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By Roderick K. Davey

Introduction in any business, it is essential that we select those alternatives which are not only technically feasible, but will be the most profitable to the business m terms of corporate objectives Some of the studies and activities requiring support of financial analysis to select the alternatives which most closely satisfy the management objectives and technical considerations are as follows 1 ) Cutoff grade determination. 2) Ore reserve estimation. 3) Production forecasts (long- and short-term) for an operating mine, development of a new mine, or expansion of an existing mine. 4) Evaluation of haulage alternatives. 5) Evaluation of equipment.

Jan 1, 1979

By J. K. Abraham

In recent years there has been interest displayed in the precipitation phenomena occurring in austenitic alloys based on the binary Fe-Ni system.1'2 Previous communications have been extended covering the kinetics of clustering in the Fe-Ni-Ti system and the effect of cluster size on subsequent mechanical properties.3'4 To a great extent the question of coherency, which has a dominant role in determining the mechanical properties, is dictated by the degree of misfit between the precipitate and matrix. Thus, the change in lattice parameter of the matrix with composition is an important variable that should be examined. The lattice parameter, ao, of austenite was determined for Fe-29.75 Ni-1 to 6 Ti and Fe-27 Ni-1 to 10 Ti alloys in this study. The alloys used in this investigation were melted in a vacuum induction furnace from a starting stock of plast iron (99.95), electrolytic nickel (99.9), and ti- tanium sheet (99.5) in a MgO crucible. The alloy compositions in atomic percent are given in Table I and their relative position in the Fe-Ni-Ti ternary is indicated in Fig. 1. The chemical analyses were taken from the homogenized bulk materials treated in a manner related earlier.3 A diamond file was used to obtain powder from the bulk. To remove the stresses induced by filing, the powders were austenitized at 1875°F for

Jan 1, 1969

By C. K. Leith

THE purpose of these conferences has been to find some basic principles to guide us in the chaos which confronts us, to arrive at elements of a national policy. None such exists, nor, as a matter of fact, have there been any concerted efforts by the public or the mineral industry to evolve one. In the absence of such a policy the administrative agencies of the government can hardly be expected to be consistent or really effective in their handling of mineral problems, domestic or foreign. Basic Facts MINERAL reserves are unequally distributed among the nations. The principal world supplies are grouped around the north Atlantic, and nearly three-fourths of the world's total is controlled politically and commercially by English-speaking people. No one nation has a complete supply of the minerals necessary for modern industry; specialization, reciprocity, and large-scale movements between the nations are inherent in nature's unequal distribution of them. The principal sources of supply are relatively few compared with the nations to be served-a fact which determines broadly certain natural outlines of the world flow. National isolation with regard to minerals can be accomplished only through backward steps in industry and lower standards of living even in the nations possessing the most of the mineral supplies. A certain measure of unified commercial control, international in scope, is a natural consequence of the limited number of the large sources of supply. Mineral supplies for the world as a whole are so large that there is little need for concern about early exhaustion, but every nation has important problems of early exhaustion of particular minerals, which must affect their policies.

Jan 1, 1933

By D. K. Crampton

INTERNAL stress and season cracking in brass have been studied for many years and the technical literature contains many data on various phases of the subject. A résumé of the literature shows certain fairly well established points upon which various authors are mostly agreed: 1. Season cracking is primarily caused by the presence of relatively high internal stresses, or possibly in some cases by external stresses, or both. 2. A specific corrosion is also essential to season cracking: Certain agents, particularly mercurous salts, mercuric salts and ammonia salts; have such a specific corrosive effect. 3. The tendency to season crack under given conditions increases with increasing zinc content, brass containing less than 20 per cent zinc being relatively immune. 4. In alpha brass the season cracks are intercrystalline; in alphabeta brass they lie between the two constituents; and in beta brass they are transcrystalline. 5. The ordinary impurities or added elements in special brasses have little or no effect on the tendency to season crack. 6. Internal stresses can be eliminated, or at least reduced to a safe 'limit by annealing treatments at temperatures that do not produce visible microstructural changes or appreciable softening of the worked brasses. There are, however, some points in connection with the phenomenon of season cracking which have not been thoroughly investigated, or wherein different authors and investigators are not entirely in agreement. The investigations described in this paper were undertaken to determine: 1. The correlation between mercurous nitrate tests and actual season cracking on long-time atmospheric exposure. 2. The quantitative effect of type and degree of drawing operations on the intensity of internal stress and on the tendency to season crack in brass tubes. 3. The possibility of drawing high-brass tubes in such a manner as to prevent the occurrence of harmful stresses rather than to find a remedy for such stresses once having obtained them.

