J. C. AXDERSOK AND 81. SCHIEBER
1838
solvent. By eq. 5 , 10, and 11, these acidity functions can be related to the experimental quantity pH pa^ =
pmH
=
pH -
pH - 6
pas"
=
(3
Ej
+ log
s
~
a
pH - 6
(12) (13)
Inasmuch as log s~~ in eq. 12 cannot be obtained for "unknowns" and the liquid-junction potential error EJ (eq. 5 ) cannot be derived independently of m ? ~ ,PWZH and pa^ are not suitable units for interpreting the measured acidity (pH) of alcohol-water media. On the other hand, subtraction of a constant (6) from pH gives an approximate value of pa^* (eq. 13). I t is recognized that pH in the aqueous medium approaches the PUH scale of the standards oiily under ideal conditions; similar restrictions apply to the interpretation of pH - 6 in terms of pa^". It is therefore suggested that the "practical" unit, related to pa^* in the may that pH is related to pa^, be designated pH*. The operational scale of pH* is thus defined as
TTol.67
range of water-methanol solvei~ts.~Different forms of the Debye-Huckel equation were used in the two investigations to evaluate ycl in eq. 4. Values of pa^* have been calculated likewise from the data given in Tables I and 11; they are listed in the last coluinns of these tables. Equation 8 TT-as adopted as the convention by which the activity coefficient of chloride ion in each medium was evaluated. The pa^* values in Tables I and 11, together with those from the l i t e r a t ~ r e2,2~can be considered as provisional standards. pH*(S), for a scale of pH* in alcohol-water media. The pH* becomes equal to pH when the medium is diluted continuously with pure water. Furthermore, the pH* determined under ideal conditions (low ionic strength, pH* neither too high nor too low) plays a welldefined role in chemical equilibria, for equilibrium constants in alcoholic solvents usually are also referred to the standard state in each particular medium. As eq. 11 shows, pH* is, under ideal conditions of measurement, actually a pmH value corrected for "salt effects," EYH.
where pH*(S) is identified with the conventional pall* values of certain selected suitable standard buffer solutions. The standard values can be assigned in the same way as the KBS standards for pH.20 This procedure for calculating pa^*, which involves essentially the addition of the logarithm of a coiiventioiial single ion activity coefficient to the experimental ~ ( a g y c J *has , ~ ~already been used to determine pa^* values for certain citrate-chloride and phosphatechloride buffers in 10 and 20% methanolz2and for an oxalate buffer and a succinate buffer oyer the entire (20) Reference 15, chapter 4. (21) Compare eq. 4. P ( ~ H ? C ~ )is* calculated by eq. 3, using sEO instead of wEo (see footnote 18). (22) R. L. Paiks, H. D. Clockford. and 9 . B. Knight, J . EEasha Mztchell Set. Soe., 73, 289 (1957).
The pH* is, however, a plurality of scales, different for each solvent composition, rather than a single universal scale of acidity. Solutions of equal pH* in different media, therefore, may prove quite different with respect to acid-base behavior. A single scale for all alcohol-water mixtures, based on pal$ (ey. l l ) , will become possible only when the validity of an independent nonthermodynamic evaluation of log m~~ or EJ,such as that of Gutbezahl and Grunrvald5or that of I z m a i l o ~ .is, ~generally ~ accepted. (23) The simplest means of obtaining pH* mould be to measure pH and subtract the appropriate value of 6 for the particular solvent composition and temperature Some 1 ariahihty in the as) mmetry potential of the glass electrode probablr can be expected a h e n the electrodes are transferred from the aqueous standard solution t o the alcoholic medium, hence, st'ndarda of the same sollent composition as the' unknowns'' are t o be preferrede If difficulty is evperienced i n adjusting the pH meter to the standard p H value a t high alcohol concentrations, a n aibitiaiy refeience point can b* chosen for measurement of the difference p H x ( X ) - pH*(S) (24) K -1 Izinailoi, Dokl Akad Nauk SSSR,127, 104 (1959).
CRYSTAL GROWTH I N THE SYSTEM E I T H I I X OXIDE-BORON TRIOXIDE-FERRIC OXIDE BY J. C. A
4AKD M.SCHIERER ~ ~
~
~
~
Department of Electrzcal Engzneerzng, Iinperzal College of Sczence urd Technolugy, London SW 7 , England Received lilarch 18, 2963 Phase equilibria in the ternary system Li?O-B20a-Fe2Q3 in the temperature range 950-1200" were studied and approximate phase stability boundaries m r e established. The study was carried out by slowly cooling (2"/hr.) ternary melts from the above temperatures to 800'. The crystals were identified by chemical, X-ray, and microscopic analyses. From these data, recommended mixtures, soaking temperatures, and detailed procedures are given for producing LiFeOz, LiFea08, and a-FeeOs single crystals.
