The Journal
KOo,5-Si02 at High Temperatures
plots of eq 4 and 7. As shown in Table 111, the value obtained by the total absorbance method (eq 7 ) is in satisfactory agreement with that derived from the usual method2i5 (eq 4). The €,(OH) obtained in this work is remarkably similar to that of Harris and Hobbs2even though our K, are widely different (Tables I and 111). The difference in K , may be attributed to the fact that Harris and Hobbs did not take into account the overlapping of monomer and dimer bands.
of Physical Chemistry, Vol. 82, No. 9, 1978 1021
(4) Y. Nagai and 0. Simamuru, Bull. Chem. SOC.Jpn., 35, 132 (1962). (5) J. Steigman and W. Cronkright, Spectrochim. Acta, Part A , 26, 1805 (1970). (6) D. P. N. Satchel1 and J. L. Wardell, Trans. Faraday Soc., 61, 1199 (1965). (7) G. M. Barrow and E. A. Yerger, J. Am. Chem. Soc., 76, 5248 (1954). (8) T. C. Chiang and R. M. Hammaker, J. Phys. Chem., 69, 2715 (1965). (9) S.D. Christian and R. M. Stevens, J. Phys. Chem., 76, 2039 (1972). (10) C. J. Bellamy, R. F. Lake, and R. J. Pace, Spectrochim. Acta, 19, 443 (1963). (1 1) E. S. Hanrahan and B. D. Bruce, Spectrochim. Acta, Part A , 23, 2497 (1967). (12) G. Allen, J. G. Watkinson, and K. H. Webb, Spectrochim. Acta, 22, 807 (1966). (13) H. E. Affsprung, S. D. Christian, and A. M. Melnick, Spectrochim. Acta, 20, 285 (1964). (14) M. Kirszenbaum, J. Corset, and M. L. Josien, J . Phys. Chem., 75, 1327 (1971). (15) T. S. S. R. Murty and K. S. Pitzer, J. Phys. Chem., 73, 1426 (1969). (16) R. C. West, Ed., "Handbook of Chemistry and Physics", 48th ed, The Chemical Rubber Co., Cleveland, Ohio, 1967-1968. (17) H. F. Shurvell and J. A. Faniran, Can. Spectrosc., 13, 110 (1968). (18) J. T. Bulmer and H. F. Shurvell, J . Phys. Chem., 77, 256 (1973). (19) C. C. Snead, J . Phys. Chem., 76, 774 (1972).
Acknowledgment. Financial assistance from the University of Ibadan Senate Research Grant is gratefully acknowledged.
References and Notes (1) J. A. Faniran, K. S. Patel, and M. A. Mesubi, Spectrochim. Acta, Part A , 31, 117 (1975). (2) J. T. Harris, Jr., and M. E. Hobbs, J . Am. Chem. Soc., 76, 1419 (1954). (3) R. E. Kagarise, J . Chem. Phys., 27, 519 (1957).
Vapor Pressure Measurements, Thermodynamic Parameters, and Phase Diagram for the System Potassium Oxide-Silicon Oxide at High Temperaturest Naomi Eiiezer, R. A. Howald, M. Marinkovic, and
I. Elierer"
Department of Chemistry, Montana State University, Bozeman, Montana 597 15 (Received October 7, 1977) Publication costs assisted by the U.S. Department of Energy
Potassium oxide vaporizes from solutions in SiOzprimarily according to the equation: KOo.5(l)-,K(g) + '/dOZ(g) (1). We have studied this equilibrium using atomic absorption in a constant temperature graphite furnace to measure the absorbance of the steady-state distribution of potassium atoms. The potassium atoms were introduced at the center of the furnace by diffusion through an orifice in sample cells which we developed for vapor pressure measurement. We have shown both experimentally and theoretically that, at a particular temperature, the absorbance we measure is proportional to the rate of diffusion of potassium atoms from the vapor pressure sample cells. For a particular cell with a particular orifice, there is a temperature-dependent conversion factor between absorbance and pressure which we evaluated from measurements with potassium aluminate standards in the cells. The pressure of potassium atoms was measured over a sufficient range of temperature to obtain the heat of vaporization. The Redlich-Kister coefficients for the enthalpy of mixing of liquid KOo,5and SiOz ( A = -86, B = -53 kcal mol-') were obtained from this data and estimates for the thermodynamic properties of (liquid),literature data on the enthalpy of glasses, the temperature dependence of rate of vaporization, and the variation of the activity of SiOzwith temperature. Redlich-Kister coefficients for log were obtained from the concentration dependence of the vapor pressure of potassium over the KOo,5-Si0zsamples: A = -10.9923, B = -6.7801, and C = -1.0101 at 1500 K. From these values, together with the enthalpy coefficients above, activities of both KOo,5and Si02 can be calculated over a wide range of temperatures and compositions. The KOo.5-Si02phase diagram was calculated from these values. Thermodynamic quantities are given for tridymite and for two potassium silicates, K2Si409and K2Si205.
