Tangent Sphere Model An Analog to Chemical Structure Ethel L. khuilz' Marblehead High School. Marblehead, MA 01945
The Tangent Sphere Model (TSM) is a useful tool for discussing chemical structure a t an introductory level. Physical models of the atom's kernel and valence electrons are easily constructed out of various sized Styrofoam spheres. Students can construct and manipulate the models as they seek to understand the basic ideas of structure, geometry, and bondine. Two-dimensional representations can be drawn utilizihg circles of various sizes. Charge and symbols are also used to designate the type of species involved. The TSM may be viewed as a "concrete" representation of the Kimball charged-cloud model. I t was first discussed in THIS JOURNAL by Henry Bent in a series of articles in the mid-1960's (see bibliography a t end of article). The key noints of this concentual model are oresented here as a Hummary of the w o k done by the parkcipants of the first Dreyfus Institute held during the summer of 1982 a t Princeton University under the sponsorship of the Woodrow Wilson Foundation. This article will serve as a reminder of the usefulness of this model especially for beginning students a t both the secondary school and college levels. Assumptions and Principles
Below is a list of the assumptions and principles upon which the tangent sphere model is based. Although the general terminology will he familiar to most readers, i t is presented here to complete the general picture of the model. Atomic kernel. The kernel of an atom is a spherical region containing its nudeus and all nonvalence, inner electron pairs. Its charge is alwayspositive,since thenumber of protonsin thenucleus always exceeds the number of innerelectrons.Hydrogen is unique in the faet that it has but one oroton and one electron and therefore eonrnini m>"innrrrlwrrcm"as inother atoms I t is reprrsrnrrrl by irs nucleuc and n me-el~ctronsphere surnunding it, a protonawd sphere \'ole nrr electron*. The talencr rlcctrons are rhuxe hund 1,eyond the kernel Valence electrons are avn~lnbefor chemwal lrmdmg. Poult Exrluemn F'rinriplu. Because ofthe Exclusion I'rinriple, no more thnn twr elertnms may uccupy the same region n t the snme rune, thus limitmg the charge un any electron sphrre t u one o r two negative charges. Charge interactions within molecules and crystals. Coulombic repulsion occurs between kernels within a molecular or crystalline structure. Similarly, repulsion occurs between valence electron pairs, in faet, between all electrons. Attraction occurs between the positively charged kernels and the negatively charged electron pairs or between ions of opposite charge. This force of attraction accounts for chemical bonding and the stability of compounds. The lowest energy is approached when the forces of repulsion and attraction between charged entities are balanced. R~lnliwW P S of kernels and valrnre electron rphcrr*.'l'hrdiamrtw of rhc kernel, lor example C ur F, ts very amall ron~parrdto the diameter of a valence electron sphere. Tangent sphere condition. Spheres representing pairs of valence electrons cannot overlap; however, they will be strongly attracted, because of coulomhic attraction, toward kernels within the molecule or crystal. This attraction will resdlt in the tangency of the electron spheres, a condition that brings the negatively charged electron spheres as close as possible to the positively charged kernels. Two-dimensionalrepresentations. A solid circle is used to represent a kernel with its net positive charge; a hollow cirele represents
valence electrons, usually paired, with a net negative charge of two; and a hollow circle with Ht written inside represents a protonated sohere: a two-electron sphere containing a proton, Ht, thus having a net charge of negative &e Appllcallon of the Tangent Sphere Model: Structure
boelectronic Systems: Single Bonds T h e tangent sphere model allows students to predict the molecular geometries of several different isoelectronic systems. These predictions can be derived without resorting to hyl,ridization or promotion, hoth of which require for their understanding a knowledge of quantum mechanics. 'l'hr TShl easilv denicti the eeometrv of 10 electron svsterns, such as CHa, NH3, and ~ a c h othese f systems can be reoresented bv the same structure, a tetrahedron comprised of lour tangrnt valence electron spheres surn~unding a small kernrl where (see Fir. 1).This tetrahedral coniiguration represen& the lowestenergy structure for the n@atively charged valence electron spheres and positively charged kernel. While each system looks basically the same, the specific molecules have their own unique characteristics. In each structure the charge on the kernel, which is equal to its group number, may be determined by adding the nuclear A
.
HZ.
Figure 1. TSM structure for some single-bonded isoelectronic molecules.
0 =
pair of electrons
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= protonated
two-electron sphere with a net charge of -1
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Figure 2. TSM shucture for some leelectron molecules.
charge to the total charge on the inner electrons. This gives the net positive kernel charge. For example the charge on the nucleus of the oxygen atom, 8+, is added to the inner electron charge, 2-, to give a net 6+ for the kernel charge. Note also that the protonated spheres in each of the structures will have a net charge of 1- as compared to the 2- charge on each of the nonprotonated spheres. The actual hond angles for NH3 (107.3") and Hz0 (104.5") are close to the tetrahedral angle of 109.5- found in the symmetrical CHI molecule. (Please note that most of the diagrams also include the more traditional symbolic representations of the molecules to assist readers in their interpretation of the TSM.) Consider next a second group of isoelectronic molecules, those having 18 electrons and two non-hydrogen kernels. This family will include, among others, H202, N z H ~and , C2H6. Figure 2 illustrates these systems. As in the previous system, the protons from hydrogen atoms are shown in the outer electron s ~ h e r e as s H+. All of the abo;c two-kernel structures maintain their symmetrical tetrahedral confiauration about both kernels. Hond angles in these compound; are also compatihle with experimental data, confirming the reliability of the TSM. This model allows students to make predictions ahout the polarity of such molecules and others that are similar, the details will he outlined a t a later point in this article.
