mind itcan be shown that

ti + P+ T(ln (1- p) + (l-l/r) p) = O (1)

where p, P, and T are the reduced density, reducedpressure and reduced temperature, defined by thefollowing equations

p= pl p* = l/v P= P¡P* T=TfI'* (2)

where p, P, T are the density, pressure and absolutetemperature, respectively. The macroscopic parametersp*, p* and T* are: the close packed density, thecharacteristic pressure and the characteristic temperature,respectively. These parameters can be expressed as afunction of the so-called microscopic parameter of thefluid as:

p* = Mlrv* p* = E*/V* T* = E*/k (3)

where r is the number of occupied sites by one atom(liquid metal) within the gas lattice, v* is the averagedvolume of asite in the lattice and &* is the interactionenergy between nonbonded occupied nearest neighborsite. M is the liquid metal atomic weight and k is theBoltzmann's constant.

To understand better the meaning of Sanchez-Lacombe parameters it is convenient to make a fewcomments that iIIustrate better the physical meaning ofthem.

Comment abouJ e- this parameters is defmed aboveas the nearest neighbor interaction energy. However, thisdefinition can be generalized. The configurationalpotential energy E of a lattice fluid model [2] can beexpressed as:

E=~(rN)<E> (4)

Where,

where < & > is the averaged interaction energy of a merwith all other mer in the fluid, &(R) is the intermolecularpotential between mers separated a dístance R. g (R) isthe radial distribution function and p is the density inmers per unít volume. An attractive potential for theínteractíon between mers is assumed i.e.

E (R) = ex> for (RIv*II3) < 1

(6)

& (R) •••-So(v*l13/Rt for (R I V*ll3) > 1

Revista Latinoamericana de Metalurgia y Materiales, Vol. 20, N°1, 2000 81

if used a hard core potential, the pair distribution functionin the mean field approximation is given by

g (R) = O for (R / V*l/3 ) -c 1

(7)

g (R) = 1 for (R / V*l/3 ) > 1

Substitution of equations (6) and (7) into equations (4)and (5) produces

E=-rN&*p (8)

and

&*=27t s, / (0-3) (9)

For the usual value of o = 6 for attractive potential,

&* =21t Eo /3 (10)

Thus, &* is proportional to the depth of the potentialenergy well.

Comment about P* the lattice fluid theory (LF)intent to describe the fluid as a disorder system, not as aordered system Iike a crystalline state. The closedpacked state should be disordered like the glassy state.As an example are compared the values of density of acrystalline (face centered cubic) system , and closedpacking of spheres that are respectively 0.74 and 0.637[8]. When close packed densities (p*) are comparedwith well known crystalline densities it is found thatmost p'" densities are usually around 10% smaller. Thusfrom equation (3) rv* can be identified as the closedpacked molecular volume of'the disordered fluido

Comment about re» and rv" it is also instructive toexamine the variations in n;* and rv* for thenormal alkanes. Between propane and tetradecane, rv'"increased from one member to the next by 15.0cm3/mole. This suggest that is each CH2 groupcontributes a constant amount to the molecular volume.The total molecular interaction energy re= , that isequal to the energy required to convert one mole of thefluid from closed packed state (reduced denslty = 1) to avapor of vanishing density (reduced density = O)increased between propane and tetradecane in asystematic way of 4.35 KJoule/mole for each CH2 [8].The value of r for each normal alcane can be determineby dividing its molecular weight by 14. The ratio forP* equation (3) is defined as the characteristic ptessureand is equal to cohesive energy density oí the fluid in theclosed packed state, because cohesive energy density(DEC)

DEC=AEvapfV=r &* / rv*=P* (11)

Thus P* measured the cohesiveness of the fluid or theintermolecular interaction.

