CoreChem:5.3 Electron Waves in the Hydrogen Atom

From ChemEd Collaborative

5.3 ELECTRON WAVES IN THE HYDROGEN ATOM

An electron in an atom differs in two ways from the hypothetical electron in a box that we have been discussing. First, in an atom the electron occupies all three dimensions of ordinary space. This permits the shapes of the electron waves to be more complicated. Second, the electron is not confined in an atom by the solid walls of a box. Instead, the electrostatic force of attraction between the positive nucleus and the negative electron prevents the latter from escaping.

In 1926 Erwin Schrodinger (1887 to 1961) devised a mathematical procedure by which the electron waves and energies for this more complicated situation could be obtained. A solution of the Schrodinger wave equation is beyond the scope of a general chemistry text. However, a great many chemical phenomena can be better understood if one is familiar with Schrodinger’s results, and we shall consider them in some detail.

The distribution of electron density predicted by the solution of Schrodinger’s equation for a hydrogen atom which has the minimum possible quantity of energy is illustrated in Fig. 5.6. A number of general characteristics of the behavior of electrons in atoms and molecules may be observed from this figure.

First of all, the hydrogen atom does not have a well-defined boundary. The number of dots per unit area is greatest near the nucleus of the atom at the center of the diagram (where the two axes cross). Electron density decreases as distance from the nucleus increases, but there are a few dots at distances as great as 200 pm (2.00 Å) from the center. Thus as one gets closer and closer to the nucleus of an atom, electron density builds up slowly and steadily from a very small value to a large one. Another way of stating the same thing is to say that the electron cloud becomes more dense as the center of the atom is approached.

A second characteristic evident from Fig. 5.6 is the shape of the electron cloud. In this two-dimensional diagram it appears to be approximately circular; in three dimensions it would be spherical. This can be illustrated more readily by drawing a circle (or in three dimensions, a sphere) which contains a large percentage (say 75 or 90 percent) of the dots, as has been done in the figure. Since such a sphere or circle encloses most of (but not all) the electron density, it is about as close as one can come to drawing a boundary which encloses the atom. Boundary-surface diagrams in two and sometimes three dimensions are easier to draw quickly than are dot-density diagrams. Therefore chemists use them a great deal.

Orbitals

A third characteristic of the diagram in Fig. 5.6 is that it has been assigned an identifying label, namely, 1s. This enables us to distinguish it from other wave patterns the electron could possibly adopt if it moved about the nucleus with a higher energy. Each of these three-dimensional wave patterns is different in shape, size, or orientation from all the others and is called an orbital. The word orbital is used in order to make a distinction between these wave patterns and the circular or elliptical orbits of the Bohr picture shown previously in Fig. 5.2. The electron density distributions for the 14 simplest orbitals of the hydrogen atom are shown in Figs. 5.7 and 5.8, both as dot-density diagrams and as boundary-surface diagrams. Note the unique label for each orbital.

At ordinary temperatures, the electron in a hydrogen atom is almost invariably found to have the lowest energy available to it. That is, the electron occupies the 1s orbital, and the electron cloud looks like the dot-density diagram in Fig. 5.6. At a very high temperature, though, some collisions between the atoms are sufficiently hard to provide one of the electrons with enough energy so that it can occupy one of the other orbitals, say a 2s orbital, but this is unusual. Nevertheless a knowledge of these higher energy orbitals is necessary since electron clouds having the same shapes as for hydrogen are found to apply to all the other atoms in the periodic table as well. In the case of a particle in a one-dimensional box, the energy was determined by a positive whole number n. Much the same situation prevails in the case of the hydrogen atom. An integer called the principal quantum number, also designated by the symbol n, is used to label each orbital. The larger the value of n, the greater the energy of the electron and the larger the average distance of the electron cloud from the nucleus. In Figs. 5.7 and 5.8, n = 1 for the 1s orbital, n= 2 for the four orbitals 2s, 2p_x, 2p_y, and 2p_z, while the remaining nine orbitals all correspond to n = 3.

Because a greater number of different shapes is available in the case of three-dimensional, as opposed to one-dimensional, waves, two other labels are used in addition to n. The first consists of one of the lowercase letters s, p, d, or f. These tell us about the overall shapes of the orbitals.¹ Thus all s orbitals, including the 1s, 2s and 3s shown in Figs. 5.7 and 5.8, are spherical. All p orbitals, such as 2p_x, 2p_y, 2p_z and 3p_x, 3p_y, 3p_z, have a dumbbell shape. The d orbitals have more complex shapes than the p orbitals, while f orbitals (a type not shown in the figures) are even more complicated. The third kind of label used to describe orbitals is a subscript. These subscripts distinguish between orbitals which are basically the same shape but differ in their orientation in space. In the case of p orbitals there are always three orientations possible. A p orbital which extends along the x axis is labeled a p_x orbital (for example, 2p_x, 3p_x etc. ). A p orbital along the y axis is labeled p_y and one along the z axis is a p_x orbital. In the case of the d orbitals these subscripts are more difficult to follow. You can puzzle them out from Figs. 5.7 and 5.8, if you like, but we will not use them a great deal in the remainder of this book. You should, however, be aware that there are five possible orientations for d orbitals. It is also important to know that there are seven different orientations for f orbitals, since the number of orbitals of each type (s, p, d, etc.) is important in determining the shell structure of the atom.

Another important point is that only a limited number of orbital shapes is possible for each value of n. If n = 1, then only the spherical 1s orbital is possible. When n is increased to 2, though, two orbital types (2s and 2p) become possible, while if n equals 3, then three orbital types occur. The same pattern extends to n = 4 where four orbital types, namely, 4s, 4p, 4dand 4f, are found.

¹ The letters s, p, d and f originate from the words sharp, principal, diffuse and fundamental which were used to describe certain features of spectra before wave mechanics was developed. They later became identified with orbital shapes.