The atomicdensities of the noble gases are valuesextrapolated to 298 K, for comparison with the other values. It is seen that the values, which represent the effect of the van der Waals forces, are somewhat smaller than those for the adjacent univalent nonmetals, but larger than those for the adjacent univalent metals, which have an outer electron in a new outer shell. T h e effect of going to a new outer shell, with increase of one unit in the value of the principal quantum number, is to increase the radius of the atom by about 35% and to decrease the atomic density by about 60%. Acknowledgment
Llteralure Cited (1) Mey.r.L.,AnnaienderChem.,Sup~l.Bond~,354(1~70). (2' Fm 8" intemting dim=ion of the history of the p r i d e tsble 8.8 Weeks, M E. '"Dismueryofthe Elements: Journalof ChemiesiEdueation,Ea*ton, PA, 1966,pp. 652 ff. "ande"Bmek~A~~Phya~z~14~32(1913). (4) Singman, C. N., J. CHEW EDUC.61,137 (1984). ( 5 ) pa.iing. L. .-college chemistry: lat ed., W.H.F
(61Donohue. J.,"T~sstructureof the ~
~
~
~sari ~ ~ 1950, ~ ~ p. 491. ~~ J~ O ~w~Di k y~a n d t ~ oan s ,:~ n cN, ~ york, W
1974. (7) Pauling.L., "The Nature of theChemicai Bond:3rd
d., Cornell Univ. Press, Itbaa,
An Easily Constructed Cuboctahedron Model Shuklchi Yamana Faculty of General Education, Kinki Unlversity, Kowakae, Higashi Osaka 577. Japan
A model of a cuboctahedron which is useful for teaching stereochemistry (for example of an intermediate in the interconversion between 1,2- and 1,l-dicarhadodecaboranesl) can he made easily by using a sealed empty e n v e l ~ p e The .~ steps are illustrated in the figure and given below. 1) The envelope is folded down the center lengthwise. 2) Two middle points of the base AB and the top DC, are marked
as E and F, respectively. 3) Two middle points of the left- and right-hand sides of the envelope ( D A and CB) are marked as G and H , respectively. 4) The intersection of the two lines GH andFE is marked as I. 5 ) The left lower part of the envelope is folded rightward at G so that the line G A falls on the line GH. The point an the line G A
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Journal of Chemical Education
correspondingto the I,is marked as J. 6 ) The left lower part of the envelope is unfolded. 7) A horizontal line Der~endicularto the left-hand side of ~- the ~
~
~~~
~~~~
envelope at J,is drawn so that it crosses the line FE and the rieht-hand side of the envelooe at -Knnd 7.. .rcsnertivdv -~ ---, ..r..... -.,. 8) ~ i intersection e of the two lines G K and JI is marked as M,and that of the other two lines IL and K H is marked as N. ~
'
~
~~~
.~ ~
~~~
Kaszmarczyk. A., Dobrott, R. D., and Lipscomb, W. W.. Proc. Nail. Acad. Scl. U. S.. 48, 729 (1962): Hoffmann.R.. and Lipscomb. W. N., horg. Chem. 2, 231 (1963). For other models made by this method, see the December 1984 issue of THIS JOURNAL. p. 1053 11.
,
~
9) The middle point of the line MK is marked as 0. 10) A line perpendicular to the line GK a t 0, is drawn so that it crosses the left-hand side of the envelope at P. 11) The upper part of the envelope is folded down at M so that the G falls on the line OP, the corresponding point being marked as Q. 12) The upper part of the envelope is unfolded. 13) The lower part of the envelope is folded up at K so that the Q falls on the M. The new corner on the right-hand side of the envelope is marked as R. 14) The points on the original front of the lower part of the envelope corresponding to the I and N, are marked as S and T, respectivelv. ~~, 151 The lower part of the envelope is unfoldrd. I61 The right luwrr part of the envelope is tolded upat Kso [hot t h ~ line KII parses the A l , the curresponding purnt being rnard~dns ~~
I ' -.