Introduction The high sensitivity of atomic absorption for atoms in the gas phase suggests its use for vapor pressure measurements. A number of studies of this type are available in the literature for samples which can be confined in quartz cells.'-2 However, there is a substantial need for vapor pressure measurements for samples and for temperatures which are incompatible with window materials. Effusion measurements in vacuo are commonly used for vapor pressure measurements a t high temperatures with
KO0.5(1) -+ K(g)
'This work was supported by the Energy Research and Development Administration under Contract No. EF-77-C-01-2524.
The data obtained are used together with published values for a thermodynamic analysis of this binary system, in-
0022-3654/78/2082-1021$01 .OO/O
a variety of detectors for the material vaporized from a Knudsen cell. A similar experiment can be performed with atomic absorption detection of the vapors provided an inert atmosphere is present to slow the transfer of vapor away from the cell. In this case, material leaves the sample cell by diffusion as opposed to effusion. We have developed this new method for vapor pressure measurements and applied it to the measurement of potassium pressures in equilibrium with liquids in the KOo.5-Si0zsystem (eq 1).
+
'14
O,(g)
0 1978 American Chemical Society
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cluding a calculated phase diagram. The samples are placed in molybdenum or platinum cells with a small drilled hole through which vapors can diffuse at high temperatures. These cells are inserted into the center of an electrically heated graphite tube furnace in an atmosphere of argon (Woodriff furnace3), and the relative amount of potassium atoms in the graphite tube is measured by atomic absorption. The measured absorbance normally increases within 5-15 min to a steady-state value, where the rate of diffusion out of the cell equals the rate at which potassium atoms leave through the ends and walls of the graphite tube. For a particular sample cell, the steady-state absorbance is proportional to the pressure of potassium atoms inside the cell. There is, however, a substantial exponential temperature dependence for the proportionality constant:
sample. However, for loading platinum cups, the bottom was sawed off, the sample was introduced, and the bottom was welded back on. Calibration was carried out for each cup, before and several times between measurements. A K00,5-A101,5 sample was put in the furnace and plateau transmittance values were recorded. The same procedure was followed for measurements of KOo,6-Si02samples. Two different procedures were used in collecting data. With one furnace, each sample was measured at a series of temperatures once it was inserted into the furnace. In the other furnace, the usual procedure was to adjust the furnace temperature and then take readings on a series of samples. Both procedures gave accurate vapor pressures and no distinction is made between them in the analysis of the data.