Introduction The flux-melt method for growing single crystals, especially of ferrite materjals, is widely used. The components of the crystal required are held in a molten flux, in which they dissolve, and are slowly cooled over an extended period. Under suitable conditions nucleation and growth of the desired crystals occurs. The most commonly used flux for growth of ironbearing crystals such as ferrites and garnets is lead
oxide.1,2 This suffers from the disadvantages of loss of flux through volat,ilization, corrosion of the platinum crucible by the moltsenmaterial, and the high density of the flux, which leads to the crystals tending to float to the surface. Various methods such as the addition of PbF2, T120a, or P 2 0 5 to the flux have been tried3-5 to S. P. Remeika, J . Am. Ckem. Sne.. 78,1259 (1956). S.IT. Nielson and E. F. Dearborn, J . P h y s . Chem. S o l i d s , 5, 602 (1958). J. W.Nielson, J . A p p l . Phys., 31,515 (1960). H. Makrom and R . Iirishnan. Compt. rend., 254, 3340 (1962). ( 5 ) D. G. Wickham, J . BppE. P h ~ s .38, , 3597 (1962).
(1) (2) (3) (4)
~
(CRYSTALGROWTH IN
Sept., 1963
overcome these disadvantages, with varying degrees of success. In addition, LinaresGhas used barium and other borate fluxes and Ballman' has used molybdenstes and vanadates. The major objection to any flux method is the possibility of cheimical substitution of ions from the flux in the crystal grown. Blackman* has found evidence of Pb2+ ions in Fez03grown by this process. Ions particularly able to substitute for Fe in a crystal are S a + , K+, Ba2f, Co2+,and even F-; one of the authors (31.S.) has developed a method for substituting fluorine for oxygen ions in spinels, which suggests the possibility of F- substitution in crystals grown from lead oxidelead fluoride melts. It was required that single crystals of lithiurn ferrite be produced and, in view of the above objections, a new flux was sought. The ionic radius of boron is much smaller than the ionic radii of either lithium or iron, so that it is unlikely t o substitute for either of these. It was decided to use Bz03 as a flux, it having the additional advantages of being noncorrosive and having a low density. Experimental Material Preparation.-The starting materials were reagent grade B.D.H. LizB,07.5H20 and Baker C.P. H3B03,Li2CO3, and Fe203. It has k e n found that the flux must be prepared from materials from which chemical water and carbon dioxide have been removed. This was readily done with the Li2B407b y heating overnight in a furnace a t 550" in a platinum dish. With H3B03,however, ,3imilar treatment produced a glaze which could not be removed from the platinum. Alao, LizC03,when molten, readily attacks platinum Both these difficulties were overcome by first producing LiFeOz by firing together Li2C03and Fe20a at 800". For melk rich in B203, the chemicals Li2B,07, LiFe02, and FenOawere miixed together with in a planetaihm mill and then kept overnight in a furnace a t 300" in a silica beaker. From loss of weight measurement it could be calculated that all leaving the chemical water had been driven off from the B203. The resulting material contained lumps of glaze admixed with the material but it was easily removable from the breaker. The material then was reground to a uniform poq-der. Where lithium-rich melts were required, Li2B407was replaced by LizBzOs. This was produced by heating anhydrous Li2B407 with Li2C03in a platinum crucible a t 800" overnight. Measurement of weight 1 0 ~ and s absence of COZ on addition of acid to a small quantity of the material confirmed completion of the reaction. The powder for a given composition wa6 melted into a 50-ml. platinum crucible in a large muffle furnac-e, additional powder being added until the crucible was 607, full, containing about 80 g. of material. At this stage an approximate melting point was determined. Crystal Growth -Reliable production of crystals of a given composition can only be obtained if the ternary phase diagram is known in some detail. The preparation of a phase diagram a t several temperatures between 1000 and 1200" for the system therefore was undertaken. Afl er soaking a t a high temperature for 20 hr., the slow-coohng program was carried out for a large number of different compositions. The resulting crystals were examined by X-ray powder diffraction and single-crystal photographs and their composition was identified. I n each case the matrix in which they were embedded also was examined. The furnace used was a 3-kw. tubular one, with a "crucilite" element capable of a maximum temperature of 1E00", povered from a saturable reactor. The slow coolirg was controlled b y a suitable cam on a standard "Ether" programmed proportional controller, using a Pt-Pt 13% R h thermocouple. The rate of cooling was 2"/hr from about 1200 to 700". To reduce the number of nucleation site? and thereby produce larger crystals, the crucible was continuously rotated imide the furnace a t 15 (6) E . C. Linares, J . Am. Ceram. S o c . , 46, 307 (1362) (7) See footnote in ref. 6. (8) M. Rlackman, private oommunioation.
