J. Phys. Chem. 1981,85,3859-3863
data, the following conclusions can be drawn. (a) From the experimental and theoretical results for the fully decationated zeolites it can be concluded that positions H3and HIare the most favorable sites for protons. (b) The experimental results on the Na-Y zeolite have shown that the most populated cation position is the SI1 site. This does not contradict the theoretical conclusion that the SI site is the most convenient, since its occupation is limited due to population of SI’ positions. As the pop-
3859
ulation of the SI’ site, with increasing decationization, decreases, this restriction disappears and the occupation of the SI achieves the maximum value. Observed occupation factors for SI and SI1 sites are approximate equal in partially decationated samples. (c) The population of SI positions prevents the occupation of neighboring H3positions. This explains the low population of H3positions even in weakly decationated samples.
Thermodynamic Model for Binary Aluminate Systems Shi Xue Doul Northeast Institute of Technology, Shenyang, Liaoning, People’s Republic of China (Recelved: February 5, 198 1; In Final Form: June 25, 198 1)
The thermodynamics of basic oxide-aluminate systems are approached in terms of statistical polymer theory. A linear chain model is presented for binary aluminate systems, in which the ion AlO3&is treated as the monomeric unit. The activities of the constituents can be expressed as functions of the A1203 content under the assumption that the systems exhibit Temkin behavior and do not produce ring anions. The activities predicted for several systems are in good agreement with experimental values and those calculated from phase diagrams. The anionic distribution, the mean chain lengths, and the mean number molecular weights for aluminate polymer chains are calculated from the activities of constituents. Values of the equilibrium constant for aluminates are regularly related to some bond parameters, similar to those in silicates and phosphates.
Introduction Aluminates are important in ceramics, the glass industry, metallurgical slags, and refractory materials. Like silicates, phosphates, and other oxygen-containing inorganic polymers, aluminate melts contain an array of anions of varying degrees of polymerization. Considerable progress in understanding the constitution of liquid silicates and phosphates has been made in both theoretical and experimental directions. On the theoretical side, thermodynamic models based on specific structural assumptions have been proposed by Meadowcroft and Richardson,l Masson et al.,293 and others. This work has indicated that theories for organic polymers can be applied to inorganic polymers. However, lack of experimental information about the behavior of inorganic systems at high temperatures and the small molecules and cations remaining as condensation products in the system presents difficulties in approaching inorganic polymers from polymerization theory. Thus, it is somewhat surprising that Masson et al. successfully modeled the MO-Si02 melts for silica concentrations up to -40 mol % Si02 by using Flory’s4 theory. On the experimental side, structural data of phosphate glasses and silicate melts have been obtained by various methods, which include chromatographic analysis of aqueous solutions of quenched phosphate melts,5 mass determination of the trimethylsilyl derivatives of silicate ions,6and high-temperature X-ray diffraction techniques.’ These studies demonstrate that polymers of varying degrees of polymerization are present in the melts and provide direct evidence for the foregoing models. The present study introduces a model which attempts to extend polymer theory to aluminates of the type MO-A1203, where MO is a basic oxide. Visiting scientist, National Research Council of Canada, Atlantic Research Laboratory, Halifax, Nova Scotia, Canada. 0022-3654/81/2085-3859$01.25/0
Activity in Binary Aluminates In aluminate melts, oxygen can be considered to be distributed between three species: 20- = 0 0
+
02-
(1)
Oorepresents an oxygen bonded to two aluminium atoms, 0- represents an oxygen bonded to only one A1 atom, and 02is free oxygen anion. The assumptions made in deriving the distribution function are similar to those of conventional polymer theory. (1)Only linear chain polycondensation reactions can occur in aluminate systems. Thus, velocity constant as well as equilibrium constant for each step of the polycondensation are identical. Functional groups are equally reactive; i.e., reactive sites on the aluminate chains are chemically equivalent, regardless of chain length. (2) The monomeric unit in polycondensation may be regarded as A102- anion, from which a series of polymers of aluminates can be derived. (3) The activity of the oxide and the fraction of ions are related by Temkin’s law.8 For a binary melt, MO-A1203, the activity of the oxide MO is given by a M O = NMz+NOZ= Noz(2) where Noz- is the ion fraction of 02-in the anionic assem(1) T. R. Meadowcroft and F. D. Richardson, Trans. Faraday SOC.,61, 54 (1965). (2) C. R. Masson, Proc. R. SOC.London, Ser. A , 287, 201 (1965). (3)S.G.Whiteway, I. B. Smith, and C. R. Masson, Can. J. Chem., 48, 33 (1970). (4)P.J. Flory, “Principles of Polymer Chemistry”, Cornel1 University Press, Ithaca, NY, 1953.(5)A. E.R.Westman and R. Beatty, J. Am. Ceram. SOC.,49,63 (1966). (6)J. Gotz and C. R. Masson, J. Chem. SOC.,Dalton Trans., 1134 (1978). (7)Y.Waseda and J. M. Toguri, Met. Trans., 9B,595 (1978). (8)M.Temkin, Zh. Fiz. Khim., 20,105 (1946).