Jan 1, 1930

By C. K. Leith

THE interdependence of nations in regard to mineral supplies has grown apace with the expanded needs of industry, with depletion of reserves, and with advances in technology. This increased mutual dependence raises many questions of foreign economic policy of special importance in our postwar planning. Realization of the urgent need for the development of a general mineral policy broadly based on national requirements led to discussions of the problem between officials of the Department of State and myself, as a result of which it was suggested that I make an informal and unofficial canvass of the views of mineral specialists of the country on foreign mineral policy.

Jan 1, 1946

By Alan K. Burton

An industry-wide demand for bigger and more efficient trucks, with their supposed economies of scale, is well established. Some trucks have been, and often are brought "off the shelf," with the manufacturer offering only certain minor options, much as in the private automobile industry. The market is well established; there is keen competition; in recent years there have been few significant changes, and few are expected. With large trucks (100-ton-capacity and over), however, the situation is different. Here, with an individual truck costing several hundred thousand dollars, there is an ever in- creasing trend to engineer a truck for a specific application. The body will be designed and sized for the material that the truck will carry; the engine and transmission will be geared to the haul profile, and tires will be designed to suit road conditions and work requirements. Early Truck Development Haulage trucks designed expressly for open-pit mine service were introduced during the 1930's. The first models to arrive on the scene hauled about 15 tons, and easily outperformed the modified highway trucks that had been used for mine haulage up to that time.

Jan 8, 1975

By D. K. Crampton

The object of this work was to obtain and correlate information on effect of internal stress in variously drawn tubes on several properties. Also, a simple approximate method of comparing types of stress distribution in such variously drawn tubes was sought. There is also included some work on comparison of single and polycrystalline materials and effect of different types of operation and preferred orientation. Preparation of Material A series of tubes was drawn to study the effect of the type and degree of final reduction on various physical properties and the approximate distribution of stress intensity across the tube wall. The material used all came from a single heat of 70-30 brass melted in a regular production Ajax-Wyatt induction furnace. The analysis of the metal as cast was: copper, 70.36 per cent; lead, 0.04; iron, 0.02; zinc, balance. The material was cast in "shells" with 4 in. outside diameter and in. wall thickness. It was drawn part way down according to the usual standard mill schedule and then by suitable individual schedules to finish by a variety of types and degrees of drawing to the same final cases in every instance; namely, 0.750 in. outside diameter by 0.080 in. wall thickness. The exact final draws, together with the percentage area reductions, diameter reductions and gage reductions are given in Table 1. In fabricating these tubes, rather unusual precautions were observed to insure the elimination of ail variables not under immediate study. All intermediate anneals were carried out at about 1200" F. down to the so-called "cut-in size"; namely, 17/16 by 3/16 in. From this point on, all anneals were made at from 1050' to 1100" F. The greatest care was used all through to have the tubes as free from minor surface defects as possible. When such defects were visible on intermediate sizes they were carefully filed out. On the final draw to 3/4-in diameter the die was aligned with the greatest possible precision in order to insure as straight a tube as possible

Jan 1, 1933

By C. K. Leith

THE outstanding fact of the mineral world today, at home and abroad, is the surplus of current production, and particularly of capacity for production, over current requirements. This is not by Any means a new condition, but never before has it loomed up in such' menacing form. - Those of us who have devoted most of our professional efforts to the discovery and development of 'new mineral supplies now suddenly find ourselves with the job apparently accomplished, waiting for some one else to find an outlet for excess mineral production before we can further. The management of the industry is in much the same boat. The psjrcho16gy of the industry has been so grooved in the direction of acquisition of, minerals that at the present moment the mineral industry confronts the new problem of surplus in more or less of a daze.