will result. Note that coulomhic repulsion will prevent the hydrogen kernels from residing in the bonding electron spheres (those nonprotonated spheres hetween the kernels). Knowing that ethene has a double hond is not a precondition for the construction of the model. With sixouter spheres about two kernels, the only way that an Octet-Rule, tetrahedral structure of electron pairs ahout each carbon kernel can he achieved (lowest energy structure) is to position the spheres as shown. What results is a representation of a double-bonded molecule. A comparison of the bond lengths and hond angles in the actual molecules, CzHs and CzH4, to the relative lengths and angles in the model illustrates the usefulness of the TSM. The double bond in this physical model for ethene can he seen and measured to he shorter than that of the single hond in the physical model for ethane. Analogous compounds containing atoms larger than hydrogen may he considered briefly here. CZHZFZ can he constructed with the cis-trans isomerism clearly visible and demonstrahle due to the restriction of rotation of the carbon-carbon double hond. The bonding of 14-electron systems such as CzHz and HCN may be represented in a manner similar t o that of the nrevious svstems studied. Two kernels with their four inner electrons will require an additional 10 electronr, or five valence electron airs (both hondina and nonbondina). Tetrahedral geome&y of electron pairs about the ''heavy atom" kernels would suggest one possible structure with the two kernels joined by three electron pair spheres. Figure 5 illustrates these molecules. Note the position of the hydrogens in the acetylene, C2H2. Coulombic repulsion would again suggest that the hydrogens should be placed as pictured. The nonprotonated spheres represent the bonding electrons. The models indicate that the relative hond lengths for the triple bond (three two-electron spheres between two kernels) is shorter than that of the double bond, as observed. The hond angles for C2H2 indicate that this molecule is linear. Indeed, this is in agreement with experimental data.
lsoelectronic Systems: Multiple Bonds
Let us next look a t a predicted structure for ethene, CzHa. This molecule must have two kernels (two carbon atoms with their inner electrons) and enough electron spheres to satisfy the total of its 16 electrons. Simple computation will indicate that with two kernels containing two inner electrons each, 12 valence electrons, or six pairs of electrons are needed. Therefore, a six-sphere structure about two kernels will be needed to represent this molecule. In placing the electron spheres in a tetrahedral arrangement tangent to each of the two small kernels, a structure like that pictured in Figure 3
@ H+
H
in tetmkdral hole
CtHt
HCN
H:C::C:H C2H4
Figure 3. TSM fm elhane. C2Hl
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r01
H
c'+
Figure 4. TSM sbucture for cis and hansditluoroethene,
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Figure 5. TSM struchnes fatwo l k l e c t m n molecules
Figure 7. Possible TSM struelure for the ionic and metallic crystals. Naqs)and Mas).
Figure 6. TSM srmctures for CHSCI.CS2, and HOCi.
Structures Analogous to These lsoelectronlc Famllles The tangent sphere model is most useful in predicting the neometw of molecules such as CHICI. CS?. and HOCI (Fie. . u 6 ) and oiher molecules containinga larger number of electrons. It also nicely illustrates the polarity of molecules, a very useful piece of insight in predicting reaction mechanisms. In unsymmetric structures, such as methyl chloride or hypochlorous acid, the charge distribution of the molecules can be predicted. By taking into consideration the coulombic attractions and repulsions of the kernels and electrons in the molecule, a low-energy arrangement may be predicted that is consistent with experimental data. Students can assess the differences in nuclear repulsions of carbon and hydrogen as compared to chlorine and hydrogen. The arrangement of atoms in a nonpolar molecule such as carbon disulfide is also predictable. In CS2, the nuclear charges on the carbon and sulfur kernels allow for the prediction of the arrangement of the three kernels. Note that the kernels of atoms in the third period, such as chlorine and sulfur, would have ten inner electrons surrounding their nuclei with net positive charges of 7+ and 6+, respectively, again equal to their respective group numbers. Structures Other Than Those wlth Electron Octets I t is valuable to note that not only are compounds with stable octets visualized by the TSM, but structures with expanded valence shells, such as PCls and SF6, or those that are electron deficient, such as BeH2 or BF3, may also be constructed. Aggregate Structure: Ionic Compounds and Metals Consistent with other models, the tangent sphere model for metals is based on the concept of electrons serving as the bonding mechanism between kernels of metallic atoms. Ionic compounds are seen as regular arrangements of positive and negative ions, each represented by an appropriate model. This concept enables one to show the relationship between ionic bonding and metallic bonding in crystals. Possible structure for the isoelectronic systems NaH(s) and MgW, are shown in Figure 7. The kernels of Na and Mg, third-period elements, will contain 10 inner electrons along with their nuclei. Note that the negatively charged spheres in the two struc-
Figure 8. H20/NH,
system according to the TSM.
tures are different. The compound, NaH, has a positive ion, Na+, and a neeative ion, H-. These unlike ions are strongly attracted to each other, stabilizing the crystal. ~ a g n e s i u m ' i structure consists of positive metallic ions, Mg2', and unprotonated electron heres. "electride ions" that serve a