3. PARAMETERS

3.1 Determination

The method that has been used to obtain the SanchezLacombe State Equation (SLSE) parameters is based on acomparison. The experimental curve of saturated vaporpressure versus temperature is compared with atheoretically calculated curve, of reduced saturated vaporpressure versus reduced temperature, using a middlepoint of coincidence between both curves and the rparameter as a variable. Thus, it is possible to find thebest value of the r parameter by using as criteria theminimum value of deviation between the experimentaland calculated curves. In this way are obtained, for agiven r,· the macroscopic parameters T'" and p* andusing equation ( 3 ) are obtained the microscopicparameters p*, v'" and &*. It is important to realize thatonly three parameters are independent variables.

4.RESULTS

The determined macroscopic parameters andmicroscopic parameters are shown in tables 1-6.

The best way to study the 33 (SL) parameters ofliquid metals is by using the periodic table, whichclassifies the metals by groups and periods. Thus allpossible linear correlation's between metals in the samegroup or period are well examined as can be shown infigures 1-12.

In order to make a more systematic discussion of allpossible lineal correJation coefficient studied in thisarticle, is shown table 7 in which all correlationcoefficient R2 are indicated for the different correlation'sconstructed.

S. DlSCUSSION

In order to make a more systematic presentation of .SLSE parameters of liquid metals with the parameters oforganic, inorganic and surfactant compounds two tablesare constructed table 8 and table 9.

The surfactant considered here are alkyl-ethoxy-alcohols whose general formula is given by:

First, are compared the values of r parameters. Anexamination of tables (1-6) indicates that the range ofvariation of r for liquid metals is (0.85 -5.49). These rparameters are smaIler than the parameters of compoundslisted in table 8 (organic and inorganic compounds) and,of course, a lot smaller than the parameters of surfactants

82 Y. von Bergen and R. van Bergen /Revista Latinoamericana de Metalurgia y Materiales.

given in table 9. This is a reasonable result because themolecules have bonds and the bonds increase the size ofmolecules and in the liquid metals there are not anybonds presento In others words, a big molecule needs tooccupied more site in the lattice than a small molecule.

Second, ~ compared the values of 8* parameters.An examination of tables 1-6 indicates that the range ofvariation of &*for liquid metal s is (4 - 77 .53) Kcal/rnole.This s* parameters are a lot larger than the &*parametersof compounds listed in tables 8,9 that are around unit. Itis reasonable result because liquid rnetals have higherboiling point ternperatures and evaporation enthalpiesthan the other compounds considered here.

Third, are compared the values of v. parameters thatis .the average volume of sites in the lattice Anexamination of tables 1-6 and tables 8-9 indicates thatthe parameters v* are comparable in all the caseconsidered here.

The macroscopic parameters p. y T* of liquidsmetal s are greater than the parameters here presented fororganic, inorganic and surfactants compounds. Theseresults are reasonable because ofthe equation (3) and thefaet that the microscopic parameter 8*, in the liquidmetals is greater than the other cases.The results of all intended correlation coeffícients areshown in table 7 and there are good linear correlation' s(R2 $OOj 1) and poor linear correlation's, that are not linearat all (R2 «1). A detail examination of table 7indicates:

1. - When the transition rnetals are present, it isimpossible to establish any good correlation coefficient(see periods 4* and 4***).

2~ - When the transition metals are excluded in period 4,the linear correlation enhances (see period 4*·). As canbe observed in columns 2,3,5 and 6, not all correlationcoefficients are good.

rabie 1 SLSE miaoscopic p8IlII1Ieters

GroaD Elemetlt r v*(mllmol) E·(Kcallmol)lA Litbium 1.34 12.8 22.33lA Sodium 3.58 8.4 7.01lA Potassium 235 24.8 7.48lA Rubidium 4.03 17.1 4.88lA Cesium 2.S4 35.8 6.412A Beryllium 1.10 5.4 68.932A Magnesium 1.66 9.9 16.202A CaIcium 2.13 14.8 16.052A Stron~ 3.22 12.2 10.172A Barium . 2.49 17.3 15.193A AbninÍlm 1.77 8.7 31.153A Qillium 1.83 8.0 27.073A Indium 1.70 11.8 26.893A ThaDium 3.28 6.2 11.724A Gennanium 1.21 12.9 50.324A Tm 1.47 14.2 41.574A Leed 2.46 9.5 17.33

