Determination of Molar Gibbs Energies and Entropies of Mixing of Melts in Binary Subsystems of the System CaO Si0 2. Si0 2

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Determination of Molar Gibbs Energies and Entropies of Mixing of Melts in Binary Subsystems of the CaO Si CaO Al Si CaO Al Si I. Theoretical Part. CaO Al Si CaO Al Si I. PROKS, J. STREČKO, L KOSA, I.

1 Determination of Molar Gibbs Energies and Entropies of Mixing of Melts in Binary Subsystems of the CaO Si CaO Al Si CaO Al Si I. Theoretical Part. CaO Al Si CaO Al Si I. PROKS, J. STREČKO, L KOSA, I. NERÁD, and Kl ADAMKOVIČOVÁ Institute of Inorganic Chemistry, Slovak Academy of Sciences, CS-84 6 Bratislava Received 6 December 99 Method for calculation of component activities in high-temperature binary melt systems is presented. For this purpose, phase diagrams and enthalpies of mixing were used. The latter ones were determined indirectly using primary composition and temperature dependence of relative enthalpy, found by means of multiple linear regression. The method was applied to the melts in the CaO Al Si CaO Al Si system to find out that within the temperature range 7 К to 95 К and limits of experimental errors for enthalpy of mixing these melts behave as athermic solutions. Component activities are equal to equilibrium activities at any composition. The determined activities enabled to calculate the composition and temperature dependences of Gibbs energy and entropy of mixing. Heats of fusion of pure components, as well as of all three binary eutectic mixtures and ternary eutectic mixture are known in the CaO Si (CS CaO Al Si ( AS CaO Al Si (CAS system. General temperature and composition dependence of relative enthalpy can be calculated using relative enthalpies of chosen melts in this system []. Relative enthalpy of the phase at given temperature is defined as a negative sum of the heat of cooling from the given temperature to the temperature of dissolution and heat of solution of the cooled sample, both measured by calorimetric methods []. These pieces of information together with the known phase diagrams were used to determine Gibbs energy and entropy of mixing of melts in binary subsystems of the ternary system as a function of composition and temperature. This paper is dealing with the general thermodynamic analysis of melts in binary systems and its application to the system. THEORETICAL In the following text we use symbols X and Y for the system components, subscripts с and eq for the quantities chosen and those related to multiphase equilibria. Thermodynamic analysis is made for phases in equilibrium state. Composition'Dependence of Equilibrium Activities aeq.xmtxv for B o t h Components of the Melt in the the Components of which do not Form Solid Solutions The temperatures 7^IX (Y(^Y,C at which the melts of chosen composition x Yc coexist with pure crystalline components X, Y were taken from phase diagrams. Corresponding equilibrium component activities at these temperatures were calculated from the Le Chatelier Shreder equation П^Х(У>(Г^*.о = J 7fus,X(Y AH^X(Y(n dr { where a^x^t^x^yj is the equilibrium activity of component X(Y at the temperature Т еял у, 7"eq P x(y(*y,c G <7 "fus,x(y. "^eut> is the temperature at which the melts of chosen composition x Yc coexist with pure crystalline components X(Y, r fusix( Y is the equilibrium temperature of fusion for component X(Y, 7~ eut is eutectic temperature in the X Y system, AH fusx(y (T is enthalpy of fusion of X(Y component at the temperature T. The temperature dependence of AH fusix (Y(7 was derived using primary relative enthalpy dependence of melts in the CS system and temperature dependence of relative enthalpy for the crystalline component X(Y. Chem. Papers 47 ( -7 (99