p=
Results and Discussion 1. Enthalpy Data. In analyzing a two component system such as KOo.&3i02, it is desirable to have enthalpy data available as well as activity data. However, there are only a few measurements of the high temperature enthalpies or heat capacities of liquid potassium silicates, and even the heats of fusion of potassium silicates are not well-established. The enthalpies at 25 "C for two potassium silicate glasses have been measured6 and reported7 although the details promised8 were apparently never published. These values can be combined with our selected values for liquid KOo,6 and Si02 at 25 "C to obtain Redlich-Kister coefficientsgJO(A(H) = -82, B(H) = -61 kcal mol-l) for the enthalpy at 298 K. The values at high temperatures should be similar; in fact, this is confirmed by measurements on the temperature dependence of activity coefficients. Recent measurements of the heat capacity of a potassium silicate glass1' indicate that there is a substantial positive excess heat capacity of mixing around 1000 K in the KOo,S-Si02system. These data are fitted by a Redlich-Kister coefficient of about A(C,) = 8.0 cal mol-' K-' above the glass transition temperature. Thus, we can expect the coefficient A(H) to increase with increasing temperature, and the vaIues for 1500 K should be about A(H) = -75.4 and B(H) = -61 kcal mol-l. The uncertainty in these values is around f20 kcal mol-', and we have attempted to improve the accuracy by considering additional data from the temperature dependence of vaporization and solubility. It is hard to obtain accurate vapor pressure measurements on KOo,5-Si02liquid of a particular composition over a wide temperature range, because a t low temperatures the liquids are quite viscous. They may be present as homogeneous glass, but it is also possible to have a liquid saturated with a solid such as tridymite or even an intermediate nonequilibrium state. This is illustrated by the calculated lines in Figure 1 for a sample with the overall composition XKop= 0.144. The lower solid line shows the pressure for liqui$ of 0.1444 mole fraction KOo,5,and the higher line at lower temperatures corresponds to liquids in equilibrium with solid tridymite at the various temperatures. If the lower temperature values are included in a least-squares analysis, the apparent heat of vaporization is low, and if they are omitted, one is left with a temperature range of only 111K. We have omitted points at temperatures where a solid would be present at equilibrium from the initial treatment of the data. This leaves five samples for which the measurements span a temperature range of at least 200 K. One of these ( X = 0.4719) is in error for reasons considered below and must be dropped from consideration. Table I shows the values for
cell e-3YIRT
(absorbance)
The value of 39 kcal mol-1 is the average obtained with two separate graphite tubes with cells filled with a mixture of corundum and potassium P-alumina. The equilibrium vapor pressures for this system are known from the work of Plante et and these values serve to calibrate the various cells used in this work. Equation 1 also fits the temperature dependence observed in a second graphite tube furnace. The agreement between the data in the different furnaces is a clear indication that the value 39 kcal mol-' depends more on the nature of the potassium-graphite interaction (diffusion coefficients and the equilibrium constant for compound formation) than on the geometry of a particular Woodriff furnace. The mechanisms for transport of potassium out of a graphite tube furnace are considered el~ewhere.~
Experimental Section 1. Preparation and Analysis of Samples. Samples in the system KOo.5-A101,5were prepared by grinding together and then heating weighed amounts of A1203and KHCO, in covered platinum crucibles in a gas furnace. The temperature was gradually increased to around 1000 "C during 1-4 h, then to 1000-1100 "C where it was kept for 8-12 h, and finally to 1250 "C for another 8-12 h. About 600 mg for each sample was prepared. KOo.6-Si02samples were prepared by grinding mixtures of potassium silicate and either potassium bicarbonate or silica, heating to 630-650 "C for about 5 h, grinding again, and storing in a desiccator. The potassium content of samples was determined by dissolving them and measuring the potassium concentration in the solutions using the Varian atomic absorption spectrophotometer, Model 1100. Some samples dissolved in 49% HF, but other samples had to be treated with mixtures of HF and "OB, H2S04,or HC1, and evaporated to dryness before a suitable analytical solution was obtained. 2. Potassium Vapor Pressure Measurements. These measurements were carried out by atomic absorption using a graphite furnace. The essential features of the furnace are described in ref 3, and its modifications for vapor pressure measurements are reported in ref 5. The temperature of the area where the sample is introduced was measured with an optical pyrometer. The sample cells were made of platinum or molybdenum. Molybdenum has sufficient strength at high temperatures to allow the use of cells with a screw closure. Molybdenum has a tendency to stick, and to prevent sealing the cell into the furnace, the pedestal on which the molybdenum cells sit was rotated during the time the cell was in the furnace. With the molybdenum cells, different tops with different hole sizes could be screwed onto the base holding any particular
KO,,,-SiO2
The Journal of Physical Chemistry, Vol. 82, No. 9, 1978
at High Temperatures
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TABLE I: Partial Molal Enthalpy of KO,, from In P vs. 1/T
X 0.21537 0.28548
r?, -82.7 - 63.2
X 0.33497 0.34025
Rl - 59.4 - 63.0
TABLE 11: Redlich-Kister Coefficients for Enthalpy in the KO,,,-SiO, System a t 1500 K
A+ Data this work Plante data', Rate of vaporization' Joint, ref 13 and this work Enthalpy of glasses, 25 oC6-8 Corrected t o 1500 K6*11 Selected, including log y z data
B(H) 0.5519B
A(H) -132 -116 -97 -118 -75
-9 -103 -24 -5 -61
-137 -173 -110 -121 -109
-71 -86
-43 -53
-95 -115
IO4/ T Figure 1. Potassium vapor pressures over a K,O-SiOp sample with X K O o s = 0.1444.