3. - Correlation coefficients in which the vaporizationenthalpy appears are:a) E* vs MIvap (column 3)b) e*/v* vs MIvap (column 4) = p. vs MIvap (column 8)e) re* vs MIvap (column 11)

The correlation coefficients including MIvap aregood except in the cases in which there are presenttransition metals.

The vaporization enthalpy is a molar quantity and sois r 6*; therefore, probably for this reason, the bestcorrelation coefficient corresponds to the curve re. vsMIvap (colurno 11).

4. - The correlation in the third period is good except inthe cases of columns 2 and 10.

5. - The correlation ofrv* vs atomic number are not goodexcept for group ] A and 8 B.

6. - The correlation of rv* vs atomic radius are goodexcept for group 8B in which are present transitionrnetals.

7. - Correlation's in which the boiling point ternperature(Tb) appears are:E*vs Tb (coturno 5) = T* vs T, (column 7)p. vs Tb(colurno 9)

In these cases, the best correlation coefficient is forthe curve P* vs Tb. The worst correlation coefficientsare in groups in which are present transition rnetals.

8. - The correlation coefficient of the curve of thecharacteristic density (p*) vs atornic weight (AW) areexcellent because p* =A WIr v*.

Table 2 SLSE maaoscopic parameters

Groap EIe_t 1'*(1() P*(At) o· (g/mI)lA Litbium 11237 71754 0.404lA Sodium 3529 34390 0.764lA Potassium 3766 12461 0.67]lA Rubidium 2456 il78~ 1.242lA Cesium 3224 7397 1.4632A Be!yDium 34687 525830 1.512A Magnesium 8153 67363 1.4732A Calcium 8076 44S58 1.2742A Strontium 5472 368S4 2.232A Barium 7642 36349 3.1983A Aluminum 15677 148284 1.7543A GaJIiwn 13622 140649 4.793A Indium 13533 93973 5.7323A Thallium 5895 77891 10.044A Gennanium 25322 161156 4.6414A Tm 20920 Ul687 5.7014A Leed 87]9 7S137 8.86

Revista Latinoamericana de Metalurgia y Materiales, Vol. 20, N°1,2000

Table 3 SLSE microscopic parameters

GJ'Oup Elemeot r v·(ml/mol) s*(KeallDlot)m Copper 0.95 9.8 77.53m Silver 1.06 12.4 45.25lB Gold 5.49 2.3 17.012B Zinc 1.60 6.7 14.162B 'Cadmíam 3.92 3.7 6.142B Men:ury 3.00 5.2 4.288B Nickel 1.10 8.4 56.988B PalJadium 1.14 9.6 58.898B Platinum 5.07 2.9 22.95

Table 5 SLSE microscopic parameters

Period Eleme--t r v*(mIImol) &*(Keallmol)m Sodium 3.58 8.4 7.01m Magnesium 1.66 9.9 16.20m Ahnnioum 1.77 8.7 31.15IV Potassium 2.35 24.8 7.48IV CaJciwn 2.13 14.8 16.05IV GaIIium 1.83 8.0 27.07IV Gennanium 1.21 12.9 5032TraosltioB metal (period IV) t

Elemellt r v*(mlfmol) &*(KeallmoI)TIIm1ium 1.26 lO.! 54.82

Vanadimn UO 9.6 62.92Chromium 0.85 10.1 68.59Mangaoese 1.07 103 46.02

Iron 0.91 103 7l.53Cobalt 1.02 9.8 58.99Nickel 1.10 8.4 56.98Copper 0.95 9.8 77.53