Jan 1, 1931

By E. A. G. Colls

Discusses corrosion in Trail chemical plants producing ammonia, sulphuric, nitric and phosphoric acids, ammonium phosphates, sulphate and nitrate, together with miscellaneous allied material problems and their solution using erosion or corrosion proof materials. Conclusion, that for work of this nature, most costly material may well be cheapest in the long run.

Jan 4, 1950

By K. J. Rosengren

INTRODUCTION Mount Isa Mines Limited operates a major mining complex at Mount Isa in northwest Queensland, Australia. The location is arid and remote, some 600 miles distant from the nearest port of Townsville on the eastern coast (Fig. I). Current mine production is 6 million long tons per annum comprising 3.8 million tons of copper ore and 2.2 million tons of silver-lead-zinc ore. Copper ore production is scheduled to increase to 5.8 million long tons per annum by 1973 and a new silver-lead-zinc mine is being developed at Hilton, 13 miles north of Mount Isa. The Mount Isa and Hilton orebodies occur in a single stratigraphic member, termed the Urquhart Shale. This shale is conformable with several other sedimentary formations striking north-south and dipping approximately 65º west. The silver-lead-zinc orebodies occur in bands conformable with the shale bedding and tend to be of tabular shape, varying in width from a few feet to nearly 200 ft. The copper orebodies occur in breccia bodies, locally known as "silica-dolomite", within the Urquhart Shale and tend to be more irregular and massive with widths up to several hundred feet. The detailed geology is described by Bennett (1 2). At present, production comes almost entirely from underground operations with only about 25,000 tons per year of carbonate ore from open pit operations. The mining method used in the copper orebodies is sublevel open stoping with pillar recovery by pillar blast before stope filling or by sublevel caving after stope filling. A large primary sublevel caving operation was also carried out until recently in one of the orebodies. The wider lead orebodies are mined by sublevel open stoping, with or without pillar recovery, while cut and fill stoping is used in the narrower orebodies. Details of the mining methods are given by Davies (3). The company's interest in open pit mining stems from a large open pit which was mined out in 1965 and a further large open pit planned for future production. In addition, the company is currently evaluating several large ore deposits in Australia and New Guinea. All of these orebodies would be mined by open pit methods and rock mechanics studies of slope stability are now a standard feature of all open pit evaluations.

Jan 1, 1972

By K. G. Davis

Dilute alloys of silver and of thallium in tin have been solidijzed unidirectionally under controlled conditions, to study the segregation associated with a cellular interface under conditions where both thermal and solute convection are present. Autoradiography and radioactive tracer counting techniques were combined with electron-probe microanalysis to study both macro- and microsegregation. It was found that, for concentrations giving only small amounts of constitutional supercooling, cell formation had little effect on the macroscopic distribution of solute along the specimen. At higher concentrations the effective distribution coefficient was higher than that expected for a smooth interface. Node spacing was independent of initial solute content at lower concentrations, becoming greater as keff increased. Silver content at the segregation nodes of silver in tin alloys was independent of initial concentration and considerably in excess of the eutectic composition. SINCE the investigation of cell formation at advancing solid-liquid interfaces by Rutter and Chalmers,' a large volume of work has been dedicated to the determination of solidification conditions under which a planar interface will break down into cellular form. Early experiments were explained satisfactorily by the concept of constitutional supercooling,2 but, due to poor measurement of temperature gradients in the liquid, lack of accurate data on liquid diffusion and equilibrium distribution coefficients, and uncertainty about the effects of thermal and solute convection, these experiments cannot be used as proof for the theory. More recent work, however, has shown that under conditions where convection is eliminated or can be ignored good correlation is observed.3,4 Investigations into segregation at cell caps5 and at cell nodes6-'' have been made, but no measurements appear to have been done on the overall, macroscopic segregation down a unidirectionally solidified rod of material which has solidified with a cellular substructure. This has practical importance in casting, where regions of material with cellular substructure are often encountered, and also in zone refining where the thermal conditions necessary for a planar interface are unattainable. Further, as will be shown, the macroscopic segregation can give information on the following question. Granted that a cellular solid-liquid interface develops from a planar one when the conditions for constitutional supercooling are exceeded, how much supercooling is present after the cells have formed? EXPERIMENTAL PROCEDURE AND RESULTS Specimen Preparation. Specimens 25 cm long with a square cross section 0.6 by 0.6 cm were grown in graphite boats by solidification from one end. Alloy compositions are given in Table I. Two specimens of each composition were grown. The tin was 5-9 grade and the silver and thallium both 4-9 grade. Ag110 and Tl204 were used as tracers. Each composition had the same quantity of tracer so that auto radiographs of specimens containing different concentrations of the same element could be easily compared. Thermocouples inserted through the lid of the boat into a dummy specimen showed that, over the first 10 cm of growth, thermal conditions were quite steady, with a rate of interface advance of 5.8 cm per hr and a temperature gradient in the melt ahead of the interface of 3.0°C per cm. The specimens were seeded from tin crystals of a common orientation to eliminate orientation effects. Dilution of the specimen by seed material was minimized by the provision of a narrow neck between specimen and seed crystal. Macrosegregation. After growth, the specimens were sectioned with a spark cutter. The rods of silver alloy were cut into 1-cm lengths and analyzed for Ag110 using a y -ray counter with fixed geometry. The specimens containing thallium were cut into 2-cm lengths and analyzed for T1 204 by taking 13 counts from each end of the cut lengths through an aperture in lead sheet approximately 0.4 cm square. The results are summarized in Figs. 1 and 2. To find the effective distribution coefficient for the silver in tin alloys under smooth interface conditions, the region of substructure at the bottom surface of one of the 10 ppm specimens, see Fig. 3, was removed by spark machining before counting. Autoradiography. For both alloy systems the samples were polished on sections taken alternately parallel and perpendicular to the growth direction, and autoradiographed by placing the polished surfaces in contact with Kodak "Process Ortho" film. Figs. 3 and 4 show the structures revealed. The alloy containing 10 ppm Ag showed substructure only after a few centimeters of growth, and then substructure was limited to a narrow layer at the base. The "speckled" substructure reported previously in this system4 is here clearly seen to be an intermediate stage between planar and cellular interface conditions. The other samples show a remarkable similarity considering