2 I. PROKS, J. STREČKO, L KOSA, I. NERÁD, K. ADAMKOVIČOVÁ Dependence In a^xco^y was obtained from the calculated values of In a^ xco^eq, X(Y(*Y, C according to eqn (7. Component Activities а хсп &у,с in the Melt at Chosen Temperature and Composition To calculate both component activities at chosen temperature 7~ c from equilibrium activities it is necessary to know partial molar enthalpies of mixing AH mjx X(Y(*Y, c» " as a function of temperature and chosen composition. Activity of component X(Y in the melt at composition x YC and temperature T c is given by where In a X( Y(X Yi с T c = In aeq, X(Y(XY, C + /X(Y(*Y, о T c ( /x (Y (* Y r c = - Ar W^Y, c.d d7. ( 7"eq.X(Y(*Y.c Enthalpies of mixing of melts in binary systems were calculated using primary temperature and composition dependence of relative enthalpy of melts in the CS system, from the equation = H rel - ( -x Y H? el(x (r c - -*унг е.,у(7~с (4 where A/-/ mix, 7~ c is enthalpy of mixing when one mole of mixture at the temperature T c and composition x Y has been arisen, H re i, 7" c is molar relative enthalpy of melt at the same temperature and composition, H%\ t x(y(7"c is molar relative enthalpy of pure component X(Y melt at the temperature 7" c^ Using the method of intercepts, the values of,x(y(*y, с T c for several chosen values of T c were calculated from eqn (4. Temperature dependences of AH mixix (Y(*Y,c. Л w e r e obtained from these values by regression analysis. The values, 7" c are loaded with considerable errors because of the high numerical values of relative enthalpies occurring in eqn (4. Partial molar enthalpies A/-/ mix(x(y (C exhibit much higher errors because they are obtained using the derivative of composition dependence of. The quantities >X(Y >c, calculated in such manner, did not give monotonous isothermal composition dependences of activities resulting in_nonequilibrium states of melts. To calculate A/-/ mix X(Y c, 7, substitute function was selected, the course of which was identical with original equations within the limits of determined errors. Substitute function gives monotonous dependences of a X(Y corresponding to equilibrium states of melts. Dependences a X(Y are deformed due to errors in A/-/ mix. Deformations occur in particular along the compositions with high values of derivative d( / dx Y. Therefore the empirical developing function (5 was introduced WXY.O T C = - In a eqx c [ - exp (- к(т с (Г с - - T ôqix >c ] (5 к(г с is composition independent parameter. The parameter K(T C was selected so that the dependence In a x(*y> T c had negative derivative for x Y G <, X Y ÔUt>. If we divide the temperature range (7~ eq x c - T c for integration to temperature intervals АГ, so that the ratio AT/T C, would be sufficiently small, we can calculate A/-/ mjx xc, Г с / according to the formula >x C, T C J = K(T C JRT CJ In a eqx c exp [- к(г с>,(г С>, - T ôq> x i c ] (6 where T ci is the temperature of the /-th interval centre. In this thermodynamic analysis the temperature range 575 К 975 К was divided into eight intervals, every of 5 K. The quantities x c / were calculated from eqn (6 for the temperatures at the centres of these intervals. Using these values the regression functions at compositions x Yc were determined and substituted into eqn (. Forx Yc G <, x Y ÖUt> we obtained discrete function values using the method of intercepts and eqn (6. For x Yc G <x Y eut, > the substitute function was represented by a polynomial, the values and derivatives of which at compositions x Y eut and x Y = were identical with corresponding values and their derivatives of substitute function and (XY, 7 C according to eqn (4, respectively. The substitute function has to satisfy the condition that its values have to lie within the error limits of the function AH mjx, Г с given by eqn (4. Isothermal Composition Dependences of Component Activities X (Y(XY, 7"C These dependences have to satisfy the following conditions: a if standard states of chemical potentials are related to pure components, the following expressions have to be valid a X{Y (xx(y=, Г с = (7 a X( Y(^x(Y=,Tc = (8 b in the case of low contents of admixture M in solvent N, A/-/ mix is given by the formula where Eqn (9 also implies lim A/-/ mix = x M M (9 x M -> _ lim M = const ( limn = ( 4 Chem. Papers 47 ( -7 (99