17) The right lower part of the envelopk is unfolded. 18) The lower part of the envelope is folded up along the line MN. On the front of theenvelope, the points corresponding to the Q, S, T, and U, are marked as Q', S', T', and U', respectively. 19) lower Dart of the envela~eis unfolded. --, The -~~~ 20) lnwcr nnrt o f the envelo~e is cut off alone the ziezae course ., The ..........r--..-. ~ - ~-~ - ~ ~ JQSTUL composed of the five lines JQ, QS,%T, TG, a i d UL. 21) The upper part of the envelope is cut off along the zigzag course GQ'S'TU'H composed of the five lines GQ', QS',' S T , TU', and U'H. 22) A line is drawn so that it passes the point Nand parallel with the line HL. This line is labeled (N). 23) The lower part of the remaining portion is cut along the lines JM, TK, and LN. The three corners J , T, and L of the lower zigzag course JQSTUL are relabeled JI and J., TIand T,, and LI and L,, respectively, according to the side on which they are situated. 24) On the reverse of the remaining portion, the points corresponding to the S', T , Q', I, U', M, N, J,, Q, K, U,LI,S, TI,and T,, are Q'*,* I*,, U'*,M*, N*, J,*, Q*, K*, U*, LI*, marked as Sr*, 'I" ~~
~~
.~ ~~
~
~
.~~ ~~
~~
~
~
S*, TI*, and T,', respectively. Now, a line is drawn so that it passes the N* and parallel with the line HL,. This line is labeled (N*). 25) Oblique lines are drawn to shade 12 triangles (QG ' M, MJ,Q, TIU', U'NH, NULI, and KT.U on the front, and Q'*M*G, M*Q*Jra, T*Ur*I*, Ur*HN*, N*Ll*U* and U*T,*K* on the hack) and two squares (T'*I*Q"Sf* and K*T1*S*Qs on the hack). 26) The remaining portion is folded both backward and forward along the lines MQ' (i.e., M'Q"), Q? (i.e., &"*I*), IU' (i.e., I*Uf*),MI (i.e., M*I*), I N (i.e., I*N*). NU' (i.e., N*U'*), QM (i.e., Q*M*), MK (i.e., M*K*), K N (i.e., K*N*), NU (i.e., N*U*), QK(i.e., Q*K*),KU(i.e.,K*Ur),S'M(i.e.,S'*M*),and (N) (i.e., (NW. 27) The right- and left-hand sides of the remaining moiety are ~ u s h e dtoward each other so that the four lines (MN, NN*, NaM*, and M*M) form a square. 28) Eight paints (Q, S , TI,and U on the front and Us, TI*,S*, and Q* an the hack) are pressed together and the six shaded triangles (MJ,Q, KT& ,' and NULl on the front; and N*L1'Ua, U*T,*K*, andM*Q*J,* on the hack), andasquareK*Tt*S*Qa, are folded inside. Thesides are taped together to form thelower half of the model given in the title. Similarly, t h e upper half of the cuboctahedron can be made by the following procedure. 29) The front, the left- and right-hand sides, and the hack of the upper half of the model are cut along the six lines (T'I on the front, HN and HN* on the right-hand side, T*I* on the back, and GM* and GMon theleft-handside). The four corners T',H, T*, and G of the upper brim of the model S'TU'HU'*T'*S'*pGQ', are relabeled TIand T., HI, H,, and HI*, T a l and TI*,, and G,*, GI,and G,, respectively, according to the original side of the starting envelope on which they are situated. 30) Eight points (Q', S', T'I, U', U'*, T*I, S * , and Q'*) are pressed together and the six shaded triangles ( T , IU', U'NHI, H,*N*Uf*, U'*I*T*,, Q'*M*G,*, and G,MQ') and a shaded square T*lI*Q'*S'*, are folded inside. The sides are taped together to form the upper half of the model named in the title.
Volume 62
Number 12
December 1985
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