Jan 1, 1969

Report of the Secretary TO THE BOARD OF DIRECTORS OF THE AMERICAN INSTITUTE OF MINING AND METALLURGICAL ENGINEERS Gentlemen:-To a Board of Directors keeping in as close touch with all of the affairs under their care as do the Board of Directors of the A. I. M. E., through their monthly meetings, a report of the year's progress is almost superfluous. To the members who read the monthly reports of. the Board Meetings and of the various activities of the Institute as published each month, a yearly report cannot carry much fresh information; yet for both it may be well, as is the custom, to glance over the field of our activities during the past year, and to make an estimate for the year to come. The beginning of 1924 was not propitious for the members of the mining profession, though tonnage production for 1923 had been high and the gross value higher than for several years. The margin of profit had been narrow, prospecting and development were limited, and many of our members were unemployed, or were working in other industries. Toward the end of the year, higher metal, prices were in sight. Old-time optimism began to grow, and it now brightens the outlook for 1925. An excellent barometer of the prosperity of the industry, and that of the Institute, for they go hand in hand, is the work of our Employment Bureau. During the past few months, the percentage of our members for whom openings have been found has been increasing to a marked extent, and many members who had fallen in arrears are placing themselves in good standing. The number of new members joining has remained about as it was in 1922 and 1923. The Directors have reason to congratulate themselves on the interest maintained during a dull period, for it is not humanly possible to take a keen interest in the technology of an industry when our hands and minds are not actively engaged in it Now, with a much brighter outlook, we can look forward to a year of increasing prosperity for the industry and for the Institute.