3 ENTROPIES OF MIXING OF MELTS. I Eqns (, ( and (77 imply at any temperature T c two further conditions (according to selection of M and N d In a N (x N = = d In аед,м(*м =, Т^.м ( d* M dx M The right side of eqn ( was obtained by deriving n Qeq.N^M f r o m t h e first section at x N =. c at Г с ' = regime, the values of a^^c are given by X(Y(^X(Y, ci 7~c = eq, X(Y(*X(Y, c ( d isothermal dependences a^ta. 7"c have to satisfy the Gibbs Duhem equation din a Y _ -x Y din a x ÓXy ÓXy (4 Function a X (Y(*Y. T c was expressed by the Margules equation [] In a X(Y,T c = ax(y(7" c In x X(Y + ^}(Т С + + jk X{Yi x Y (5 / = where / and r are natural numbers. Eqn (75 identically fulfils the conditions (7 for definite values jx(y(7" c and K X{Yti. Substituting eqn (5 into (4 and comparing coefficients with the same powers of mole fractions we get next r + conditional expressions among coefficients oc X(y, / X(Y (T C, and K xcn,,. e Eqn (75 has to be fulfilled for n chosen values of both activities a X(Y (x X(Y p c calculated from eqns ( and (.The condition (7 is within the errors of activities determination for sufficiently large n fulfilled and therefore it was not considered. Comparing number of conditions ( x + r n and number of coefficients ( x ( + r in the equations, we get unambiguous relation between r and n r='\+n (6 Composition and Temperature Dependences of Gibbs Energy of Mixing AG mix, T and Entropy of Mixing AS mix, T in the Melt Composition and temperature dependence of component activities a X(Y, 7 was calculated at suitably chosen temperatures T c from the isothermal dependences of a X(Y l 7" c. AG mix, T were calculated from the formulas that are by the definition AG mix, 7 = RT [( -x Y In a x l x Y In a Y, 7] (77 and AS mix,t = - a(ag mix > 7 dt APPLICATION General Composition and Temperature Dependence of Relative Enthalpy in the CS (8 To determine this dependence, the arithmetic means of relative enthalpies of eleven melts of various compositions measured at several temperatures were used as the input data []. This dependence was determined by multiple linear regression [4] in the form He. = X ß /MCaOf[x(AI ] c 'T d ' (9 i Parameters in this equation are shown in Table. The weights of the input data were proportional to I/o and normalized to the number of input data. Errors of relative enthalpy and enthalpy of mixing were determined according to the formulas ( and (, respectively Table. / \ш, [ст(в,] + v / ЭН дн ч гв г ( +ЕЕ^^.в у Г cr(ah mlx = - v ЗНге j у(дан т Vel, к [cr(ŕu] + [otf«uf xv ( Exponents b c d h Coefficients B if and Standard Deviations a(ß, of Regression Polynomial (9 in the CS b с d ß kj moľ К" d СТО kj moľ K~ d Chem. Papers 47 ( 7 (99 5