THE
SYSTEMLi20-B2a03-Fez03
[;
1839
A A'\
0
'\ \
\
\ \
Li,O
Li e 0,
Li e,O,
Fig. 1.-Ternary diagram Li20-B20?-Fe208 showing approximate phase boundaries a t 1100". I n region I, LiFeO2 and LilBzOa are obtained together with little glass content. Region I1 is the stability field of LiFe,08 and LiZB204 with glass. Region I11 is the stability field of a-Fez03, Li2B407, and larger amounts of glass. Region I V is almost only glass with little Li2B407,and legion V is a-FeZOBwith binary lithium-boron oxides. r.p.m. throughout the cooling. After completion of the cycle, the crucible was mounted upside down on a silica tube inside the furnace. The temperature was raised until the contents dropped out, through the furnace tube, into a container full of water located below it. After examination of the resulting material optically and by X-rays, the crystals were extracted by boiling the matrix in concentrated nitric acid or, in some cases, a 1 to 1 mixture of nitric and acetic acids. I n every case the p1:ttinum crucible was closely covered during cooling to reduce losses by volatilization. Below about 40 molecular % Li20, the loss was 3% or less. I n the IJi2C-rich regions much greater losses-up to 207, by weight-were encountered a t first. A special close-fitting lid was manufactured and the losses were reduced to not more than 5 7 , .
Results and Discussion Figure 1 summarizes the approximate phase boundaries of the ternary E,ystem Li2O-B2O3-FezO3 obtained by the above method, at 1 1 0 0 O . Only identified crystals have been included in this diagram; many Li20-B203 binary crystals which have an incongruent melting point have been omitted from the diagram. The widths of the different phase regions do not appear to change appreciably in the range 1000 to 1200'. However, in region I1 of Fig. l , a-Fe203was recognized to be present a t 1150' and the LiF508disappeared. This is taken to be evidence that the ferrite melts incongruently in a ternary metal oxide system. Anderson, et U Z . , ~ have found similar behavior with silica, the ferrite being transformed to a-Fe302by prolonged heating at llOOo in contact with SiOz. It should be noted that the LiFe508crystals grown in the Li20-rich region were partially in the ordered phase, while those m the Fez03-rich region were very small, only partly molten, and in the disordered state. A glass, amorphous under X-ray examination, having a composition 5Fe2O3~25Liz0.7OBzO3 (molar yo) is found over a wide range of composition. This is similar to Hummel'slO observations in the system Li20-B203-A1203. Different amounts of this glass were found in different regions as indicated in Fig. 1. (9) J. C.Anderaan, P. Ndckah, and &I. Schieber, to be published. (10) K. H.Kim and A. Kummel, J. Am. Ceram. Soc., 45,487 (1962).
Iiomwr A . PIEROTTI
1840
The LiFeOz crystals grown by this method proved to be much more suitable for order-disorder heat treatmentll than polycrystals of the same composition. Table I gives the compositions cf some of the mixtures used, together with approximate melting temperatures and initial soaking temperature. The approximate yields and compositions of the crystals obtained after dissolving the flux also are given. TABLE
Initial mixtures, molar %
Approximate melting temp., O C .
501,i20.42Bz03.8Fez03 55Li20 37B20d8Fe203 25L1~0.70R&~ 6Fe203 29Li20 43B20129Fe2O3
750 800 950 1050
1 Max Approximate soaking % yield, b y nt., of temp., "C. iron-containing phase
1100 1090 1160 1200
15y0 LiF508 870 LiFeOz 1007, glass 10% G-FenO?
