R. H. Pelruccil Western Reserve University Cleveland, Ohio
Three-Dimensional Models in Phase Rule Studies
Through the Science Teaching Equipment Development Program of the National Science Foundation, the author has been engaged in a project having this two-fold objective: to illustrate and encourage the use of three-dimensional representations in the teaching of phase rule principles, and to study methods of preparing prototype three-dimensional models from which copies can be made. The results of the project are reported in this paper. Three-dimensional phase diagrams have received only limited discussion in most physical chemistry textbooks, but are considered in greater detail in several textbooks on heterogeneous equilibrium (1-6). In a recent series of articles Schweitzer and Wales (6) have described phase behavior in single and multicomponent systems through three-dimensional diagrams and their two-dimensional sections. The aim of this project was to develop prototype models which could be copied in a nonbreakable, lightweight plastic material. Several such prototypes and various of their sections for typical one-, two-, and three-component systems have been prepared from plaster of Paris, and it is hoped that copies of these will become available.% However, even if these models do become available commercially it is unlikely that a wide variety of types can he offered; it would seem appropriate therefore to describe some of the methods developed in this project so that they may be used by others. The major emphasis of this paper will be on methods of const.ruction although some attention will be given to the space model for a simple binary eutectic system, since this type of three-dimensional represent* tion has been less widely described than 0thers.j General Method for Constructing Models
Many individuals have constructed three-dimensional phase diagrams and by a variety of methods. Among those described previously in THIS JOURNAL are carvings from blocks of soap (7) and the use of transparent sheets of glass or celluloid to represent isothermal planes (8). One method described elsewhere employs
Presented in part at the 145th Nationrtl Meeting of the American Chemical Society, New York City, September 1963. ' Present address: California State College, Srtn Bernardino, Califarnirt. The author p l m to make the results of this work mailable in one of two forms (or possibly both): stereophotographs of the several models, together with a comprehensive descriptive booklet; or lightweight nonbreakable copies of the models themselves. Because the type of commercial production which is undertaken will depend on the market potential, readers are urged to communicate to the author their interests in the possibilities mentioned.
plaster of Paris (9). In the present work the following approach was taken: First a system was selected which would illustrate well the particular phase behavior to be described and the most reliable data obtained from the literature. The literature data were used to fabricate structural elements which were assembled to produce a "skeleton" model. Whenever possible the data were plotted to scale, but in some cases they could be employed only to give a correct schematic representation. Next the skeleton model was filled with plaster of Parisa in such a way as just to cover the structural elements and a t the same time produce reasonably smooth surfaces within the model. Any gross irregularities in the surface were removed, either by filling the low spots with plaster of Paris or by sanding down the high spots with a coarse grade of sand paper. Minor surface imperfections were not removed at this point because their effect was largely eliminated by using a rather thick coating of parting compound in preparing a negative of the prototype. To prepare plaster negatives of the prototype models it was necessary to cover completely all of the surfaces of the prototype with a parting compound. Several types of parting compounds were tried but the one which seemed to work best was a solution of stearic acid in petroleum ether, saturated a t about 50°C and applied to the surfaces of the model with a good quality brush. The solvent evaporated from this solution very quickly, leaving an adherent white deposit of stearic acid. Any small loose pieces of stearic acid were removed from the model with a stream of compressed air. A close inspection of the prototype was always made a t this point to insure that the surfaces were completely covered with parting compound. The prototype model was next placed into a mold, which was usually nothing more than sheets of '/,-in. Plexiglas, fastened together with screws and with the seams between sheets sealed with several thicknesses of masking tape. Plexiglas is especially good for this purpose since plaster of Paris does not adhere to it; extremely smooth surfaces result where the plaster is in contact with Plexiglas. With some of the larger models, for which a considerable mass of plaster was needed, special straps or braces were also used to support the mold. An amount of plaster of Paris mixture known to be more than sufficient to cover all the surfaces of the prototype was prepared and added to Full details on the preparation and uae of plaster of Paris for industrial molding can be obtained from the United States Gypsum Company, Industrial Division, 300 West Adams Street, Chicago 6, Illinok. The plater used in this work was U.S. Gypsum, HydraealB-11.