Zinc 1.60 6.7 14.16

Table 7. CorreIation Coefficient R' for aIl possible linear Correlations StUdied

83

Table 4 SLSE macroscopic parameters

Gmup ElellleDt T*(K) p. (Al) p. (gl mi)

m Copper 39013 326545 6.81m Silver 22767 149747 8.19m Gold 8559 304200 15.532B Zinc 7128 87724 6.112B Cadmium 3089 68084 7.fH12B Memuy 2152 34293 12.9828B Nickel 28671 279894 6.378B Palladium 29636 252486 9.78B Platinum 11548 332307 J3.S

Table 6 SLSE macroscopic parameters

Period Elelllellt T* (K) P* (At) o· (g I 011)

m Sodium 3529 34390 0.764m Magnesium 8153 67363 1.473m AIumÍllum 15677 148284 1.754IV Potassium 3766 12461 0.671IV Calcium 8076 44858 1.274IV GaIIium 13621 140649 4.79IV Gennanium 25322 161156 4.641

Tnuitioa metal (period IV) IElemeat T*(K) P* (At) p* (g 1mi)TIIanium 27585 224278 3.781

Vanadium 31664 271744 4.831Cbromium 34517 282332 6.08Mangaoese 23158 184096 4.98

Iron 35993 286396 5.98Cobalt 29685 249594 5.901Nicke! 28671 279894 6.37Copper 39013 326545 6.81

Zinc 7\28 87724 6.11

Grwp "" MaoI*_E •••.•.11&•••• "'* •• .11&_ e••• Tb 'VI~nd T-vsT, p*v.Mlv ••• P*v.T, •VI lItomic weit rs· ••.•Mlvap"

lA 0.932 0.8841 0.99T7 0.9602 0.9751 0.9602 0.99T7 0.955 0.9338 0.9119lA 0.848 0.9513 0.9567 0.8433 0.9513 0.8433 0.9567 0.8594 0.8923 0.99993A 0.7065 0.9073 0.8954 0.923 0.8151 0.923 0.8944 0.8536 o.m 0.96694A 0.8822 0.9971 0.9261 0.9589 0.9752 0.9589 0.9284 0.9998 0.9901 0.9547lB 0.4847 0.1496 0.7229 0.1334 0.9864 0.1334 0.725 0.7455 0.9664 0.9998za 0.7572 0.6642 0.9848 0.6642 0.9998 0.6642 0.9839 0.9855 0.9839 0.87668B 0.9992 0.9945 0.9466 0.9214 0.7005 0.9214 0.951 0.9998 0.9823 0.9646

Period ,.•••MaoI*_" •••••\1"'••••"'. nAu... E.nT~ ••.•.a-k •.•• T*YOT• P*YOMl ••ap P·vsT. ....-"' ....•• rs· ••• Ml,,1I'3 0.8058 0.940S 0.9791 0.9375 0.9185 0.9375 0.9791 osrn 0.8193 0.99254* 0.3465 0.6921 0.6816 0.6414 0.9499 0.6474 0.68S6 0.638 0.473 0.87514- 0.7582 0.8962 0.9838 0.8443 0.9381 0.8443 0.9838 0.9989 0.9836 0.9871..- 0.17 o.5m 0.5624 0.483 0.8136 0.4714 0.5569 0.4714 0.613 0.7m

•aIl mdaIIic dement oflbe fourtb period•• metallic elcment of!he fuurtlt period excludirlg tnmsiIion meIaIs••• onIy tnmsition metals offourtb period

Table 8 SLSE parameters of severa! compounds(2,5,6)

o.powac: r T*(K) p* (At) E*(kallm.} v*{ ••••••••• } MoIee.weiñt D"(¡/ml)n-pentane 8.41 434 3100 0.862 11.5 72.15 74.7i-pentane 8.05 430 3000 0.854 11.8 72.15 76.2n-bexane 931 457 3200 0.908 11.7 86.18 79

