Alan D. Adler and William J. Steele1 universitv of Pennsvlvania Philadelphia
II
Precise Atomic and Molecular Models
The first set of molecular models was probably that constructed by a carpenter for Dalton about 1810 (1). Since that time, the expanded educational use of simple models, along with the increasing denland for improved models in research, has promoted much interest in the design of atomic and molecular models (Z-5). While much of this effort has gone into the design of very simple and inexpensive models for instructional purposes, several authors have concerned themselves with increasing the usefulness of nlodels as precision research tools (6-18). The minimum requirements for versatility and sensitivity in precision model linlcages demand continuous but fixable control of the observable structural parameters, i.e., internuclear distance, bond angle, and relative hindrance to rotation about a bond. Units for precision models must identify the atoms or atomic groupings and either locate the centers of mass for skeletal or lattice models, or deformably designate some "electronic" shape of the unit in space-filling designs. As conveniently as possible, models should further convey by color coding, markings, etc., any other information pertinent to quantum, kinetic, or bond theory commonly depicted in models; e.g., bond order, partial charge, etc. Skeletal Models
Figure 1 exemplifies such a precision design for skeletal or lattice models. It is a modified combination of two previous designs (12, 18). The linlc is comprised of two clampable ball-and-socket joints connected by a sliding rod and cylinder controllably fixed by a central set screw. This set screw is plastic, of the "ball plunger" type, permitting adjustable control of the relative hindrance to axial rotation. The ball joint also carries a short threaded rod for firmly connecting the linlc to a solid spherical unit along a given axis. Since angular adjustments are made at the unit's surface and not at its actual center, there is an apparent displacement of the "nucleus" of the unit away from this center. This produces an error in visualization of
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the structure. For a symmetric angular adjustment %Am made equally about each linlc axis of the fiducial bond angle a, the nucleus is apparently displaced by an amount r(sin%Aa) {csc %(a Am)} along the angle bisector. The positive sign is for an increase of angle, the negative for a decrease, and r is the radius of the unit. Similarly for a uni-axial change about one angular leg only, the displacement is r(sin Aa) (csc(a +Aa)) along the axis of the unchanged leg. When the scale of the models, S(e.g., in./A), is large with respect to r, this displacement error is minimized and usually negligible. That is, a low ratio of r/S more closely approximates the desired point mass depiction. If this displacement error cannot be ignored, the units can be properly repositioned, either through the above equation, or empirically with calipers and a goniometer. Space-Filling Models
Precision linkages for deformable space-filling models are exemplified in Figures 2 and 3. These links have been primarily adapted for use with semi-space-filling designs (10, 14, 19). However, by suitably modifying relative dimensions and countersinking the linkage sites into the units, these links may be accommodated for use with deformable units of the completely spacefilling type (6, 15, 16). The conceptual and practical advantages of semi-space-filling models and the disadvantages of completely space-filling models have been discussed previously (10, 17). However, the fidelity of representation gained by semi-space-filling designs may be even further enhanced by the use of deformable units, as a closer and truer approximation to a completely space-filling design can be taken without resorting to the "shaping errors" colnmon to this type of design. The inherent resilience of the units permits this.The link of Figure 2 is used with hollow deformable units, where the relative ease of deformability may be
Bsylor University, Houston, Texas.
Figure 1. Skeletal model link and unit. A indicates the set rcrew controlling and fixing both bond length and rotational freedom. B indicates the set rcrew and clompoble bail and rocket ioint for control of angular adiushnent.