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t,he mold while the plaster was st,ill in a freely-flowing liquid condition. After filling with plaster i,he mold was vibrated gently for several minutes, either on a laborat,ory shaker or by hand on a laboratory cart. The purpose of the vibrating was to release air trapped in the mixture which would have produced flaws in the surface of the negative. After t,he plaster negative had set. t,horoughly, t,he mold was disassembled and the negative removed from the positive. Usually no more effort than gentle tapping of the negat,ive with a rubber niallek was required lo separat,e t,he negative and positive. Any surface inrpcrfcctions in the negative were removed and when the negative had been brought to its most perfect condition it was coated with stearic acid parting compound, replaced in the Plexiglas mold, and covered with plast,er of Paris. In this way any desired number of positive copies could be made. T o put the positive copies into their final forms, several operations were generally involved. The surfaces were cleaned thoroughly, sprayed with a clear varnish, aud painted with a high quality white paint. Lines aud curves which were t,o be emphasized were outlined in India ink with a ruling pen. Lett,ering was accomplished with a mechanical lettering set,. In some cases the letlering was done direct,ly on the surface, in ot,her cases on a sheet of posterboard which was t,hen cut aud rubber-cemented to the surface, aud in still olhers on st,rips of paper which were cemented to the surfaces. Where isobaric or isothermal sections were desired, a positive copy of the model was marked off carefully and sectioned with a large nret,al band saw.4
Figure 1. Conrtruction of model for ternory eutectic system. o-skeleton, b 4 l l i n g skeleton, =--finished prototype, d-prototype in mold, enegative model, f-negative in mold, g-negotive and positive, hpositive copy, i-finished positive copy.
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The general procedure is perhaps best illustrated in the next section vhich deals with the construction of a simple ternary eutectic phase diagram. The P-V-T model for water also mas prepared by this general procedure. A variety of methods were used in oonstructing the complete three-dimensional phase diagram for a binary eutectic system; t,hese are considered in the last sect,ion of the paper.
A Simple Ternary Eutectic System
A simple ternary eutectic system for which sufficient data can be found in t,he literature is t,he system, KNOrNaNOsLiKOl (10). A number of isotherms were plotted on triangular coordinate paper cemented to a Plexiglas base. Several holes were drilled along each isotherm and 3/z2-i~~. ahmiiuum rods were cut to scale and mounted from these holes. The faces of the p r i m model, representing the three binary systems, were cut from '/4-in. Plexiglas sheet and fastened t,o the base. Figure 1 depicts the steps of the procedurc.
The One-Component System:
Water
Because of the extrenie variations in temperature, pressure, and specific volume involved in a complete representation of the system water, this model cannot be made conveniently t , scale, ~ but a schematic representation is possible. To construct the portiou of the model representing the various phase relationships iuvolving liquid, vapor, and ice I, constant ten~peraturesections were d w ~ r on ~i posterboard and cut out. These sections were then transferred to perforated aluminum sheet5 and cut out with a metal band saw. The isothermal sections were mounted a t measured it~tcrvals in a Plexiglas frame. This was accomplished by milling l/ls-iti. grooves in the Plexiglas. The resulting skeleton model is shown in Figure 2. The prototype, negat ivel and positive models were obtairied by thc 7 general procedure already described. The skeleton niodel for the portion of the system involv- Figure 2. The system, water. ing high pressure modifications skeleton model. of ice was made from isobaric sectious cut from pcrforated aluminum sheet. Figure 3 is a photograph of the complete three-dimensioual model representing variations in the specific volume of water as a function of temperature and pressure. The high pressure modifications of ice are depicted separat,ely ill Figure 4, and Figure 5 shovs several pressure-volume isotherms.
In this work a model \IL saw from the I)oi11 Company, Des Plaines, Illinois, was used. 'Because plaster uf Paris does not adhere to metal it xns necessary to use perforated sheet far structural elements. TI& material was obtained in 4 X 8-ft sheets, '/,sin.thick.