It, calculated from our data. These values correspond to
Redlich-Kister values of A(H) = -132 and B(H)= -9 kcal mol-'. One can supplement our enthalpy values with others from the literature, but there are problems in all these cases. The latest NBS measurements12 are not given in detail. The earlier values4 have been analyzed without any allowance for a composition dependence of the heat of vaporization. Preston and Turner13 have accurately measured the rate of evaporation of potassium oxide from KOo,5-Si02glasses over a wide temperature range, but the heats of vaporization which they found are too low to correspond to the process
K00,5(so1)
K(g) + '/'loZ(g)
(2)
The most reasonable interpretation of their data is that they were measuring evaporation by the process KOo.&sol) -t '/zH,O(g)
4
KOH(g)
(3)
Using values selected for the thermodynamic properties of KOH(g)14gives R1values consistent with ours and with the one average value in ref 4. These values correspond to Redlich-Kister coefficients of A(H) = -97 and B(H)= -24. A combined least-squares treatment of our values and those in ref 13 gives A(H) = -118, B(H) = -5 kcal mol-l. It is clear from these determinations, summarized in Table 11, that A is around -100 kcal mol-l, but the value for B is not fixed very closely. A good value for B can be obtained, however, from the temperature dependence of log yz. Log yz is known a t several temperatures and compositions from the phase diagram measurements.16 We have selected T = 1784 K, X = 0.102, K = 0.94293, log y2 = 0.0212, because we have a substantial amount of vapor pressure data at this temperature. The other point selected is T = 1150 K, X = 0.297, K = 0.78139, log y2 =
0.0459. Using our values for log y1 as a function of composition at 1784 K, one can get log y2 = -0.012 at X = 0.293, T = 1784 K by a Gibbs-Duhem integration. This corresponds to a value of R2 = 860 f 700 cal mol-l for X = 0.297. This establishes the equation: A - 1.812B = 9.7 f 8 kcal mol-l. The vector A 0.5519B is orthogonal to A - 1.812B and, therefore, we have evaluated A 0.5519B = -115 from the values in Table I1 and combined this with the value for A - 1.812B, obtaining A(H) = -86, B(H)= -53.5 kcal mol-l. The new measurements on the heat capacity of a KOo,S-Si02glass with XKOo6 = 0.390 provide additional enthalpy data. Kracek's measurements on room temperature glasses can be interpolated to give an excess enthalpy of mixing for the liquids of He = -16.3 kcal mol-l - H298 from for this particular composition. Using Hlooo Table I11 for the liquids and a graphical integration of the heat capacityll for the mixture converts this to He = -16.0 kcal mol-l at 1000 K. Assuming that C, for the mixture remains constant to 1500 K, this value will change to He = -15.0 at 1500 K. The Redlich-Kister parameters we have selected correspond to He = -17.7 for X = 0.390 so the discrepancy is less than 3 kcal mol-l. Figure 2 shows the comparison of R, values calculated from the selected enthalpy coefficients with the individual values from the temperature dependence of the vapor pressure or rate of vaporization. 2. Thermodynamic Data for KOo.6. Pure liquid KOo,s is extremely difficult to prepare and handle, and its thermodynamic properties have not yet been successfully measured. It is, however, possible to make reasonable estimates as shown in Table 111, and these estimates were used as the basis for the calculations in this paper. The standard16-17heat of formation of solid KOo,sat 298.15 of
+
+
TABLE 111: Thermodynamic Properties for the Pure Materials Material
wZ98.15)
cal mol-'
H" 1 o o o ) a
cal mol-' -36200 -32128 - 206860 - 206640 - 206432.6 -205399 - 1003060 -557657.4
@olooo,
C, equationb
cal mol-' K-' a 103b 1O6C 109d KO,.,(c) - 43200 18.090 9.097 0.96 "0.5(') -40628 19.4256 11.45 1.80 SiO, (quartz) -217700 16.899 16.48 2.4 SiO, (cristobalite) -216417 18.1108 16.72175 2.05366 -1.65179 0.6463 SiO, (tridymite) - 217000 17.5259 16.72175 2.05366 -1.65179 0.6463 SiO, (1) -215740 18.1766 16.61 3.2268 1.2384 KZ O9 - 1059818 89.8337 85.081 9.0876 -3.1173 KzSizO,(S) -609691.4 67.1074 51.638 5.504 -1.55864 The power series for C p (generally valid for 1000-2200 K) is C, a is defined as A H ~ O ~ , , .+, , ( ~ , , ,-H",,,,,,). , = a + b ( T - 1000)t c ( T - 1000)z + d ( ~ - 1 0 0 0 ) ~ .