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Figure 2. Space-Rlling model link: external odiu.tment style. A lndlcater one of the set rcrewr controlling angular adjustment, B indicoter one of the set rcrewr cantroliing rotatonal freedom. and C and D, in combination, indicote screws controlling length odiurtment.
controlled by thickness, composition, and/or perforation of the arcuate surfaces (20). The link consists of a disc and a half disc, each carrying a central linking collar and flange, both mounted oppositely on a threaded rod, whose excess length projects into the hollow unit. Appropriate set screws in the circumferences and on the faces of these discs enable continuous but fixable adjustment of length, angle, and frictional control of ease of rotation. The screws on the half disc adjust the fiducial angle by hearing on the unit wall and using the linking aperture as a pivot. Small angular changes are given in radians by d/R, where d is the length of screw projected from the disc and R is the radial distance of the screw from the pivot. The link of Figure 3 is used with solid deformable units such as sponges, plastic foams, etc., where the relative deformahility is controlled by density or composition (20). Discs similar to those of the previous linkage are interconnected by a set of three threaded rods and cylinders whose heads are seated ball-andsocket fashion in the discs. Springs mounted over the rods keep the discs in place. A tapered screw projecting into the linking collar and flange enables continuous control of the ease of rotation by varying the frictional fit of the link in the linking aperture of the unit. With two of the rods and cylinders adjusted to a length dl and the third to d2,the link length along the axis is given by (dl d2)/2 and A a = 2 sin-' ((dl - d2) /4R} from the fiducial angle a, where again R is the radial distance of the rods from the disc center. Again, since angular adjustments are made a t the faces of the units with these two linkage designs, displacement errors similar to those discussed for the skeletal links occur. These may he estimated with the previous equations and corrected in the same fashion. Again the scale of the models affects the magnitude of this displacement error. The scale also affects the over-all sensitivity of the models (81). "Thermal" radii, r, (16), have usually been chosen in the design of atomic models, rather than van der Waals' radii, YVdW. The problems thus incurred and the advantages of van der Waals' radii have been given by several authors (7, 10, 17). However, it has not been generally appreciated that by choosing deformable materials for model units, one may not only readily adopt van der Waals' radii in the design of the units, hut can also arrange to have the units relatively deformable to one another as are the real atoms they represent. The compressibility of a spheroid is a cubic function of the ratio of the compressed radius to the uncompressed radius. Therefore we may choose the ratio r,/rvdw as a measure of the relative deformability of the elements a t room temperature and one atmosphere for incorporation into deformable model design
+
Figure 3. Space-filling model link: internal adjustment style. A indimter o tapered screw controlling rotatiand freedom. B indicates one of the threoded rod ond cylinder arremblier with spring for rimultmeour control ond odjvrtment of bond angle and bond length.
(see Table 1). This ratio is a periodic function of the atomic number and when plotted as such gives a complex, though interesting pattern. Table 1
..
~~
~
~
~-
~~~~
~aa,l;;' radii (see text). Note that the members of the oxygen family consistently have the lowest values in each period. Discussion