The liquid-vapor region of Figure 6 consists of a lens-shaped volunie included between two surfaces, a liquidus above and a vaporus below. One may consider t,he volume to he filled a i t h isothermal, isobaric tie-lines and the liquidus and vaporus surfaces to originate along the vapor pressure curve on one face of the diagram and terminate along the corresponding curve on the opposite face. In the region of t,he triple points the lenticular volume is cut off from the sides by the surfaces OaRPE and OASPE, which represent the effect of pressure on the two-phase solidliquid equilibria. With decreasing temperature the liouid-vanor reeion shrinks to a sinnle tie-line. EF. The temperature a t which this occurs is the eutectic temperatsure, t,he lowest temperature a t which a stable liquid can exist. The composition a t point E is that of the eutectic solutiou, the composition a t point F, the eutectic vapor. The phenomenon of azeot,ropism in a binary system is represented by a pressure-temperature-composition region of the liquid-vapor loop, in which for each isobaric or isothern~alsection the liquidus and vaporus coincide a t one point. The locus of such points gives rise to a curve representing t,he variation of azeotrope composition wit,h temperat,ure and pressure. There are several possibilit ies for the azeotrope locus; it may intersect t,he vapor pressure rurve of one of the pure com~onents (e.e.. , , this locus intersects the curve OanI in Figure 6) or it may intersect one of the freezing point curves. The hypothetical systenr described hy Figure 6 is one in which the liquid-vapor loop is zeotropic at, lower temperatures and becomes azeot,ropic at higher temperatures. A three-diliie~isional phase diagram is
-
"
-
,>
Figwe 3.
P-V-T model for the s y s t e m , woter
A Binary System of the Simple Eutectic Type
The conlplele represeutatiou of the phase behavior of a two-conlponent sysl,em requires a three-din~ensional diagram, involving the variables temperature, pressure, and composit,ion. This diagram can be constructed by using two parallel temperature-pressure planes t,o represent the phase behavior of the pure components, and the region between for mixt.ures of variable composit,ion. Perhaps the most. familiar behavior in binary systems is t,hat. which rcsul1.s if the component,s are miscible as liquids and gases hut immiscible as solids, i.e., the eut,edic behavior. Figure 6 represents a hypothetical binary eutectic sysf.em in which both components are volat,ile. In the front plane OA represents the triple point of co~nponentA ; OALis the vapor pressure curve of component A ; OAA, t,he sublimation curve; and OAS, t,lle fusion curve. The corresponding curves for cornpo~icntR in the rear plane are OeM, OBI, and OBR.,respectively.
Figure 4 lleftl.
Figure 5
High p r e s u r e forms of ice.
(right). I r o t h e r m d
sections of t h e P-V-T model for water.
Figure 6.
Schematic three-dimenrionol phase diagram for a binary eutectir system (with ereotropim in the liquid-vapor equilibriuml.
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eutedic system are the relationship of a condensed phase diagram to the more general three-dimensional diagram, and the role of the vapor phase in eutectic fusion. The terms "condensed system" and "condensed diagram" have been very clearly defined by Ricci (19). The conventional two-dimensional phase diagram for a binary eutect,ic system, usually drawn as a T / c plot, is not a section of a three-dimensional model, but is instead a projection onto the T / c plane. The usc of such a T/c projection in forn~ulatinga vapor phase mechanism of eutectic fusion has been described previously in THIS JOURXAL (13). Figure 7 shows a portion of the three-dimensional model for a binary eutectic system and Figure 8 shows how t,his model can be considered in terms of its subregions-solid-solid-vapor, solid-solid-liquid, solidliquid-vapor, and liquid-vapor. Construdion Methods
Figure 7. Three-dimemionoi model of a binary eutectic
rydem.