0 1981 American Chemical Society
3800
Dou
The Journal of Physical Chemistry, Vol. 85,No. 25, 1981
bly. The standard state of the oxide is defined to be pure liquid oxide at the temperature being considered. On the basis of these assumptions, the chemistry of the aluminate melt is constructed by following a series of polycondensation reactions:
-
+ A1033- K11 N2Ob4-+ 02A1033- + A1205" -% A130,5- + 02A1033-
-
+
Kin
A1033- Aln02n+l(n+2)-
Aln+102n+3(n+3)+ 02- (3)
where Kll, KI2,...,K1,are equilibrium constants for each step of the polycondensation, respectively. For the simplest reaction, the equilibrium constant is given by K11
= N A ~ ~ O ~ ~ N O Z - /=( N2Nos/N12 N A ~ O ~ ~ - ) ~(4)
where N1 and N 2 represent the ion fraction of monomer and dimer, respectively. Based on the polymer theory, the theoretical expression for the anionic distribution in polymers derived from bifunctional monomer is
Nx=
flx-l(l
- 01)
(5)
where N, is the mole fraction of x-mer and a! is the extent of reaction which is defined as number of A1-0- bonds that have reacted (6) total number of A1-0- bonds During the polycondensation, free oxygen ion 02-accumulates in the system until equilibrium is attained, and its presence cannot be ignored in the evaluation of ion fractions. So eq 5 should become
N A I ~ O ~ Flgure 1. Theoretical curves of am vs. NAIRs for linear chain polymerizatlon of MO-A1203 wlth various values of K,,.
Then, from the relationship between ct and R derived by Flory: ct = 2 / f - 2 / 2 f (12) one obtains
fl=
N , = aX-'(l- a)(l - NOZ-)
(7)
- -1 "203
-2+-
2 - flf
1 - No2-
Substituting for ct and Noz- in eq 13 from eq 9 and 2 f - -1 -2+----- 2 (14) 1 - UMO U M O N&os - 1- UMO K 1 1
Substitution for N1 and N 2 in eq 4 yields
+
For f = 2, we obtain the theoretical expression of the activity of MO as a function of composition for linear chain polycondensation as follows: 2 2 - =1 2+-(15) NA1203 1 - a M O 1 + aMO(l/K11- 1)
The mole fraction of alumina in the binary system may be expressed by means of the ion fraction of the constituents of the melt by the following equation: NA1203= (moles of A1203from aluminates)/[(moles of free MO) + (moles of MO from aluminates) + (moles of A1203fromaluminates)]
Theoretical curves of UMO vs. NAlz03for various values of Kll are presented in Figure 1. For comparison of the present work for aluminate melta with that for silicate melts, the theoretical curve of U M O vs. the mole fraction of Si02 and the theoretical curve of uMO vs. the mole fraction of A1203are presented in Figure 2 at the same value of KI1. From Figure 2, it is seen that the difference between them is marked.