Yol. 67
It should be noted that the regions with less than 15 molar % LizO have melting points greater than 1100' and therefore are not recommended for crystal growth. The cornpositions given in the table are examples of some of the mixtures used for growth of the crystals specified. The largest crystals produced were generally of about 1.3-mm. length and weighed up to 8 mg. and the smallest were 0.2 min. in length and of weight about 1 mg. They were larger where there was little volatilization and longer soaking times were used. Acknowledgment.-This research was carried out with the aid of a research grant from the Clothworkers Guild of the City of London. (11) J. C. Andersonand
>I. Scliieber, t o b r published.
THE SOLUBILITY OF GASES I N LIQUIDS' Rr ROBERTA. PIEROTTI Georgia Instiwle of Technology, Atlanta I S , Georgia Received M a r c h 19, I963
B method is developed for predicting the solubility, the heat of solution, and the partial molar volunie of simple gases in nonpolar solvents. These properties are predicted for a number of gases dissolved in benzene, carbon tetrachloride, and liquid argon. The agreement with experiment is very good. The method uses equations derived by Reiss, Frisch, Helfand, and Lebowitz for calculating the revervible work required to introdure a hard sphere into a fluid. An estrapolation procedcre based upon esperimental solubilities is developed which is a direct test of the reliabiiity of their equations. It is found that their equations are in escdlent agwement with experiment.
1. Introduction I n a series of papers Reiss, Frisch, Helfand, and Lebovitz2 have developed a statistical mechaiiical theory of fluids based upon the properties of the exact radial distribution functions 'ilhich yields an approximate expression for the reversible work required to introduce a spherical particle into a fluid of spherical particles. They consider the case of a system of ( N - 1)particles obeying a pairmise additive potential and couple one additional particle obeying the same potential to this system by the procedure of distance scaliiig. The coupling procedure is used to obtain an expression for the chemical potential of the fluid in terms of a function related to the radial distribution function for the fluid. Although the radial distribution function is not known, it emerges that for hard sphere particles the only part of the radial distribution function which contributes to the chemical potential of the fluid is that part which determines the number density of particles in contact vith the hard sphere particle. While i t would be desirable to use their method to treat soft sphere molecules, the theory rapidly becomes too complicated. Instead of the more rigorous approach, it is possible to treat the soft potential as a perturbation to the treatment of hard spheres. The present paper mill show that the expression derived by Reiss, Frisch, Helfand, and Lebowitz for the reversible m-ork required to introduce a hard sphere iiiio a fluid is an excellent approxi(1) Thia rrork i i a s snpportrd in part b y a grant from the Petroleum Research Fund of the American Chemical 3omety. ( 2 ) (a) H Rriqs h. I,. Frisch and i L Leboritz. J Chem Phws. 31, 369 ( l U i 9 ) , (b) H Pric., I3 L. Frisch E. I h l f a n d , and J. L. Lebonitr, z b d . 32, 119 (1960). (3) r. Helfand, I1 R ~ i s sH. L. Fiisch, and J . L. Lebowitr, z b d , 33, 1379 ( 1960)
mation and that it can be used along with the molecular properties of liquids and gases to predict Fvith good agreement to experiment the solubility, the heat of solution, and the partial molar volume of simple gases dissolved in numerous solveiits. Alternatively, the hard sphere approximation may be used along with a siiigle measurement of the gas solubility in order to determine the solute-solvent interaction energy. 2. Theory Thermodynamic Equations.-The chemical potential of the solute pz', in a very dilute solution of nonelectrolytes is given by4 PZ1 = - xz
+ Pa, - kT In h 3 j 2+ kT In ( N z / V )
(1)
where -xz is the potential energy of a solute molecule in the solution relative to infinite separation, P is the pressure, f i g is the partial molecular volume of the solute, VXZ3and j , are the partition functions per molecule for the translational and internal degrees of freedom for the solute, Nz is the number of solute molecules in the solution, and V is the volume of the solution. For very dilute solutions, V = N l f i l , where N I is the number of solvent molecules and fil is the partial molecular volume of the solvent and also NP,"~ = Q, the mole fraction of the solute. The sum of the first two terms on the right of eq. 1 represents the reversible n ork required to introduce one solute molecule into a solution of concentration NZ/V. If the solution is sufficiently dilute to ignore solute-solute interactions, then the reversible work of adding a solute molecule to the solution is equivalent to (4) R. H. Fowler and E. A . Guggenheim, "Statistical Thermodynamics," Cambridge, 1939, paragi aph 823.