Jan 3, 1925

By S. K. Nowak

The lithium solubility limit in solid aluminum was determined by the use of micro-graphic techniques. The solubility limit thus established was shown to be a true equilibrium by checking the reversibility of AlLi phase precipitation and dissolution around the equilibrium temperature. The experimental results fit a straight line by plotting log N(LI) against 1/T. The metallurgical reasons explaining the lack of agreement among previous investigations are discussed. THE solubility limit of lithium in solid aluminum was determined by Shamray and Saldau1 by micrography and by Grube et al.' and also Voss-Kiihler- by measurement of electrical resistivity. Since the addition of lithium induces extremely small changes in the aluminum lattice, X-ray techniques could not be employed for this purpose; in fact, the addition of lithium does not cause an expansion of the lattice as might be expected from a consideration of atomic radii, but causes a small contraction from 4.040 to 4.037A3, 4 when the solvent aluminum is saturated with this element. An analysis of the results from previous investiga-tions'" shows that with improvement of investigation techniques the solubility limit tends toward a lower lithium content. Again, some of the author's own experimental results on the action of atomic hydrogen on A1-Li alloys seemed to indicate that the solubility of lithium should be much lower, i.e., about 0.3 pet at 300°C: instead of 1.15 pet, the lowest published value.4 For the foregoing reasons it was decided to investigate this problem. The aluminum and the lithium used for this purpose were of the highest purity. The lithium solubility limit was determined by metallographic techniques, in particular by the use of a reagent especially developed for the purpose of revealing the AlLi phase. The solubility limit thus determined was shown to be a true equilibrium by checking the reversibility of AlLi phase dissolution and precipitation around the equilibrium temperature. All of the experimental work was carried out in 1951 at the Centre d'Etudes de Chimie Metallur-gique at Vitry, France, under the direction of Georges Chaudron.

Jan 1, 1957

By C. K. Leith

CERTAIN ideas on iron and steel sanctions to follow originated in a series of conferences held under the joint auspices of the War Department and Brookings Institute in Washington last spring. The views presented may be regarded as those of substantial segments of the steel industry. In view of the fact that steelmaking capacity can be so quickly converted to the production of armament, any effective control must necessarily include the entire steel industry and not solely the parts devoted to the production of implements of war. It must also be exercised all the way from production and shipment of raw materials through their ultimate conversion into finished products and use. Control will be dis- cussed under two main headings: (I) control of raw materials at the source; (2) control of steel capacity within the Axis countries.

Jan 1, 1944

By C. K. Leith

WE face the postwar problem of the use of minerals as sanctions to control the armament and the re-armament of the Axis powers at the source, minerals being the raw material of armaments. That is the subject of today's discussion. It is one thing to talk about controlling minerals in the interest of rearmament, and it is another thing to put any such ideas into effect. A provision for min- eral sanctions-not by that name but under broader terms-was contained in Article 16 of the League of Nations, and an attempt was made to apply min- eral sanctions when Italy cut loose in Africa. That was a failure because the plan was badly thought out. They did not have the right minerals included in the list. Also, they did not have the machinery set up at all. Somebody said to the League of Nations: Let's apply the sanction, and it was assumed that that was all that was necessary. It turned out to be a rather sad perfor- mance. However, that certainly cannot be taken as an example of what will happen if a properly thought-out pro- gram is adopted.