If both numerator and denominator are divided by the total sum of polymer anions, EN,, and it is remembered that the mean chain length, 2, is given by C x N J E N , , eq 10 can be rearranged to yield
Experimental Evidence In order to examine whether the theoretical expressions of activity derived above are actually valid, one must compare the theoretical curve of u M O with experimental data. Activities of CaO in the system Ca0-A1,03 at 1500 O C have been measured by Sharma and Richardsong and Chipman.lo Activities of MnO in the system MnO-A1 0 at 1650 and 1600 "C were measured recently by Jacog,fB
1 CXNX
(9) R. A. Sharma and F. D. Richardson, J. Iron. Steel Inst., London,
198, 386 (1961).
(10) J. Chipman, ''Physical Chemistry of Process Metallurgy", Interscience, New York, 1961.
The Journal of Physical ChemMy, Vol. 85, No. 25, 1981 3881
Thermodynamic Model for Binary Aluminate Systems
TABLE I: Values of Heat of Fusion, Melting Temperature, and Equilibrium Constant for Systems MO-Al,O, kcal/mol
BaO
13800 16700 19000 18500 17000 13000 7400
SrO CaO
MgO Be0 MnO FeO a
T,,”C 1910a 2435a 2615 2825 2530 1785 1378
K,,
T, “ C
0.045 0.068 0.123 0.61 0.82 2.50 5.90
1425-1900 1690-2435 1500 1995-2825 1980-2250 1600-1650 1400
From phase diagram. 1.01
NAjtv,
AH,,
MO
A
I
I
1
I
NSSO,
Figure 2. Comparison of activities calculated from the model for (1) Masson model,, silicate; (2) MO-AI2O3 to those for MO-SiO,: present work, aluminate. 1.
I
I
t
e,+
//-
2183
/2200
0.
/
0.
/ /
0
/
U
/
0.
/
/
/
/
/
f
I N,i,o,
Flgure 4. Activities of MO plotted against mole fraction of A1203for three systems; the curves are theoretical. (- -) MnO-AI,O,; (0) Sharma and Richardson,’* 1650 OC (experimental); (0)Jacob,” 1600-1650 OC (experimental); : (0)calculated from phase diagram of Novokhatskil.ls (- - ) Bs0-Ai,O3; (0) calculated from phase diagram of Galakhov.” (-) Sr0-Ai203; (A)calculated from phase diagram of Massaz~a,’~ a point at 1750 ‘C from extrapolation by the relation TR In rSm = NAhO,a.
-
N,\i20,
Figure 3. Activity of MO vs. mole fraction of A1,03 for various binary FeO-AI,O,; (0) systems; the curves are theoretical. (---) Ban-Ya and Shim,“ 1400 O C (experimental). (-- -) MgO-AI,O,; (0) from phase dia ram of Aiper et ai,’’ (- - -) CaO-AI,O,; (A)Sharma and Richardson (experimental), 1500 O C ; (A)Chipman (experimental), 1500 O C . (-) Ba0-AI2O3; (X) calculated from phase diagram of Purt.’6
8
whose results are in reasonable agreement with those of Sharma and Richardson1* in the liquidus region. For comparison with theory, these activities must be converted (11)K. T. Jacob, “Proceedings, International Symposium on Metallurgical Slags, Halifax, Nova Scotia, 1980”, Can. Metall. Q.,in press. (12)R. A. Sharma and F. D. Richardson, Trans. Metall. SOC.AIME, 233, 1586 (1965).
from the solid standard state to the pure liquid standard state by eq 16. The heats of fusion and melting temT, - T In uMO(l) = AHrn( RT,T aMO(s)
-)
(16)
peratures from ref 13 for some binary systems of MOA1203are presented in Table I, including values of Kll. The values of acaO and am0 thus obtained are reproduced in Figures 3 and 4, respectively, along with theoretical curves corresponding to eq 15 for selected values of Kl1. The agreement between the theoretical curves and the (13) 0. Kubaschewski and E. LL. Evans, “Metallurgical Thermochemistry”, Pergamon Press, Elmsford, NY, 1958.