especially useful in illuslraling that ill some systems azeotropism may develop over only a range of temperatures and pressures but not over ihc elllire liquid range. An isothermal section of the spacr model in Figure 6 in the azeotropic region would rcvcal a maximum in the liquid-vapor loop-a maximum in vapor pressure. An isobaric section would show a minimum in the liquid-vapor loop-a miriin~mnin boiling point. AB is a segment of thc sublimation curve of compollent A and IH the corresponding segment for component B. I t can getmally be assumed that the pressure of a vapor above a mixture of immiscible solids is simply the suin of the vapor pressures of the pure solids. The identical curves C D and J G represent the s u ~ nof the vapor pressures of the solids as a function of temperature. The curves CD and J G can be connected by isothermal, isobaric tie-lines which produce the ruled surface CDEFGJ. Each of these tie-lines has three points of interest,, the endpoints which lie in the planes of A and B and desigoatc that the solids in equilibrium are pure, and a point on the curve ZF which gives ihe coolposition of the vapor in equilibrium with a solid mixt,ure. The rulcd surface O,FEDO, represents the threephase equilibrium, solid ASliquidSvapor. Composition of the liquid is given by points on the curve OLE,and vapor by t,he curve 0 3 . The corresponding three-phase equilibrium, solid BSliquidSvapor, is represented by the ruled surface OeEFGOB. The special significance of the ruled surfaces of three-phase equilibrium is t,hat t,hey depict not only variations in equilibrium teinperature (freezing point) wit,h composition but also variations in equilibriunl pressure. Equilibrium vapor pressure and vapor conlposition are not generally represent,ed in convent~iorraltwo-dimensional diagrams of binary eutectic systems. There are several additional possibilities for the shapes of solid-liquid-vapor ruled surfaces which are discussed by Wales (11) and by Zernicke ( 5 ) . Among the intere~t~ing ideas which can be illustrated by using a thrce-dimensional representation of a binary 326
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The complete three-dimensional representation of a binary eutectic system was not a simple matter, for the system had to be considered in terms of its several subsections which fit together in a particular way. Because of this requirement it was necessary to depart considerably from the general procedure in preparing this model. Construction details can best be discussed in t,erms of the subsect,ions designated by Roman numeral in Figure 8.
Figure 8.
Several rubsectionr of the model f o r o binary eutectic system.
The Solid-Solid-VaporSection ( I ) : To construct this sect,ion it was decided to begin with a skeleton of the portion of the model below the vaporus surfaces. This was done by erecting isothermal sections of perforated aluminum sheet from a Plexiglas base as shown in Figure 9. The Plexiglas sides correspond t,o the sublimat.ion curves of the two pure components. Each section originates a t a point on the sublimation curveof one pure component, curves upward to a point represent,iog t,he sum of the vapor pressures of the two solids
Figure 9. Binary eutectic ryrtem. Solid-solid-vopor section (I). o skeleton negative model; b, preporation of positive; c, final poritive.
aud then back downward to a point on the sublimation curve of the second component. Aft,er filling the skeleton inodel with plaster and covering its surfaces witah parting compound, the Plexiglas sides of the original were replaced with two new ident,ical sides representiug the sum of the sublimation pressures of the two pure solids. A p1ast)er of I'aris ~nixt~ure of just the right consistency was now poured onto the original and the height of the curve was checked using a block of wood as a screed. This opera(.ion is suggested by Figure 9b and the relationship of the finished model t,o the original is brought out in Figure 9c. The Solid-Liquid-Vapw Seclims (II and IZI): The prototype models for these sections were easily prepared from skeleton models in which isothermal sections mere made of perforated aluminum sheet. The only difficulty encountered in making positive copies was that of preparing negatives ~vhichcould be freed easily from t.he protot,ypes. A negative made in sections solved the problem. A small box of plywood was constructed around the protot,ype model, which itself was mounted on a wooden base. The prodotype was coated with part,ing 'ompound and a layer of plaster of Paris mixture about 2 4 cm deep was poured into t,he box. After this section had hardened it was covered with parting compound a d another section poured ont,o it. This process was repeat,ed four or five times. The relationslip of one sect,ion of a negative to the prototype is shown in Figure IOU. Figure l o b shows a complete negative consist,ing of several sections and the prototype from which it was made. The Liqui&Vapor Sections (IV and V ) : The liquidvapor equilibrium surfaces were prepared in two sections ' 1 ure aud by two different met,hods. Sertion I V (see T'g 8) originates a t the eutectic kmperature as a single tie-line and enlarges into a lent,icular region extending from one face of t,he diagram t,o the other in the region of the melting points of the pure components. The skeleton model for section I V was prepared in two parts, one representing t,he region below the vaporus surface and t,he other the region above the liquidus. These two p a r k and t,heir relat,ionship to each other are shown in Figure 11. The skeleton model was filled with plaster of Paris, the surfaces were coated with parting compound, the two parts set together and sealed along the out,side edges with putty, and the interior region filled with a plaster of I'aris mixture. This interior lenticular region is Section IV.
