Locating Bond Positions in the Construction of Molecular Models-An Alternative Approach A survey of the principal books on the construction of molecular models, such as Sanderson.' B a ~ s o w Dahhv.3 . ~ .. O r m e r ~ d .and ~ the Nuffield Chemistrv, Proiect's , "Handbook k r ~ e a r h e ' r s . " as ~ well as n large number oi journal anirlea on thrs ~~~
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R I I ~ ~ reveals I , that [here exwts a rernarkahle similarity in approach as far as the actual construction of molecular models is concerned. One of the problems involved in the construction of a reasonably accurate molecular model, and particularly in a fairly large-scale one in which defects are all too apparent, is the location of the bond positions on the central atom, i.e., locating on the surface af a sphere a number of points between which a certain angle is subtended a t the center of the sphere. Thus, for example, in building a model of a water molecule two points subtending an angle of 104.5' a t the center of the sphere have to be located in order to position accurately the two smaller spheres representing hydrogen. This problem is usually solved by making use of a variety of mechanical aids, such as compasses, wooden or cardboard jigs, temdates. etc. he following method, however, is not only simple in execution hut also has the added advantage of applying to a 'real' problem the elements of a branch of mathematics which is often taught in introductory university mathematics courses, hut seldom applied to areal situation. This is radian, or circular, measure for angles. A radian is defined as being that angle a t the center of a circle which suhtends on the circumference of the circle a length of 1 radius. Thus for a circleof radius R, a radian may be represented as is shown in the figure. I t is clear that in a circle the ratio of the semi-circumference to the radius is the same as the circumference to the diameter, namely the value s. Thus n radians must be equal to 180'. or 3.14 radians = 180".From this it follows that 1 radian = 57.3". We now apply this result to the problem of determining the position of bonds on the surface of a sphere. To do this we first convert the given hond angle from degrees to radians by dividing by 57.3.Thus for the two O-H honds in the water molecule the bond angle of 104.5' becomes 104.5/51.3= 1.82 radians. In the same way the angle of 109.5 hetween the four C-H bonds in the methane molecule becomes 1.91 radians. We have now converted a n angular measure a t the center of the sphere t o an angular measure related to the surface of the sphere-which is clearly more convenient to work with. However, the actual length to be measured off on the surface of a particular sphere will depend upon the radius of the sphere concerned, and from the figure it is seen that the length is obtained by multiplying the angle in radians by the radius of the sphere. Thus when constructing a demonstration model of a water molecule using a sphere of diameter 8 cm to represent the oxygen atom, the distance between the two O-H honds will be given by 1.82 X 4 cm = 7.28 em, measured along the surface of the sphere. This distance must now be measured off on a curved spherical surface. It has been found that the simplest method of doing this is hy making a paper strip about 1 em in width and about 2 c q longer than the length required t o he measured off. Using a pin, two holes the required distance apart are pricked on the strip. The strip is then Laid on the surface of the sphere and the position of the two pinholes marked an the sphere using a compass point, pin, or sharp pencil. The two band positions are now located with avery fair degreeof precision. In the case of constructing models of molecules such as ammonia or methane, where more than two points have to he located on the surface of the sphere, the method is extended by using a soft sharp pencil to draw a number d a r e s . The points of intersection of these arcs will indicate the required bond pusitions. Thus to locate the hond positions of the three hydrogen atoms when constructing a model of an ammonia molecule a paper strip is prepared, as described above, for an angle of 107" and a n arc is drawn using any point, say A , on the surface of the sphere as center. Using any point B on this arc as center, a second similar arc is drawn. This arc will naturally pass through the center point A of the first arc and will also cut the first are a t a point C. These three points A, B, and C will be the required points, with the angles between A a n d & A and C, and B and C all being the same. Although a t first sight the method may appear slow and cumbersome, it is in fact remarkably quick and simple, and, using a slide rule to carry out the few calculations involved, it has been found that the four points needed on the surface of a sphere representing the carbon atom in a model of the methane molecule can be plotted accurately in 3-4 mi".
Sanderson, R. T., "Teaching Chemistry with Models," D. van Nostrand Company Inc., Princeton, 1962. Bassow, H., "Construction and Use of Atomic and Molecular Models," Pergamon Press, Oxford, 1968. 3 Dahby, R.E., "Making CrystalModels," Pergamon Press, Oxford, 1969. 4 Ormerod. Milton B., "The Architecture and Properties of Matter-An Approach through Models," Edward Arnold, London, 1970. Nuffield Foundation Chemistry Project, "Handbook for Teachers," Longmans/Penguin Books, London, 1967. 2
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Education Bureau, Transvaal Education Department Private B a g X76,Pretoria, South Africa
Volume 51. Number 4. April 1974
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