A generalized equation of state with an application to the Earth s Mantle

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1 Geofísia Internaional 49 (), 77-8 (010) A generalized equation of state with an appliation to the Earth s Mantle J. A. Robles-Gutiérrez 1, J. M. A. Robles-Domínguez 1 and C. Lomnitz 1 Universidad Autónoma Metropolitana, Iztapalapa, Mexio City, Mexio. Instituto de Geofísia, Universidad Naional Autónoma de Méxio, Mexio City, Mexio Reeived: June 9, 009; aepted: February 4, 010. Resumen En este trabajo se muestra la pertinenia de inluir en la euaión de estado de Kamerlingh-Onnes, interaiones múltiples no aditivas de fuerzas entre partíulas. Estas fuerzas son de aráter eletrodinámio. A partir de la euaión de estado así generalizada se obtienen las isotermas en la veindad del punto rítio y del punto triple para sistemas polares o no polares. Se desarrolla el ejemplo del agua. Se generaliza la euaión de estado para el manto desarrollada por Birh, y en partiular se obtiene la ompresibilidad isotérmia para el manto terrestre. Se presentan las formas que toman, bajo esta generalizaión, algunas leyes de la meánia y eletrodinámia. Palabras lave: Euaión de estado, fases, energías no aditivas, energías binarias, isotermas, ompresibilidad isotérmia. Abstrat This study analizes the pertineny of inluding in the state equation of Kamerlingh-Onnes, non additive, potentials of multiple-interatiions of partiles. These fores are indeed real and of a eletrodinami harater. From the state equation no gerenalized, we obtained the isotherms in the veinity of the ritial point, and of the triple point for polar (or no polar) systems. We developed the example of water. We generalized the state equation for the mantle developed by Birh, and in partiular, we obtained the isothermal ompressibiliy for the earth mantle. We present, the forms that, under this generalization assume several laws of lassial mehanism and eletrodinamis. Key words: Equation of state, phases, non-additive energies, additive binary energies, isotherms, isothermi ompressibility. Introdution In an earlier paper (Robles-Dominguez et al., 007) we derived the equation of state for a one-omponent fluid and in general, for a fluid of 1,, 3... phases. This result was based on introduing inter-moleular potential energies suh as additive binary, non-additive tertiary, non-additive quaternary interations and so on: p = V - b G 1 + S G q q= (v ) q where p is pressure on a ontainer of volume V, V is the molar volume, b is the molar volume of all moleules, G 1, G, are funtions of absolute temperature T. Here G 1 is the mean field of all rigid shells of a moleule, G is the mean field of all additive binary interations between pairs of moleules, G 3 is the mean field of all non-additive ternary interations and so on. The term non-additive means that the interation among, say, 4 partiles is different from the other interations so that they annot (1) be inluded in a sum of binary interations or in a sum of interations of three partiles, and so on. We also onsider the mean field approximation for all these energies. In Setion II, we derive our equation (1) in a new basis of funtions, whih is (r) r, r = 1,, 3,..., where r is the molar density. In Setion III, we obtain the isotherms in the viinity of the ritial point and of the triple point of any system, polar or not, and applied it to water as an example. Setion IV, provides exist, and it is shown that, the new fores develop a generalization of eletrodynamis, lassial mehanis, et. In Setion V, we derive the isothermal ompressibility for the Earth s Mantle after Birh (1947). Setion VI, ontains the onlusions. The equation of state in base (r ) l, l = 1,, 3 If the 1 term in Equation (1) is expanded in a Taylor V - b series in terms of V -m, where V >> b and DV = -b, we find: 77