w,,,,
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X K9a
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1978
is also expected on theoretical grounds, since the addition of alkali oxide will break Si-0-Si bonds, allowing more anharmonic vibrations above the glass transition temperature. The only additional information necessary to fix the thermodynamic quantities shown in Table I11 for KOo,5are the estimates of EhlertZ3for the melting point and heat of fusion of KOo,5(1370 K and 5.0 kcal mol-l). The values shown in Table I11 for cristobalite, quartz and SiOz (liquid) are values chosen to fit the JANAF tabled6 except that the free energy functions of cristobalite and liquid have been adjusted by 0.0262 cal mol-l K-l to bring the quartz-cristobalite equilibrium temperature to 1175 K, the minimum temperature observed by H o l m q u i ~ for t ~ ~the formation of cristobalite from quartz. This temperature must lie above the quartz-tridymite equilibrium temperature of 1150 K. 3. Redlich-Kister Equations for log y. Every measured potassium pressure can be converted to an activity of KOo.6 by assuming that PO,= ' I 4 P and ~ dividing PKP021'4 = 0.707 PK5/4 by the equilibrium constant for the reaction
10
Mole Fraction of KO,5 in SiOp
Figure 2. Comparison of R, values from enthalpy coefficients and from vapor pressure temperature dependence.
-43.2 kcal mol-l is based upon heat of solution measurements reported in 1907.18 We have estimated the ,,, + SKF - S N s = 20.958 entropy of KOo,5at 1000 K as S 31.50 - 27.370 = 25.09 using JWNAF16 values for the other solids. The heat capacity for potassium oxide estimated in the JANAF tables is much too large, and we have made alternative estimates for the high temperature heat capacities of solid and liquid KOo,5 as 9.097 0.00096(T - 1000) and 11.45 0.00180(T - lOOO), respectively, assuming heat capacity additivity in the values of Whitelg for KA1Si308 crystal and glass. The data of Whitelg are apparently the basis for the enthalpy values listed in current compilations.20J1We have not attempted to find heat capacity equations valid down to 298 K, but instead have estimated Him,- HZg8values for KOo,5from the measurements on KA1Si308. Since these equations were developed, the heat capacity of a KOo,5-Si02glass has been publishedll with the value 16.7 cal mol-l deg-I for X K O o s = 0.390 at about 1000 K. This corresponds to Cp(KOo,6(l))= 26 cal mol-l deg-l if additivity of heat capacities were correct. The heat capacity of this particular glass must be influenced substantially by interactions between the oxides. There is evidently a substantial positive excess heat capacity of mixing at this composition. It is reasonable to assign a Redlich-Kister parameter of A = 8.0 for the excess heat capacity of mixing in the KOo&-Si02system. Sodium oxide shows a similar near additivity of heat capacities in solid silicates,16 titanates,22aluminosilicates,20 and NaAlSia08 glassz0but a substantial positive excess heat capacity of mixing in the NaOo~,-Si02liquid system.l1tl6This behavior
+
(4)
-t 1 / * 0 * ( g )
(5)
and values of Rl from the Redlich-Kister equation
H, = (1- X)*[-8600 -
5300(-1
+ U)]
(6)
Our computer program permits us to easily include the temperature dependence of R, corresponding to C,, = (1- X ) 2 [ 8 . 0 ]
(7)
The values of log y1 corrected to 1500 K are plotted in Figure 3. Compositions which would have solid present at equilibrium appear twice in Figure 3, as diamonds assuming a nonequilibrium glass of the stoichiometric composition and as squares for the equilibrium liquid composition from the phase diagram of Kracek et a1.16 Most of these points are closer to the line when the equilibrium liquid assumption is used, but for some, the stoichiometric glass is clearly a better approximation. A choice has been made for each point, and the corresponding square or diamond is shown in solid black. An
I
I
I
I
I
2
.3
.4
.5
Figure 3. Log y, vs. mole fraction.