4 I. PROKS, J. STREČKO, L. KOSA, I. NERÁD, K. ADAMKOVIČOVÁ where к denotes the components of the CS system. Relative errors of Н гы were less than.5 %. Molar Gibbs Energy and Entropy of Mixing in the Melt In the following text we will use symbol X for AS and Y for CAS. The base for thermodynamic analysis was the phase diagram of this system constructed by the optimization of data taken from [5 7] (Fig.. All experimental data published since 95 to 96, including information about the conditions and ways of their measurements, were taken into account. Simultaneously the binary phase diagram of this system was correlated with that of the ternary system CaO Al Si [8]. We can assume that systematic errors of data taken from various authors differ and correspond to experimental technology level of the era they were measured at. Therefore we could not process these data by means of common statistical methods. To determine the dependence In a^x^v. the relation A/~/fus, X(Y(7 = H e x(y, melt(7 ~ ^ľel, X(Y, cryst(t ( was substituted into eqn (. H^C^AS, cryst, 7 was taken from [9]. Temperature dependence of molar heat capacity C p (CAS, cryst from [] and the value of heat of solution A/-/ S,(CAS, cryst from [] were used to calculate H? e] (CAS, cryst, T. Enthalpy of mixing was calculated from eqn (4 into which the relative enthalpies of melts of the system and the pure components AS and CAS, calculated from eqn (9, were substituted. Enthalpy of mixing was within the error limits a equal to zero at any composition and temperature 7 K 95 K. The highest error limit did not exceed the value of 4. kj mol". Thus for the melts of the system the following relations are valid within the framework of regression function (9 and = ( _ AH mix(x(y ic, 7 = (4 If we substitute from the relation (4 into eqn ( we get /X(Y(*Y,C = (5 and eqn ( will be then of the form In a X (Y(XY, с T c = In Зад, X (Y(*Y, c As follows from eqn (6 the activities of AS and CAS in their solutions are equal to their equilibrium activities in the melts coexisting with their corresponding crystalline phases. The dependence In aeq (X (Y(*Y can be used within the interval x Y e <*Y, eut. > "for AS and x Y e <, x Y eut > for CAS to integrate Gibbs Duhem equation isothermally 9-7/K J " t... I I xicasj Fig.. Phase diagram of the system. D [5], + [6], x P]> [8] (taken off from the ternary phase diagram. dlna Y = -^Y d ina x (7 x Y But in our case we used the procedure described in the paragraph of of the theoretical part. To determine coefficients in eqn (5 we have considered the conditions (8 and ( for both components, nine values of activities calculated from eqn ( and presented in Table five of them (ax corresponding to compositions x Y e <, X Y eut > and four (a Y corresponding to compositions x Y G <*Y,eut» "!> Substituting for n into eqn (6 we get r being equal to. Margules' series (5 for both components have therefore 4 coefficients independent of temperature. The dependence a X(Y (*Y> Д which in this case agrees with the dependence а^хоо^, is for both components shown in Fig.. Substituting eqn (5 into (7 for both components we get for the melts of the system the dependence AG mix, T and from eqn (78 AS mix, T. The dependences AG mix and 6 Chem. Papers 47 ( -7 (99

5 ENTROPIES OF MIXING OF MELTS. I Table. Starting Values of AS and CAS Activities Used for the Margules Equation Coefficients Calculation i i i j i i i j i i i [- Activity x(cas a( AS a(cas l - L^ «i» x(cas Fig.. Isothermal composition dependences of the quantities of mixing AG mix (7 and 7AS mix ( of the system at 7 K. REFERENCES Fig.. Composition dependences of activities for AS (7 and CAS (. 7"AS mix at the temperature 7 К are shown in Fig.. As follows from eqn ( the melts of the system behave at all compositions and all temperatures under study as athermic solutions within the limits of errors for, T. This property is obviously the consequence of the fact that the melts AS and CAS have similar structure.. Kosa, L, Tarina, L, Adamkovičová, К., and Proks, I., Geochim. Cosmochim. Acta 56, 64 (99.. Eliášová, M., Proks, I., and Zlatovský, I., Silikáty, 97 (978.. Margules, M., Sitzber. Akad. Wiss. Wien, Math. Naturw. Kl. II, 4, 4( Draper, N. R. and Smith, H., Applied Regression Analysis. Wiley, New York, Osborn, E. F., Am. J. Sei. 4, 75 ( Gentile,A.Land Foster,W.R., J. Am.Ceram. Soc.46,76( Rankin, G. A. and Wright, F.E.,Am. J. Sei. (4th Ser. 9, ( Osborn, E. F. and Muan, A., Phase Equilibrium Diagrams of Oxide s. Plate, The CaO Al Si. The American Ceramic Society, Columbus, Ohio, Žigo, O., Adamkovičová, K., Kosa, L, Nerád, L, and Proks, L, Chem. Papers 4, 7 (987.. Richet, P. and Fiquet, G., J. Geophys. Res. 96, 445 (99.. Kosa, L, Žigo, O., Adamkovičová, K., and Proks, I., Chem. Papers 4, 89(987. Translated by E. Čajagi Chem. Papers 47 ( -7 (99 7

Determination of the heat of fusion of 2CaO- A1 2

Determination of the heat of fusion of 2CaO- A1 2 - SiOj (gehlenite) O. ŽIGO, K. ADAMKOVIČOVÁ. L. KOSA, I. NERÁD, and I. PROKS Institute of Inorganic Chemistry, Centre for Chemical Research, Slovak Academy