2 Geofis. Int. 49 (), = S 1 (DV d ) s 1 = 1 +b 1 + (b ) V -b s! s=0 dv V V V (V ) (V ) 3 () or 1 = S a 1 (3) V -b m (V ) m m=1 The oeffiients a m are the same found by Ursell and Mayer (.e., Uhlenbek and Ford, 1963). where: et. We replae (3) into (1) and we obtain: p = G 1 S a 1 + m SG 1 = q Sd 1 (4) (V ) m (V ) q n (V ) n q= n=1 m=1 d 1 = G 1 a 1 d = G 1 a + G d 3 = G 1 a 3 + G 3 (5) Note that the equation of state (4) ontains linear independent vetors 1, n=1,,3,... thus the Kamerlingh- (V ) n Onnes equation is derived from statistial mehanis. The parameters a m and G were alulated from additive binary potentials that ontain all additive binary interations (.e., Uhlenbek and Ford, 1963). The parameters G q, q 3, were obtained from non-additive potentials. Thus, d 3 a 3, d 4 a 4, and so on. Sine d 3 = G a + G, the quantity G a is idential to the oeffiient of van der Waals, or Ursell & Mayer. Not so G 3, our oeffiient for non additive tertiary potentials. Also, G 4 is our oeffiient for non additive quaternary potentials, and so on. If d n = 0, n equation (4) should beome the equation for an ideal gas, namely: p = RT (6) V where R is the universal gas onstant. This is beause: If we set d 1 = RT. (7) 1 = r (8) V where r is the molar density, we may write (4) as: p = Sd n rn (9) n=1 The isotherms in the neighborhood of the ritial and triple points of water Equation (9) is an appropriate equation of state whih yields the isotherms of polar or non polar fluids. Let us derive the ritial isotherm by means of the well known onditions: p = T=T p r =r p = T=T p r =r = 0 (10) = 0 (11) p = (T, r ) (1) where r is the ritial molar density and T is the ritial temperature. In equation (1), if we substitute p = p and r = r, we may use (10), (11) and (1) in order to obtain d, d 3, d 4 assuming d n = 0, for n > 4. Thus we obtain the ritial isotherm in the neighborhood of the ritial point: ( 3 r 3r 4r ) ( -RT ) 6r 1r d 3 = 0 (13) 3 4 r r r d 4 p - RT r Now we alulate the solutions of these equations for water, using data of Blak and Hartley. (1993): p =.09 (10) 7 Pa T = K V = (d) r = 1 = V R = (10) 3 J K The solution of (13) is 6 d = (10) Pa m6 () 3 d 3 = (10) Pa m9 () 3 d 4 = (10) Pa m1 () 4 For these values the non additives interations between three and four moleules are neessary to explain the Critial Point, and the isotherm in the neighborhood of the ritial point is: 78

3 Geofis. Int. 49 (), 010 p = RT r + d r + d 3 r 3 + d 4 r 4 (14) Van der Waals ould not obtain this isotherm beause he did not onsider potentials of 3, 4. moleules respetively. When a wider interval of molar densities is desired, more experimental data will be required; in this ase the ritial isotherm will be of degree n > 4 whih will enable us to ompute the values of d n to a higher preision. We onsider now the isotherm in the neighborhood of the triple point in a system, polar or otherwise. As an example we onsider water. The minimum urve near the triple point to provide stability in (, r g3, T 3 ), (, r s3, T 3 ) y (, r l3, T 3 ), must be of fifth degree. Let the right sub index 3 denote triple point, and let g, s, and 1 be the three phases. The equation (8) is redued to: where = d 13 r + d 3 r + d 33 r 3 + d 43 r 4 + d 53 r 5 (15) d 13 = RT 3 (16) R is the Universal Gas Constant and T 3 the absolute temperature of the triple point in the system. The remaining four unknown onstants in (15) may be determined as follows. Let the molar densities be r g3 < r d3 < < < r l3 where r s3 y r l3 are obtained by experiment from the solid and liquid phases of water at the triple point. The other molar densities are restrained by Maxwell s onstrutions for triple point: = r s3 + r d3 = r g3 + (17) (18) Aording to Keenan and Keyes (1978) we have, for water, pu lg3 V g3 = 330 pu lg3 V s3 = V l3 = pu lg3 (19) T 3 = F lb = pu lg and realling (r = 1 ) we find: V r g3 = r d3 = (10) 6 = (10) 6 = (10) 6 (0) = (10) 6 T 3 = k = (10) N m R = (10) 3 J K The system of equations is as follows: ) ) ( ) (rd3 rd3 rd3 rd3 p3 - (d3 d13rd d 33 = p3 - d 13 (1) d 43 p3 - d d 53 p3 - d 13 and the solution is: d 3 = (10) -16 Pa ( d 33 = (10) -6 Pa ( d 43 = (10) -13 Pa ( )4 d 53 = (10) -0 Pa ( ) )3 )5 whih yields the isotherm of water in the neighborhood of the triple point: =RT 3 r (10) -16 (r) (10) -6 (r) (10) -13 (r ) (10) -0 (r ) 5 (3) For these values the non-additive interations between three, four and five moleules are absolutely indispensable, for all fluids, to obtain the isotherm for the triple state; the expliit isotherm for the triple state for the water is equation (3), where the term of non-additive interations between three moleules is the dominant term; this equation is derived, using statistial mehanis, for the first time. Existene of an infinity of new eletromagneti fores, and generalizations of eletrodynamis, lassial mehanis. Robles-Domínguez et al. (007) derived expression (1), the first two terms of whih redue to the van der Waals equation. While van der Waals had onsidered all binary interations in the fluid, his expression does 79

4 Geofis. Int. 49 (), 010 not fit adequately the experimental data. We therefore onsidered additional interations suh as non-additive tertiary, non-additive quaternary and so on, whih are the Fourier oeffiients of the development based on Funtional Analysis (see Courant and Hilbert, 1953; and Kolmogorov and Fomin, 1957) whih reprodue exatly the experimental isotherm. The new fores are real. But in a fluid the interations are eletromagneti. Therefore these new interations are eletromagneti interations. In equation (1), G 3 represents the mean field of the non-additive interation between moleular triplets. Coulomb ould not obtain these results. Thus equation (1) ontains an infinity of additional new eletromagneti fores. If these fores are not onsidered, e.g. by making G 3, G 4, to vanish, funtional analysis suggests that we may not sueed in reproduing the experimental results. When those new fields are onsidered, in the Eletromagneti ase, the Maxwell s Laws are generalized. Thus the fourth Law of Maxwell takes the form: x (H + H 3 + H ) = J + (D + D 3 + D ) (4) t Where H is the Magneti Field, D is the Eletri Displaement Field, J is the Eletri Current and subsript is employed for binary additive fields, sub-sript 3 is employed for ternary non-additive, et. In Classial Mehanis, Newton s Seond Law is generalized to the form: F + F 3 + F = ma (5) where F is the resultant of the additive binary fores upon m, F 3 is the resultant of the non-additive tertiary fores upon m, and so on. Here m is mass and a is aeleration. Generalized onservation of energy b K a +j a +j 3a +... = K b +j b +j 3b a (F +F 3 +F ) dl (6) here K is kineti energy, a and b are two points of the path of the partile, j, j 3, are the energies of all additive binary, non-additive ternary, non-additive quaternary onservative fores, and so on, and the integral is the total work done by all binary additive non-onservative fores, tertiary non-additive non-onservative fores, et. In Quantum Field Theory are obtained: Generalized Klein-Gordon equation with eletromagneti field ( p ih t -j -j ) y = (- ih p - A - A ) y + (m 0 ) y (7) j is the additive salar potential energy operator j 3 is the non-additive tertiary salar potential energy operator. et. A is the additive binary vetor potential operator. A 3 is the non-additive ternary vetor potential operator. et. All these operate on the Wave Funtion. (for the ase with additive binary interation only see Björken and Drell, 1964). Generalized Dira equation for the eletromagneti field g 1 ( ih p t -j -j )y = g (- ih p -A -A )y + g 3 m 0 y (8) where g s, s = 1,, 3 are Dira matries (for the ase with additive binary interation only see: Björken et al., 1964). Isothermal ompressibility in the Earth s Mantle The geophysiist F. Birh obtained a state equation for the Earth s Mantle (Birh,1947) whih is: 7 5 p = 3 [( r )3 - ( r )3 ] χ 0 (9) where p is pressure, χ is isothermal ompressibility and r is the density of the Earth s Mantle; the sub-sript zero is used for quantities evaluated to atmospheri pressure. The density is: where m is the mass and V the volume. r = m (30) V The mass an be expressed in terms of the moles number n: m = Mn (31) where M is the moleular mass, and ρ an be expressed in terms of n as follow: r = M n (3) V 80