3862
The Journal of Physical Chemistry, Vol. 85, No. 25, 1981
experimental values is satisfactory for these two systems. 0 0 For the system Fe0-A1203, Ban-Ya and Shim14 have measured the activities of FeO at 1400 OC with liquid oxide as the standard state. The line compared with their experimental results in Figure 3 corresponds to eq 15 with K11 = 5.9. Activity-composition plots for binary systems BaO-, SrO-, MgO-, and Be0-A1203 are shown in Figures 3 and 4. The activities of MO in these systems are calculated from phase diagrams of Purt,16Massazza,16Alper et a1.,17 and Galakhov,13 respectively, and are converted to the liquid standard states by using the free energy of fusion with an assumption that the difference between the heat capacities of solid and liquid MO may be neglected. The values of the heat of fusion and melting temperatures of MO for all of these systems are quoted from ref 13 (melting temperatures of BaO and SrO from corresponding phase diagrams). Several points of uMnO in Mn0-A1203 calculated from the phase diagram of Novokhatskiilg are also included in Figure 4. For the systems MgO-A1203 and MnO-AI2O3, the phase diagrams show the melts in equilibrium with the solid solution, so the activities calculated from phase diagrams via free energy of fusion must be modified. It is assumed that the solid solution follows approximately Raoult’s behavior in the region of very low concentration so that the activities in equilibrium with the solid solution can be converted to those with liquid oxide through Raoult’s law. Such conversion for these two systems has already been made, as shown in Figures 3 and 4. From Figures 3 and 4,the theoretical curves from the linear chain model show good agreement with the points calculated from phase diagrams. For some systems, however, such as MgO-, CaO-, and Be0-A1203,the predicted activities become lower than the experimental values at higher A 1 2 0 3 contents. This is attributed to the restriction of the linear chain model, which predicts zero activity of MO a t the composition MO-A1203.
Dou
5
“
’
0.8
I
K,,=0.123
I
I
I
I 1 /
I
0.8-
0.6-
i
Ionic Distribution From eq 7, the ionic distributions for the binary systems MO-A1203 can be calculated from the degree of polymerization and the activities of MO. By substituting from eq 9 for a in eq 7, we obtain a useful expression for N, as follows: N, =
/
1
0
0.1
0.2
0.3
0.4
NA1t08
From this equation, ionic distributions can be calculated if the value of Kll for the system is known. Ion fractions of Nl-N6 for the system CaO-A1203 are given by eq 17 with Kll = 0.123 and are plotted against NAlzogin Figure 5. These curves indicate that the ion fraction of A Q 3 reaches a maximum at the point of composition of the (14)S. Ban-ya and J. D. Shim. “Proceedings,International Symposium on Metallurgical Slags, Halifax, Nova Scotia, 1980”,Can. Metall. Q. in press. (15)G. Purt, Rader Rundsch., 4, 201 (1960). (16)F.Massazza, Chim. Ind. (Milan),41, 114 (1959). (17)A. M. Alper, R. N. Mcnally, P. G. Ribbe, and R. C. Doman, J. Am. Cerarn. SOC.,45, 264 (1962). (18)F.Ya. Galakhov, Izu. Akad. Nauk SSSR, Otd. Khim. Nauk, 1035 (1957). (19)I. A. Novokhatskii and L. M. Lenev, J . h o g . Chem., 11, 233 (1966).