Jan 1, 1944

By F. Forward, H. Veltman, A. Vizsolyi

Activities of oxides in the ternary FeO-MnO-SiO system are calculated from data on the binaries, using the Gibbs -Schuhmann method. These activity data are used, together with thermodynamic relations, to calculate the contents of oxygen, silicon, and manganese in liquid iron, and the composition of the corresponding oxide phases in equilibrium with it. Excellent agreement was found between calculated and experimental metal compositions, indicating that thermodynamic methods, such as those presented, can supply useful data on slag-metal equilibria, and are therefore capable of wider application. When molten steel containing oxygen is treated with a deoxidizer, the product of the reaction is composed of the oxides of the deoxidizing elements together with more or less iron oxide. This assemblage of oxides is presumably very close to equilibrium with the melt since it is formed directly from elements dissolved in it. Therefore, it is likely that metal analyses corresponding to a given oxide phase composition are calculable with considerable precision where sufficient information is available on activities in both phases. The present study was made to test this hypothesis in the case of deoxidation by silicon and manganese. Here experimental data on the relationship among oxygen, silicon, and manganese in the metal phase are available, as are activity data for the MnO-SiO2, FeO-SiO2, and FeO-MnO binary systems. A further objective was to demonstrate in a specific example how gaps in the data may be filled by judicious interpolation, and how the Gibbs-Schuhmann method is applied to obtain activities in ternary systems. The rigorous derivation of the latter, requiring considerable facility in handling partial differential equations, tends to mask the real simplicity of the method, and repel the very engineers and experimenters who would benefit the most from its use. I) GENERAL PROCEDURE Deoxidation of steel with silicon and manganese results in the formation of two phases, metal and oxide. The number of components is four—Fe, Mn, Si, and 0. The phase rule then gives the number of degrees of freedom: If temperature and pressure are fixed, 2 deg of freedom remain. Consequently, it suffices to fix the concentration of two of the components of the system for all the concentrations to be determined and, at least in principle, calculable. The practical problem is then to discover a straightforward method of calculation with a minimum of trial and error solutions. The type of reactions to be dealt with are like the following: Si (in steel) + 2 O(in steel) = SiO2(in oxide phase) The standard free energy of this reaction is readily obtained from the literature, so that the equilibrium constant can be calculated by the expression: Since the metallurgist is concerned with percentages rather than activities, a real solution requires information that will permit activities to be converted to concentrations. This information is fairly adequate for the elements dissolved in molten steel, but is not available for the FeO-MnO-SiO2 oxide system. Therefore, the first step in this study was to find the relation between activity and concentration for the three oxides making up the system. The next step was the calculation of slag and metal compositions that are in equilibrium with one another. As shown by the phase rule, it is sufficient to assume the values of two composition variables in order to calculate the rest. But which two are picked makes a vast difference in how difficult the calculation turns out to be, so that it is necessary to make preliminary trial of several routes to find the easiest one. A convenient method is to begin with the slag, and to pick FeO concentration and that of one of the other oxides. Starting with the relation between oxygen dissolved in the liquid steel and FeO in the slag permits a good first estimate to be made of the oxygen content of the steel in equilibrium with FeO, because the iron content being fairly close to 100 pct, Fe has an activity close to unity. Using this first estimate of oxygen activity, approximate values of the manganese and silicon contents of the melt are readily calculated from the appropriate equilibrium constants. A minor correction for the depar-

Jan 1, 1963

By R. K. Hopkins

For a great many years fusion welding has been used in and around petroleum refineries, but it is only within six or seven years that the more important pressure vessels have been constructed by this process, because before that time the properties of fusion welds would not with-stand such rigorous service. Since the adoption of fusion welding for the construction of this equipment, the refiner has required pressure vessels suitable for operation over a very wide range of conditions. On the one hand, vessels must operate at extremely high temperatures, in the range of 950° to 1000° F., and on the other vessels must be suitable for operation at sub-zero temperatures down as low as -75° F. Also, vessels are required suitable for operation at pressures upwards of 1000 lb. per sq. in., and as low as a 29-in. vacuum. In addition, vessels are required to withstand the attack of very corrosive mediums. It has been a rather difficult task for the fabricator to meet all of these require-ments, but as a result of the extensive research carried on by steel manu-facturers and several fabricators of class I vessels, practically all of these demands have been met. In order to meet the refiners' requirements for this wide range of conditions, the fabricator has worked closely with the metallurgist. This has enabled him to choose the proper base metal for each service condition, but it has been the fabricator's problem to develop a welding rod and welding technique, as well as a heat-treatment, that would produce in the welded vessel characteristics in and around the weld comparable to those in the base metal. Where corrosion is not a serious problem, and where the operating temperatures are not too high, ordinary carbon steel has served very well. But when corrosion is serious, and/or when the temperatures are either very high or very low, it is necessary to use some form of alloy steel to give satisfactory service.

Jan 1, 1935

By R. K. Adams

The availability of digital computers costing less than $15, 000 makes them economically attractive as versatile research and experiment tools. Several such installations at Oak Ridge National Laboratory have shown that improvements in research productivity and experimental control accuracy can result from the proper application of a small digital computer. However, careful attention must be paid to several potentially expensive problems in 1) interfacing the experiment to the computer and 2) programming the computer. Two specific applications of very small computers to metallurgical testing and research are described. One is a system for data acquisition and precise temperature control associated with a thermocouple testing program, and the other is a system for pulse calorimetry determination of the specific heat of metals over wide temperature ranges and in the neighborhood of metallurgical transformations. The data acquisition system which uses a PDP-8/S computer monitors signals from up to 264 low-level thermocouple signals at two scanning speeds: 10 readings per second for long-term drift data, and 500 readings per second for transient data. The pulse calorimetry system which uses a PDP-8 computer accepts up to 6 signals of 0- to 10-v range and because of a larger analysis program, uses a bulk memory (magnetic disc) in addition to core memory.