K(g)
d log yl/d(l/T) = -H1/4.575428
I
XKq,
+
which can be calculated from the values in Table 111. The calculated activities can be divided by the mole fraction to obtain activity coefficients at the experimental temperature, and these values can be corrected to a standard temperature of 1500 K using the equation
+
+
0
KOO,5(1)
I
.6
Mole Fraction of KO5 in SiO,
I
.7
I
.8
I
I
.9
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The Journal of Physical Chemistry, Vo/. 82, No. 9, 1978
K00,5-Si02 at High Temperatures
TABLE IV: Summary of Power Series Coefficients at 1500 K for KO,,, in SiO, Redlich-Kister A
B
C
log Y H
- 10.9923 -86000
- 6.7801 -53000
-1.0101
CP
8
log Y H
-4.8045 - 35842
1025
Bale-Pelton
40
41
42
- 6.7801 -53000
-0.6734
Liquid -11.3290 -86000 8
Tridymite
increased experimental scatter for points at mole fractions above 0.4 is clearly evident in Figure 3. This is probably related to the problems of maintaining a uniform known composition when the vapor pressure of KOo.5 is relatively high. The effect is most pronounced at lower temperatures and higher KOo.5concentrations and is sufficient to prevent an accurate determination of the heat of vaporization of KOo,5from the sample at X = 0.4719 with our apparatus. The points in Figure 3 are fitted by the Redlich-Kister coefficients, A(1og y) = -11.2116, B(1og y) = -7.4673, C(1og y) = -1.6695. Figure 3 indicates a consistent tendency toward low values for log y at X = 0.06664, which is particularly pronounced at lower temperatures. Thus, it is best to drop all the points at this composition from the least-squares treatment. Just as with the enthalpy data, the data for potassium oxide alone do not determine the intercept at X = 0 accurately, and a better set of coefficients is obtained by including a value for log y2from the phase diagram15 in the treatment. With these two adjustments, we obtained the final Redlich-Kister coefficients, A(1og y) = -10.9923, B(1og y) = -6.7801, C(1og y) = -1.0101, which give the calculated solid line in Figure 3. This is a smooth curve with no sign of the pronounced break at high silica concentrations which Callow25found necessary in his treatment of this binary system. We believe that it is primarily the inclusion of the temperature dependence of log y2 in determining the enthalpy, and not just the additional experimental data, which is responsible for this qualitative difference in the final curves. The other published analysis of this binary system26is based upon the assumption that log yzis given by the equation log 7 2 = C/T (8) and no experimental enthalpy data. This assumption is qualitatively correct insofar as the large Redlich-Kister coefficients do get closer to zero with increasing temperature. For example B(1og y) is given by the equation B(l0g y) = +0.942 - 11583/T
(9) and has only a small part independent of T. This treatment26is, however, based upon the mole fraction of K 2 0 instead of KOo,5,and does not give the correct Henry’s law behavior at low potassium oxide concentration. The Redlich-Kister parameters for KOo,5-Si02 liquid mixtures are summarized in Table IV which gives the corresponding Bale-PeltonZ7 coefficients as well. With these values one can calculate the activities of KOo and Si02 at any temperature and composition. The experimental data used do not extend beyond XKOo4 = 0.55, and any use of these coefficients for solutions richer in potassium oxide than this should be recognized as an extrapolation. The calculated activities can be combined with vapor pressures for pure KO,,(l) calculated from Table I11 to give equilibrium pressures of potassium as a function of temperature and composition. The solid lines in Figure 1 were calculated in this way.