U cIimdb.!t lUIDo 838 462 2910 0.918 13 155.07 142n-beptane 9.89 480 3140 0.954 u.s 100.21 80.8n_ 10.82 494 3170 0.982 12.8 114.23 82.6benc:ale 7.77 532 4280 1.06 10.2 19033 240

melhylchloride 6.75 413 5301 0.821 6.4 50.48 111walx:r 8.99 606 moo 1.2 U 18.02 112

ammonia 8.25 3112 12170 0.759 2.fj 17JB IIU

84 Y. von Bergen and R. von Bergen /Revista Latinoamericana de Metalurgia y Materiales.

Tab1e 9 SLSE parameiers of non-polar surfactants(7)

Surlaetant r T"(K) p. (At) E*(lu:aIImole) v*(mllmole) Mole\:. wel2bt p*(ztml)C4EI 13.61 512 4930 1.0183 8.53 1I8.l7 1.02CI2ES 38.43 5053 371838 I.lISI 10.04 406.62 11.24CI2E6 43.23 4922 3S~9S2 1.1347 9.78 4SO.63 9.19CI2E8 5284 4726 333355 1.1633 9.39 538.73 8.76

"j16

1524Y- 21.678x - 14.156

22 R' = 0.971 14•'di' 20 :i"] ~ 13::: 18 e y = 26.286x • 25.312g ~ 12 R2 = 0.9999~ 16 1:

•11104

12 10

10 91,3 1,4 1,5 1,6 1,7 1,8 1,35 1.4 1,045 1,5 1,55 1.6

Atomie radius (A 0) A tom ie radius (A 0)

Fig. 1. Volume VS. Atomic radius (group 4 A) Fig. 2. Volume vs. Atomic radius (group 2B)

100 so90 • •80 40

Y = 33,631x - 33,84170 R2 = 0,9792

160 ]' 30o~::::so !!.

~ 40 ~ 20

30

20 10

la

o o1,3 1,8 2,3 2,8 1 1,2 1,4 1,6 1,8 2 2,2 2,4

Atomic radius (A") Atomic radius (A")

Fig. 3. Volume vs, Atomic radius (group 1 A) Fig. 4. Volume VS. Atomic radius (group 2 A)

Revista Latinoamericana de Metalurgia y Materiales, Vol. 20, N°1, 2000

75

65

~ 55

1

35

25+---_r--~----~--_r--~----._--_r--~30 35 40 45 50 55 60 65

MltDalI (KXaA / ¡10M;)

Fig. 5. Intennolecular Interaction Energy vs ~Hvap (grupo 2 A)

100

90

180

- 701•e 60

so

4060 65 70 75 80

AH1Dall (lI:)(al../ J.loM¡)

Fig. 7. Intennolecular Interaction Energy vs mvap (grouplB)

85

70

57

55

53

51

~49e]47~'i:! 45

43

41

39

3738 43 48 53 68 73

•y - 0,5265x+ 17,055

R'=0,9668

58

AH1!ICLlt(ltl(aA / 1101..1:)

63

Fig. 6. Intennolecular Interaction Energy vs mvap (grupo 3A)

60

55

50

145

:¡; 40!'i:! 35

30

25

2085 20 30 40 50 60 70

lI.Hlllall (Xl(aA /lloM¡)

Fig. 8. Intennolecular Interaction Energy vs Mivap (Period 3)

180000 12000

•160000 Y - 9O.474x- 108198

R''''0.9998 10000

140000

8000

~120000 g.• f-C>.6000

100000

400080000

~+------.------'------'r------r----~2000 2200 2400 2600 2800r, (1<)

Fig. 9. Characteristic Pressure vs Boiling Temperature(group4A)

3000

2000+-----,-------,-----,-----,----,-----,----,----1900 1000 1100 1200 1300 1400 1500 1600 1700

Tdk)

Fig. 10. Characteristic Temperature vs Boiling pointTemperature (group 1 A)

86 y. von Bergen and R. von Bergen /Revis~a Latinoamericana de Metalurgia y Materiales.