Figure 6. Ionic fraction of monomer and number mean molecular weight distribution plotted against N for CaO-AI,O, and Mn0-AI2O3: (1) CaO-A1203, 1500 OC; (2) Mn0%203, 1600-1650 OC; (-) ionic fraction of monomer: (- - -) mean molecular weights.
compound 3MO-A1203, that of Al2Os6 reaches a maximum at the point of composition of 2MO-A1203, and so forth. For the system MnO-A1203, the ionic distributions show trends similar to those of Ca0-AI2O3, but the ionic distributions of the former are much wider than those of the latter. The calculated ionic fractions of Nl for CaO-A1203 and Mn0-AI2O3 are compared in Figure 6. Mean Chain Lengths. According to classical theories, the mean chain length is defined as f = CxNJCN,. Combining eq 13 with eq 12 and using Temkin’s law, we derive the following equation: l/f = (1 - ~ M O ) [ ~ / ~ -N1A 1 ~ O ~ (18)
From eq 18 mean chain lengths of polyion can be calcu-
Thermodynamic Model for Binary Aluminate Systems
TABLE 11: Relations between S o m e Bond Parameters and Values of K,,for MO-A1,0,, MO-SiO,, and MO-P,O,
I
8.0-
cation
K,, of
K,, of
MO-
MO~~.
Ai,o,
sio,
K+
0.265 0.26 0.48
Na+ Li+ Bal+
Sra+
Cas+ Mga+ Cdl+
0.045 0.068 0.123 0.61
Be2+ 0.82 Pb” MnZ+ 2.50 Fez+
5.90
0.0016 0.01
0.70 0.98 1.00
0.2 0.25 1.00
1.02
Zn2+ co2+
Snz+ Ni”
K,,, o f MOP,O~
0.98 2.6 2.55 46
H’
1.22
iii
S.L*
hi;
1
I
I Cd
Pb
FI
,*I,
0.91 1.01 0.97 0.97 0.99 1.04 1.23 1.46 1.47 1.55 1.60 1.64 1.66 1.70 1.72 1.75 2.10
0.354 0.538 0.423 0.579 0.64 0.749 1.15 0.64 1.32 1.42
1.69
”
I 4.33 5.138 5.39 10.0 11.03 11.87 15.03 16.9 18.2 15.03 15.64 16.18 17.96 17.05 14.63 18.15 13.60
4.0r
2.0-
I
C“
f
riz- u ) / L \
XA
C”
d
i
I 1,
6.0
-4.0
I
Y
3
0-
-
- 2.0-
--2.0
-4.0-
--4.0
-6.0
lated in terms of activities of MO and do not depend on the functional degree. The number mean molecular weight Mn,of aluminate polyions is given by
(19)
where Mo is the mean molecular weight of the polymer unit. Values of for the systems Ca0-A1203 and MnOA1203are plotted against NAl2o3in Figure 6 , from which it is seen that a low value of Kll implies a relatively low average degree of polymerization and a narrow mean molecular weight distribution.
an
Discussion and Conclusions A linear chain model is presented for binary aluminate systems of the type MO-Alz03. The distributions of aluminate ions, mean chain lengths, and number mean molecular weights are predicted in terms of activities. Activities in the binary system MO-Alz03 are expressed as a function of the composition and the equilibrium constant. This provides an approach through which thermodynamic properties may be obtained. Activities calculated from the linear chain model are in good agreement with those of the experimental measurements (CaO-, MnO-, and FeO-A1203) and those estimated from phase diagrams (BaO-, SrO-, BeO-, and Mg0-Alz03). Aluminate stoichiometry imposes activity-composition relations different from those for silicate stoichiometry, even for the same value of Kll. As some phase diagrams and heats of fusion are rather uncertain, due to the relatively high temperatures, the agreement should be considered as approximate.
(20) Shixue Dou and Huakuen Liu, “Proceedings,International Symposjum on Metallurgical Slags, Halifax, Nova Scotia, 1980”, Can.Metall. Q.in press. (21) A. L. Allred and E. G. Rochow, J. Inorg. Nucl. Chem., 5, 264 (19.58). ~_.__,.
(22) J. C. Slater, Phys. Reu., 36, 57 (1930). (23) P. Balta and E. Balta, Rev. Roum. Chirn., 16, 1537 (1971).
