Lars Melander University of Gdteborg and Chalmers Institute of Technology F O C ~ ,5.402 20 Gateborg 5, Sweden
Rectangular-BOX Model for the Polar Bond
The present writer has previously shown how a one-dimensional rectangular-box model can be used to illustrate chemical binding without introducing non-physical concepts as Coulomb and resonance integrals.' By means of a rectangular-box model it is also possible to show qualitatively how the electron cloud will behave in a polar bond. Students are generally taught that the electrons of a polar bond are pulled toward the most electronegative atom, i.e., toward the atom in the surroundings of which an electron has the lowest potential energy. For those who think a little deeper than the majority this fact is more controversial than it may appear at first thought. It is certainly in agreement with an electrostatic model, but, on the other hand, we should and do frequently talk of electrons in terms of a dynamical model. For a classically moving body the behavior would rather be the opposite. It moves at high speed when its potential energy is low, and vice versa. Hence it would be expected to spend most of its time in regions of high potential energy. In fact, the tme distribution of the electrons in a normal bond is a result of the properties of the quantum-mechanical ground state which diverge markedly from the classical pattern. This divergence is often pointed out in connection with the study of the harmonic o~oillator.~ By means of a simple box model the above-mentioned nonclassical behavior of a particle in the ground state can be studied in principle without inclusion of any other approximations than those inherent in the choice of potential-energy function. The computational work needed is of the same kind as that in the previous article1 and can be carried through with a reasonable degree of accuracy by means of some common mathematical tablesa and an ordinary slide rule. The region within which the electrons of a chemical bond move is pictured as a rectangular one-dimensional box with inpenetrable walls. The potential energy is assumed to be constant within each half of the box and lower in the half corresponding to the more electronegative atom. The potential-energy function has consequently the appearance of Figure 1, from which the dimensions and the assumed values of the potential energy are evident. In the region - a < x < 0, V = 0, hence the onedimensional time-independent Schrodinger equation for a single particle
Figure 1.
Potentidenergy fundion.
It is obvious that $ must be a sine function in this region and have the form $
~
=t
A sin
ol(z
+ a)
(3)
as it must attain the value 0 at x = -a. For the ground state there should be no node, hence we can confine aa to the interval 0 < aa < s ( a can always be considered a positive quantity). To the right of the origin two different cases have to be discerned according to whether E > Vo or E < Vo. In the former case eqn. (1) can be written
==-
d4$
2m(E
- Vo)
Ed = -@*
(4)
and $is a sine functiou of the form =
B sin [B(z - a )
+ r] = B sin p(a - z )
(5)
because the angle should approach s from below as x approaches a from the left. Since the ground-state wave function has no node, pa can be confined to the interval 0 < pa < T (0 can always be considered a positive quantity). Since the functions $left and &ight.l as well as their first derivatives must fit together a t the origin we obtain the two equations -
MELANDER, L.,J. CHEM.EDUC.,39,343 (1962);, PAWLING, LINWS, AND WILSON, E. BRIGHT, JR., Introduction to Qunnt,um Mechanics," McGraw-Hill Book Co., New York, 1935, p. 74. a All tables needed are found in, e.g., "Handbook of Chemistry and Physics," The Chemical Rubber Co., Cleveland, Ohio. 1
becomes
686 / Journal of Chemical Education
@
A sin ua
= B
sin pa
(6)
m.4 cas aa = -6B cas pa
(7)
8_ tan an = - tan pa
(8)
u
When E < V,, eqn. (1) is written 2m(V0 - E )
dP$ =
*
=
rz*
(9)
In this case the right-hand part of the wave function must be of the type = C sinh qz
J.,,,at.
+ D cosh qz
(10)
( y can always be considered a positive quantity) and a t the right-hand wall we have the condition z ( a ) = C sinh qa
+ D oosh ?a = 0
Figure 2. PmbobiliW distribution functions for indicated values of the stop-height parameter n = (2mVo)'/lml(i.
The claim for continuity of the wave function and its first derivative a t the origin gives the equations ( A sin an
=D
(12)
which together with eqn. (11) give u
tan aa = - tanh ya
(14)
It is now convenient to introduce a parameter n related to the height of the potential-energy step V o by means of eqn. (15). From the definitions (2), (4), and (9) the following relations are now obtained 6%' = y4a~
=z .