5 Geofis. Int. 49 (), 010 and the Molar Density r is: then: and: r = n (33) V r = M r (34) r = M r = r (35) M Birh s equation an be rewritten as follows: p = 3 [( r 7 5 ] )3 )3 - ( r χ 0 (36) Our State Equation (1) is valid for gases, liquids and solids; of ourse Birh s Equation is a partiular ase of (1). We an express that equation in the base {r l, l = 0, 1,, 3, 4, 5,...}developing it in Taylor series near : ( r l d ( ) )r = ( 1 )r S Dr dr r r r =r where r an take the values 7 & 5, and Dr =r - r l d ( ) l=0 l! (37) S Drdr r r =rr +r(r -r )rr-1 + r(r-1) l! (r-r ) r r- + r = 0 0! l=0 and Birh s equation is: r(r-1) r(r-1) (r - ) 3 r (38) 3! 3] (r ) 5 / = 3 {[ r + p = 3 [ 1 (r ) 7 / 3-1 χ 0 ( ) 7 / 3 ( ) 5 / 3 χ r r ] - [ 4-10 r r and Birh s state equation is: 5 r ]} (39) 81 0 p = 3 ( r r r ) (40) χ This state equation shows, at this approximation, that the non additive interations of three bodies are relevant. The isothermal ompressibility is: Or in terms of r : χ = - r ( χ = - 1 ( V ) (41) V p T r 1 ) p = T 1 ( r ) r p (4) T Impliit derivation applied to the last equation of p gives: ( ) r = 1 = 3 8 ( r0 r r0 r + 19 r r p T p χ ) (43) and the isothermal ompressibility is expressed as follow: χ = χ 0 ( r + 8 r - 0 r + 19 r -3 0 r +...) (44) Aording to this equation, if the molar density inreases the isothermal ompressibility dereases. We an use other equations for the Earth s Mantle, but Birh s equation is more simple and we want to show only the existene of the term of tertiary non-additive interations in the Earth s Mantle. Conlusions In this paper we show that there is an infinite number of new moleular fores in a fluid; these fores are of eletromagneti nature. Here we present the orresponding generalization of the equations of eletrodynamis and lassial mehanis and quantum field theory in whih omprising these new fores. We propose that any field in nature must be assoiated with a orresponding infinite set of new non-additive fores. Also, the Earth s Mantle equations must ontain these new fores. Aknowledgments Critial review from two anonimous reviewer is aknowledged. 81

6 Geofis. Int. 49 (), 010 Bibliography Birh, F., Finite Elasti Strain of Cubi Crystals, Phys. Rev., 71, 11, Björken, J. D. and S. D. Drell, Relativisti Quantum Mehanis. MGraw-Hill Book Company, 300 p. Blak, W. Z. and S. A. Hartley, Termodinámia, Editorial CECSA, Méxio, 883 p. Robles-Domínguez, J. A. M., J. A. Robles-Gutiérrez and C. Lomnitz, 007. An Equation of State for more than two phases, with an appliation to the Earth s mantle, Geofísia Internaional 40, 3, Uhlenbek, G. E. and G. W. Ford, Letures in Statistial Mehanis in Amerian Mathematial Soiety, Providene, R. I., 30 p. Courant, R. and D. Hilbert, Methods of mathematial physis. 1 st English ed., translated and rev. from the German original. New York, Intersiene Publishers, p. 6. Keenan, J. H. and F. G. Keyes, Thermodynamis properties of water inluding vapor, liquid and solid phases. Ed. J. Wiley. New York, 156 p. Kolmogorov, A. N. and S. V. Fomin, Elements of the theory of funtions and funtional analysis. Translated from the 1 st Russian ed. by Leo F. Boron. Rohester, N. Y., Graylok Press, 18 p. J. A. Robles-Gutiérrez 1, J. M. A. Robles- Domínguez 1 and C. Lomnitz 1 Universidad Autónoma Metropolitana, Iztapalapa, 09340, Mexio City, Mexio. Instituto de Geofísia, Universidad Naional Autónoma de Méxio, Ciudad Universitaria, Del. Coyoaán, 04510, Mexio City, Mexio Corresponding author: nautilo68@yahoo.fr 8