edited by
J. DUDLEY HERRON Purdue University West Lalayette, Indiana 47907
How Does the Electron Cross the Node? Prepared from responses written by Russell H. Johnsen T h e Florida State University and Winston D. Lloyd T h e University of Texas a t El Paso As introductory chemistry courses place more emphasis on modern theories about the structure of atoms, students are frequently confused about the meaning of concepts a t the heart of those theories. The following question illustrates the problem:
Figure 2. The graph of y = s i d , a periadic function analogous to a wavetunclion.
Atomic orbitals are frequently interpreted as the probability distribution fur an electron at a given energy level. As such,all orbitals other than the 1s show a surface representing azeroprabability separating other regions of "on-zero probability. Far example, a 2p orbital has two lobes of "on-zero probability separated by a planar surface of zero probability as illustrated in Figure 1. Figure 3. The graph of the equation for the 2pz orbital.
Figure 1. Probabilitydistribution for a 2porbitai.
minimum value of -1 and a maximum value of +l.In a similar fashion the p-orbitals (one electron wavefunctions) are zero-valued in a plane, with a positive lobe above the plane and a negative lobe below the plane. For example, the Zp, oneelectron orbital ('P2pr)has the form '/4(27r)-'/2
Thoughtful students point out that it is illogical to assume that there is some finite probability of finding an electron in the left lobe and some finite prohability of finding that same electron in the right lobe but no probability of finding it between. How does it get from one side to the other? Both respondents to this question indicate that the confusion arises from an assumption that the electron is a particle, when the model that produces the orbital diagram assumes that the electron behaves as a wave. In the following comments, Johnson shows how a discussion of wave behavior might he used to clear up the confusion. The Electron as a Wave
Brought up in a world in which particle phenomena and their Newtonian interpretations are common place, the student finds it difficult to make the transition required for an understandine- of electron behavior in terms of wave mechanics. In this particular instance the student is making the fundamental error of asking a "particle question" about a model which views the problem in terms of the behavior of waves. The wavefunction, or orbital, and its square (which is the probahility distribution) arrives a t the picture of the twolobed 2p orbital utilizing mathematical functions which are zero valued a t certain values of the independent variable. A simple example of such a function is y = sin 0 for which y is zero valued at values of 0 corresponding to integral multiples of a radians, as shown in Figure 2. Note that the value of the function oscillates between a
( z / a p re-Z1/2a COSB
When r is the distance from nucleus, a is the atomic unit of distance, and B is the angular coordinate. Analysis of this function shows that for y = 0 (the xz plane) the function graphs as shown in Figure 3; i.e., the xy plane is a nodal plane in which the value of the function (and its square, the probability distribution) is zero. A two dimensional analog can be found by examining the behavior of vibrating strings. In such an analysis, it is clear that the question of the electron "crossing" the node does not arise. The picture simply represents the value of the probability distribution function a t a given angle and distance from the nucleus. Lloyd takes a slightly different approach in his response and explains why a description of the electron as a charge cloud has certain advantages. The Electron Cloud
The apparent paradox presented in this question is the result of the assumption that the electron is a discrete particle moving in such a way that the time spent a t a given position is described by a probability function. In fact, such an interpretation violates the Heisenberg Uncertainty Principle. Uncertaintv concernine the location of the electron is about as great as the size of thYe atom. What is shown in texts as the s h a ~ of e atomic orbitals is a Dart of a orobabilitv function. The prohability function describes the probability that an experiment desiened to locate the electron will eive a oarticular result. I t is"important to note, however, thacthis Gobability Volume 57, Number 9, September 1980 1 651
function is not time dependent; i t applies to all locations a t the same time. Consider what happens when we shoot electrons, one a t a time. a t a barrier which has two closelv holes in it. If . spaced . we then let the electrons impinge on a phosphor screen, there will he a tinv flash uf light each time it is hit. A time exposurr of the screen will give diffraction pattern. This diffraction pattern (probability distribution) will agree with the result expected for two spherical waves which have their origins a t the two holes. If you contrast this result with the result expected for discrete particles that travel in a straight line through one or the other of the two holes, you can see that the
a
~h , ~of , the tom," vol. 11, B ~H.A.~and ~~ ~I,.,~ " t ~~world Basic Books Inc., New York, 1966, p. 1225. Morwick, J. J., J. CHEM. EDUC., 56,262 (1979).
652 1 Journal of Chemical Mucation
electron is behaving only as a wave, not as a particle, as it passes the harrier.' It is interesting to consider the probability distribution at distances far from the nucleus. We know that the electron cannot puss~hl)be as much as one cent:merer from the nucleus, but there is a finite probability that an experiment designed to detect electrons will give positive result for that distance (the probability distribution function is not zero). Clearly the result of such an experiment should not be interpreted in terms of the location of a discrete particle. To get around this difficulty we frequently describe the electron in an atom as an electron cloud.2 This description is more intellectually satisfying than a particle description. It focuses attention on the wave description of the electron and ignores size or location which have almost no meaning in this context.
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