Figure 10. Binary euteitic ryrtem. Solid-liquid-vapor r%ctionr I l l a n d o, prototype model o n d ~ e c t i o nof the negative; b. prototype and ~ o m p i e t enegative modell.
IIII.
Figure 11. Binary eutectis ryrtem. negative model (left).
Liquid-vopor section
IIVI,
Figure 12. Binary evtectic ryrtem. prolotype model lright).
Liquid-vopor section
(Vl, skeleton
skeleton
Section V was prepared fro111 isothermal liquid-vapor loops which were cut from '/&n. balsa wood s h e e k The vaporus surface was cut from stiff paper and glued along the bottoms of the several isot,hermal sections. The resulting skelet,on model is shown in Figure 12. The model was then partially filled with plaster of Paris, the lower paper surface removed, and the liquidus and vaporus surfaces produced by filling with p1ast)er and sandpapering. A negative of section V was not prepared although if desired oue could have been made by the split section method previously described. An advant,age of t,he procedure used in preparing section V over that used in sectioos I-IV is that only simple hand tools were required. The Solul-Liqrrirl Sections (VI and V I I ) : Equilibria involving the condensed phases, solids and liquid, are represented by sections VI and VII. These were prepared by assembling isobaric sections from L/8-i~~. balsa wood sheet as shown in Figure 13. From this skeleton model were preparrd the prototype original, a negat,ive, and positive copies. Again all the operations
Figwe 13. Binary eutectic ryrtem. skeleton prototype model.
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required in producing t,he final model were performed with simple hand tools. Conclusion
This paper has dealt with several methods of preparing three-dimensional phase diagrams from plaster of Paris. In addition to t,he examples described here, models of the state regions of a simple t,ernary eutectic syst.em, a complete nrodel of a ternary eutectic system with binary compound, one of its polyt,hermalsections, and several of its isothermal sections have also been prepared. From the success encountered with the methods used to produce this wide variety of pieces, it is perhaps appropriate to conclude that any phase behavior which can be represented by a three-dimensional diagram in graphical or schematic form can be reprcscnted in model form as well. Certain of these models can be prepared with relative ease, others will require a great expenditure of effort. Each instructor must dekrlnine for himself if the improved presentation of phase rule principles which results from the use of t,hree-dimensional models will compensate for this expendit,ure of effort. Hopefully instructors may be spared the task of constructing models if some, such as those described in this paper, can be lnade available commercially.
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The financial assistance of the 'atimal Science I.'oundation, through its Science Teaching Eqaipment Development Program, is grat,efully acknowledged. Also the author wishes to express his appreciation to Mr. Louis Cordek for his assist,ance in the construction of the models, Mrs. Barbara Carter for some of the dcsign, and Mr. George Ball and Mr. Bruce Frumker for the photographic work. literature Cited (1) FINDLAY, A,, CAMPBELL, A. N., .\m &JITH, N . O., "Phme Rule," 9th ed. Dover Press, New York, 1951. ( 2 ) Krccr, J. E., "The Phase Rule and Heterogeneous Equilihrium," D. Vsn Nostrand Co.,New York, 1951. (3) WETMORE, F. E . W. A N D LEROY,11. J., "Principles of Phsse Equilibria," McGraw-Hill Book Co., New York 1951. (4) M.\SING,G. (tr&nslstedby B. A. ROGERS),"Ternary Systems," Dover Press, New York, 1944. (5) ZERNICKE, J., "Chemical Phme Theory," Gregory Louin, New York, 1957. (6)BCHWEITZER, 0. R. AND W . L L E C. ~ E., Chem. Eng., 70, No. 11. 117 (19631: ibid. No. 13. 111. W.SLES.C. E..
CLRVETH, H. R., J. Phys. Chem., 2, 209 (1898). WALES,C. E., C h m . Ena., 70, No. 19,187 (1963). Rrccl, J. E., op. eil., p. 63. R. H.,J. CHEM.EDUC., 36, 0113 (19.W). PETRUCCI,