Jan 1, 1969

By P. Somasundaran, G. E. Agar

The role of calcium, phosphate, and fluoride species in determining the zeta potential of apatite in aqueous solutions was investigated in this study. The zeta potential of apatite was determined by streaming potential measurements over a wide range of concentrations of potassium nitrate, calcium nitrate, potassium dihydrogen phosphate, and potassium fluoride solutions at various pH values. While the addition of phosphate made the apatite surface more negative at all pH values, calcium caused a significant increase of the zeta potential only when the zeta potential was negative. Fluoride addition had only a minor effect on the zeta potential, but both negative and positive values were increased.

Jan 1, 1973

By R. O. Williams

Analytical representation of the thermodynamics of solutions is highly desirable from the standpoint of accuracy, compactness, and numerical manipulations. In particular, computer calculations are greatly implemented. Mathematical considerations show that previous expressions have one or more serious defects. This investigation shows a Fourier series to be satisfactory but that it is also possible to derive a new series which fits certain additional conditions. Included examples show the value of analytical expressions in giving a simple characterization of each system using some two to five parameters, the elimination of the Gibbs-Duhem integration, and the es timation of the error for the experimental function as well as derived functions. It is further shown that the present characterization provides easy comparison between systems. IN the past, thermodynamic calculations have depended to a considerable extent on tabular and graphical methods. As the volume and precision of such data increase such methods become less satisfactory. Specifically, the selection of the optimum representation and the estimation of errors require statistical methods which in turn require analytical representation. The utilization of such data require further manipulations which are best done analytically for maximum precision. For example, phase equilibria are determined by common tangents to free-energy curves: a graphical determination is normally of low accuracy. As computers are increasingly used analytical representations become almost mandatory. Insufficient mathematical consideration has been given previously to the selection of empirical expressions. Those expressions having some theoretical justification are generally too inflexible and mathematically unattractive. We consider the problem in some detail and show that a Fourier series can be effectively used. Also a new series is defined which has certain advantages. ANALYSIS We wish to consider the analytical representation of the heat of mixing, AH, the excess free energy, ?Gxs, and the excess entropy, ?sXS, as a function of composition, X, for binary solutions relative to the pure components in the same state. When a distinction is not required, we use W to denote any one of the above functions. One may use a Taylor expansion around X = 0 to generate a power series. As the derivatives are un- known we represent the series as W = A + BX + CX2 + DX3 + EX4 + ... [l] where the constants A , B, C , ..- are to be selected to provide some optimum fit. For the extremes of composition W is necessarily zero so it follows that A = 0 [2a] B +C + D + E +••• = 0 [2b] Nonelectrolytes, which we are considering, appear to satisfy the condition that d3W/dx3 = 0 [3] in the terminal regions. This is the basis of the a, ß, and Q functions used by Hultgren et al.' and others. While this condition does not have a strong theoretical basis it does appear desirable that any analytical relation should satisfy this condition. Darken2 and Turk-dogan and Darken3 have shown that many systems exhibit this behavior over an extended range from each terminal region, departure being restricted to a limited intermediate region. Since we have no a priori knowledge as to where this transition occurs we can require that this condition be satisfied only as a limit at the extreme compositions as a general condition. We will show later how more restricted conditions can be included in specific solutions. Darken2 has called this behavior the quadratic formalism; we call our application the limiting quadratic formalism, LQF. This condition applied to the above power series requires that D = 0 [4a] 4-3-2E +5-4-3_F + 6 • 5 . 4G + ••• =0 [4b] The form of the power series normally used, due to Margules,4 is W=X(1-X)(A + BX + CX2 + DX3 + EX4 + •••) [5] where A, B, C, --. are a new set of constants. (Guggenheim5 has given a variation of this expression in a more desirable form. Since, however, it is contained in the above expression it does not require separate consideration.) This form is precisely what results by incorporating the conditions in Eq. [2] into the power series and regrouping the constants. The LQF requires that B =C [6a] and 4.3.2(D-C) +5-4-3(E-D) + ••• =0 [6b] Thus, the correct form of the Margules expression with two adjustable parameters is w =X(1-X)[A + B +X2-2/3x3)] 171 and the EX4 term must be included before three adjustable parameters are permitted.

Jan 1, 1970