- n% - ,zan
(16) (17)
Since a, 8, and y can all be assumed to be positive quantities without introducing any undue restrictions, eqns. (8) and (14) assume the following forms tan un
a=/,
=
- tan 4-
( E > Vo) (18)
un
Once a value has been assigned to the step-height parameter n it is an easy matter to solve eqns. (18) or (19), whichever turns out to be the appropriate equation, for the unknown ara, e.g., by plotting the two members of the equation as functions of ara and finding the point of intersection. The values of pa or ya can then be found from eqns. (16) or (17), respectively, and the functions (3) and (5), or (10) can be drawn, the relation between A and B being established by eqn. (6) and those between A, C, and D by eqns. (11) and (12). Since it is possible to integrate the square of the different functions, normalization could also fairly easily be carried through, if desired. Figure 2 shows the (normalized) probability distribution functions, for n = 0, 1, 2, 3, 10 and m . It is quite obvious how an increasing potential VO= n2fi2/2ma2 in the right-hand half of the box will increase
the probability of finding the particle in the left-hand half, i.e., the electron will be attracted toward the one of the two atoms which offers the lowest potential energy. One more feature of the polar bond can be demonstrated in connection with the above calculations. Pauling states that "the energy of an actual bond between unlike atoms is greater than (or equal to) the energy of a normal covalent bond between these atoms."" "normal covalent bond" means in this connection one which would exist between A and B if there was a negligible difference in electronegativity between the two kinds of atoms, a kind of average between the nonpolar bonds A-A and B-B. The extra stabilization is generally ascribed to the L'ionic resonance energy," hut it is feasible to demonstrate the existence of the former by means of the present box model without recourse to the concept of resonance. If two electrons, one in a box of length b and the other in a box of length c, are transferred to a common box of length 2a the total energy is increased by the amount
provided that the potential energy is the same in all three boxes and that the interelectronic repulsion may be neglected. If one of the two initial boxes is a t the potential V,, the other a t the potential 0, and the common-box potential is as shown in Figure 1, the energy increase will be
where E has the same meaning as in eqn. (2). Using eqns. (2) and (15) this could be transformed into
The two last terms in the brackets of eqns. (20) and (22) will make the energy increase negative for any reasonable choice of the relative magnitudes of the boxes, i.e., forb and c < 2a. This is in principle due to the delocalization of the electrons for moderate values PAULING,LINDS, "The Nature of the Chemical Bond" (3rd ed.), Cornell University Press, Ithacrt, N. Y., 1960, p. 80. Volume 49, Number 10, October 1972
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687
N-Acetylanthranilic Acid: A Highly Triboluminercent Material Triboluminemence. the emission of lieht when a crvstal is broken. has been known far mme time but remains an l though rnwly itmrga~irand orgmiv 1nnrerix1.i xrr rrlrorred rn exhil~itrt.is pmperty.' ol,wurr p l w m n r n ~ s wru The effevr hu. I w w rxplninrd hg exelrarim d rhr r d e r ilr try an elrrtric dischnrpe between the surfneb of the fmrtured crystal and subsequent fluorescence. I n the eouee of the laboratory preparation of 2-methylbenzisoxaeinone,'.s we have found that its hydrolysis product, N-acetylanthranilic acid, is highly triboluminescent and that this serves to make the experiment more intriguing than the normal synthetic preparation. In the experiment, anthranilic acid is converted t o Zmethylbenzisoxrtzinone by the action of acetic anhydride. This compound may be isolrtted (a good exercise in that the benzisoxszinane is readily hydrolyzed by stmaspheric moisturel) or converted directly t o N-acetylanthrmilic acid by a mild hydrolytic reaction.
Experimental Preparation of N-acetylanthranilic acid: Place 10 g (0.072 mole) of snthranilic acid in a round-bottom flask equipped with a reflux condenser. Add 30 ml (32 g, 0.32 mole) of acetic anhydride, bring the mixture slowly t o the reflux temperature and maintain heat for 15 min. Allow the solution t o cool and add 10 ml water through the condenser. Bring the mixture t o a soft boil once more and allow to cool slowly. Isolate the crystals of N-acetylanthranilic acid ( m . ~ 183-5'C) . by suction filtration in the hood and wash the ~ r o d u c with t a small amount of cold methanol. If the intermediate henzisoxazinone is isolated according t o Helmkamp and JohnsonPit may readily be converted to N-acetylanthranilic acid by dissolving in a hot mixture of 35 ml of acetic acid and 10 ml of water (assuming approximately 10 g of the benaisoxaainone) and allowing the solution to stand. Theseprocedures usually yield well-formed crystals but the material may be recrystallized from acetic acid/water mixtures. To demonstrate the property of triboluminescence, the crystals should be well-formed, and it is vital that they he completely free of solvent. The triboluminescence is best demonstrated by placing several crystals of the compound between two watch glasses and gently grinding. Thelight emitted is readily obeerved in a. darkened room.
and Company, 1968, p. 155. a BOGERT M. T., AND Smi,, H. A., J. Amer. Chem. Soe., 29, 517 (1907).
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