- 4.8045 -35842
4. The Thermodynamic Properties of Tridymite. There is still a lively controversy in the literature over whether or not tridymite is a stable form of pure s i l i ~ a . ~ ~ ~ ~ Tridymite only forms from quartz or cristobalite in the presence of impurities such as alkali oxides, and it is possible that it is thermodynamically stable only with a small percentage of dissolved material. For the reasons outlined below, we believe that pure tridymite is not thermodynamically stable. The calculations thus far have not made any use of data for tridymite; the solubilities used at 1784 and 1150 K in determining RZand log y2are based upon cristobalite and quartz. The enthalpy30 and heat capacity3I of tridymite are known, and the only remaining uncertainty is the equilibrium constant for the reaction SiO, (cristobalite)
-L
SiO, (tridymite)
(10)
at 1175 K where quartz and cristobalite are in equilibrium. If cristobalite and pure tridymite are in equilibrium at 1743 K, where the tridymite-cristobalite transition is observed in the presence of potassium oxide, K1175 is equal to 1.00455. On the other hand, the constants chosen for tridymite in Table I11 give KllT5= 0.99658. Thus, the total range in question is less than 1%and tridymite is either stable or unstable at 1175 K with a margin of less than 0.5%. A satisfactory phase diagram can be drawn for either of these situations; however, there are two pieces of experimental data which favor tridymite stabilized by dissolved alkali oxide. H o l m q u i ~heated t ~ ~ a mixture which was 0.00484 mole fraction NaOo.5 with quartz to 1623 K for a period of weeks. He found both cristobalite and tridymite to be present at equilibrium in this sample. This is a clear indication that the solubility of NaOo.5 in cristobalite is less than a mole fraction of 0.00484 while the solubility in tridymite is greater than this value at 1623 K. Thus, the equilibrium constant for cristobalite to tridymite at 1623 K is apparently less than 0.99526. The value K1175 = 0.99658 was chosen to make K1623 equal to 0.993, a reasonable amount smaller than 0.995. The compositions of liquid in equilibrium with tridymite are known over the temperature range from 1040 to 1743 K.15 With the Redlich-Kister coefficients we have obtained for the liquid, we can calculate the activities of solid KOo in tridymite saturated with the equilibrium liquid. With either set of equilibrium constants for tridymite, these data give a partial molal enthalpy of KOo,5in tridymite of about -36 kcal mol-l. If the entropy of solution of KOo.5 is approximately ideal and Rl = -36, the solubility of KOo.5 in tridymite at 1743 K (activity of KOo,5(s) = 0.7157 X 1 0 9 will be a mole fraction of about 0.0234. This suggests that the cristobalite to tridymite equilibrium constant may be as small as 0.98, and the choice K1175 = 0.99658 is preferable to an equilibrium constant larger than 1.00. With K1175 chosen as 0.99658 the activities of both solid Si02and KOo.5 are known for tridymite in equilibrium with
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The Journal of Physical Chemistry, Vol. 82, No. 9, 1978
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Eliezer et al.
There are surprisingly little data available on the thermodynamic properties of potassium silicates for comparison. The heats of fusion are substantially different from heats of fusion at 298 K (4.62 and 4.47 kcal mol-l, respectively) from the data of Kracek et ala6KzSi,09 appears in the phase diagram only over a narrow temperature range, and the enthalpy value for it reported in Table I11 could be substantially in error. However, the standard enthalpy of formation of KzSizO5at 298.15 K in Table I11 differs by only 11.4 kcal mol-1 from the value (-598.3 kcal mol-l) calculated from the heat of solution in HFa6 The uncertainty in the heat capacity of the solid could easily account for the discrepancy.
2000
IS00
References and Notes
L
k X,
,
T
Mole Fraction of KO,, in SiO,
Figure 4. Phase diagram of the K20-Si02 system.
liquid at 1743 and 1150 K. This fixes a single RedlichKister coefficient at each temperature (-4.0764 and -6.3938, respectively), corresponding to A(1og y)looo= -7.41556 and A(H) = -35842 cal mol-l. These values together with those of Table IV are sufficient to calculate the concentration vs. temperature lines in the phase diagram for equilibria involving solid SiOz. The calculated phase diagram is shown in Figure 4 together with the experimental liquidus points. Tridymite is shown to be stable with mole fractions of KOo,6from 0.0033 to 0.0091. The calculated phase diagram does not show the large KOo.5 solubility at low temperature suggested by H o l m q u i ~ t ,primarily ~~ because we have fixed the quartz-cristobalite equilibrium much lower than the value of 1323 K which he used. 5 . Thermodynamic Values for Solid Potassium Silicutes. The measured liquidus curves for K2Si409and K2SizO5,together with calculated activities for KOo.5 and SiOz along these curves, are sufficient to fix the enthalpy and free energy function for these solids. The selected values are shown in Table 111, and Figure 4 shows the liquidus curves calculated from these values. AHo for the reaction K,Si,O,(c)
-+
2K00.,(1) t 4Si0,(1)
(11)
is 117 kcal mol-l at its melting point, 1034 K. However, allowing for the partial molal enthalpies of the liquids at X1 = 0.3333, the calculated heat of fusion is 21 kcal mol-'. Similarly, the heat of fusion of K2SiZO5at 1325 K is 17 kcal mol-l.
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