2S

•20

;itSS."-¡lO""t

s

O+---.---r---r-~--~--~--~--~900 1000 1100 1200 1300 1400 1500 1600

Tb(K)

Fig.l l. s' vs BoilirlgpointTemperature(group lA)

6. CONCLUSIONS1.- In this artícle, to the OO8t of OUT knowledge, is firsttime that the Uquid metals are discussed in tems of alattice gas model,2.- The (SL) parameters of 33 liquid metals werecalculated and compared with (SL) parameters ofinorganic, organic and surfactants compounds, alwaysfinding logícal results as can be appreciated in thediseussíon .3.~ It ls notorious from tables (1-6) and (8,9) that theparameters s'" and T'" of liquid metals are much graterthan the parameters 8'" and T'" oí others compounds,which índicates that the interaction between occupíednearest neigbboring sítes is eíevated, which producesstrong interaction energy between fue atoms.4.· It is incredible that a simple model sueh as the latticegas permits the calculation of thennodynamic propertíesby using onl)' 'VapOr pressures data and densities canprove most useful.S. - The presenee of transition metaís with the variablesstudied here destroys sny possibiUty 10 find a good value10r the oorrelation ooefficient R2, which means tbat fuetransition metals for themseJves are no correlated,6. - (Nstute of parameters ~*). One oí fue best linearcomlation's corresponds to the CUlVe rn* vs MIvap.which indicates that the total energy of intennolecularintemctions is related lo fue Huid eohesive torees,1.- (Nature of parameters l'V*). The CUlVe characterlsticdensity (p'*) vs atomic weigbt has an excellentconelation ooefficient, which reinforces the fuct that rv*is the total volume of tbe atom, as tbe chamcteristicdensity is the atomic weisbt ( g I mole) divided by tbetotal volume (mIl mole). Thus \he to18l volume is rv*.8.- AAexoeUent oomlation ooefficient is oblained fur thecurve rv* vs atomic radius., whioh inditates tbat rv* is fuetotal volwne ofmetals

y =0,0319x+ 2,1652R2 = 0,9901

8

6

s

4+-------.-------.-------.-----~50 100 ISO 200

Alomic _ight ( g Imole)1700 250

Fig. 12.CharacteristicDensityvs atomic Weight(group 4 A)

9.- The parameters ofliquid metals here calculated, to thebest of OUT knowledge open a new field of research andapplicatíon to metallurgy by using the lattice gas modelto describe mixtures.10.- Witb the Imowledge of líquid metal parameters it ispossible determine the liquid- vapor equilibriumconditions, the spinodal region and other tbennodynamicproperties. Furthermore this model can be extended tomixtures,11.- As a consequence of the used model, it can be easilyunderstood tbat the parameters v* and E* are related tothe size and intermolecular interaction potential,respectively.

7. REFERENCES

1. H.U.Borg Stedt and G. Fries. Liquids Metal Systems:Material Behavior and Physical Chemistry in LiquidsMetals Systems 2 Ed.. Plenum Press. New York andLondon (1995)

2. C. Sanohez and R. H. Lacombe, J. Pbys. Chem. 80(1976) 2352-2362

3. Takimichi Iida and Rodenek LL. Guthrie, •• ThePhysical Properties of Liquid Metals" ClarendonPress. Oxford (1988)

4. Handbook of Chemistry and Physics. CRC Press 53th

&lition5. R. von Bergen, C. A. Hemandez, L. Cuetos ami Y.

von Bergen Anales de Química, 87 (l99I) 291-3056. R. von Bergen, E. Roge], L. Cueros and Y. von

Bergen Anales de Química, 81 (1991) 293-2967. R.von Bergen and E ..Rugel Fluid PItase Equilibria

153 (1998)63-728. ID. Bernal «Liquids StNcture. Properties. Solid

Intexactlons" Ed, T. J. Hughel , Elsevier Ansteldam(1965).