The above designs are presented in their most general form and many simplifications, n~odifications,and/or embellishments can be considered for specific purposes. For example, centering marks and calibrating scales could be directly impressed on the units and links. Adjustable springs and turnbuckles could be hooked over demountable studs on the spacefilling unit faces to allow continuous but fixable representation of hydrogen bonding. Further, deformable van der Waals' units that could only compress to a "hard core" radius (22,%9) could be designed for demonstrating the kinetic properties of gases, rtc. Several prototypes of these models have been made from a variety of materials. Their performance has been satisfactory. The links have also been successfully adapted to the new plastic Courtauld models now commercially available (24). The designs presented meet the requirements for precision models set forth a t the start. Unlike many of the available "dynamic" models, they permit small adjustments to be fixed, accurately measured, and then studied at leisure. Continuous or dynamic phenonlena therefore may he studied as a set of well characterized discrete steps. Unfortunately, one of the criteria for good model design seems to have become the mere "huildability" of a large number of structures and the gross representation of their rotational and vihrational characteristics. Careful examination of available structural data (25, 26) often shows that the fidelity of accurate representation of many complicated structures (e.g., crystalline glucose), and evcu several simple structures (e.g., phosphine), is very poor with many such models. Similar arguments can be lodged against many of the dynamic characteristics also (e.g., no available "dynamic model" can spontaneously depict the "breathing mode" of benzene). Properly used, the model designs given above suffer no such limitations and therefore offer the precision research Volume 41, Number 12, December 1964
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tools needed in productive scientific inquiry. There are no immediate plans for making these models commercially available, but the authors will be glad to supply interested readers with further details of the construction, use, advantages, and disadvantages of these model designs. The authors would like to acknowledge the support and advice of Drs. P. George, R. Rutman, and M. AIillman in the development of these ideas. Literature Cited (1) PARTINGTON, J. R., "History of Chemistry," St. Martin's Press, London, 1962, Vol. 3, p. 780. (2) PLATT, J. R., Science, 131, 1309 (1960). (3) SMITH,D. K., "Bibliography on Molecular and Crystal
Structure Models," National Bureau of Shndards Monograph 11, Washington, 1960. (4) SANDERSON, R. T., "Teaching Chemistry with Models," D. Van Nostrand Co., Princeton, N. J., 1962. (5) FIESER,1,. F., "Chemistry in Three Dimensions," Fieser. Cambridge, 1963. BRIEGLEB,G., in Houhen-Weyl "Methoden der Orgrtnisohen Chemie," 4th ed., Verlag, Stuttgsrt, 1955, Vol. 3, Part 1, pp. 545-572. See also, BRIEGLEB,Forbch?. Chem. Forseh., 1, 642 (1950). COREY,R., AND PAULING, L., Rev. Sn'. Inst., 24,621 (1953). DREIDING,A. S., Helu. Chim. Ada., 42, 1339 (1959). GODFREY, J., J. CHEM.EDUC.,36, 140 (1959). HARTLEY, G. A N D ROBINSON, C., Tmns. Far. Soe., 48, 847
,&""",.
/,Or..,>
(11) ~,*SSATIT, A., French Patent No. 1,101,229 (1955)
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Journal o f Chemical Education
(12) M c G o w n ~J., , AND lMOSS, X., J. Sci. Instl.., 29, 234 (1952). (13) PETERSEN,Q. R., Chem. Eng. N m s , 37, 99 (1959). (14) ROBINSON, R., AND AMBROSE, E. J., Tram. Fa?. Soe., 48, 854 (1952). (15) STUART, H. A,, Zeit. Phys. Chem. (B), 28,350 (1934). (16) STUART,H. A., "Die Struktur des Freien &Iolekuls," Springer, Berlin, 1952. (17) WEPSTER,B. M., Rec. Trav. Chim., 65,318 (1946). (18) WOOSTER, N., MCGOWAN, J., AND MOORE,W. T., J . Sci. Instr., 26, 140 (1949). (19) ROBINSON, C., Disc. Far. Soe., 16, 125 (1954). (20) ADLER,A., AND STEELE,W., U.8. Patents No. 2,942,357 and 2,953,860 (1960). (21) LAMBERT, F. L., J. CHEM.EDUC.,30, 503 (1953). (22) HIRSCHFELDER, J., CURTISS,C., AND BIRD, R., ''Molec~la~. Theory of Gases and Liquids," John Wiley & Sons, Inc., New York, 1934, pp. 1-52. (23) P~RTINGT~N, J . R., "An Advanced Treatise on Physical Chemistry," 2nd ed., John Wiley & Sons, Inc., Xew York, 1962, Vol. 1, 84&888. C., "New Courtauld Atomic Models," Griffin (24) ROBINSON, and George Ltd., Wemhley, Great Britain, 1962. Also, of interest, SHULMAN, S., "Modeling of Biomaleeulitr Structures," Ealing, Cambridge, 1963. L. E., Editor, "Tables of Interatomic Distances and (25) SUTTON,
Configurations in Molecules and Ions," Special Public* tion No. 11, Chemical Society, London, 1958. A,, "The Hydrogen (26) PIMENTEL,G., AND MCCLELLAN, Bond," W. H. Freeman & Co., San Francisco, 1960,255295
(27) PAULING,L., "Nature of the Chemical Bond," 2nd ed., Cornell University Press, Ithara, New Yark, 1954, 187193.
