Absolutely Small - Michael D. Fayer [58]
As mentioned above, each of the orbitals has a different shape. It is common to rename the orbital with an indication of its shape. For example, the three different 2p orbitals, rather than being called 2p1, 2p0, and 2p-1, are usually called 2px, 2pz, and 2py. The relation between the subscript and the shape will become clear when the shapes are presented.
HYDROGEN ATOM ENERGY LEVELS
Figure 10.1 shows an energy level diagram for the hydrogen atom. Levels are shown for n = 1 to 5. The spacings between the levels are not scaled properly for clarity of presentation, but as shown, the spacing between levels gets smaller as n increases. Also as n in-creases, the number of different states (orbitals) associated with the particular n increases. Hydrogen is a special case because it only has one electron. For hydrogen, all orbitals with the same n have the same energy. As discussed in the next chapter, for atoms with more than one electron, for a given n, orbitals with different l values have different energies.
FIGURE 10.1. Hydrogen energy level diagram. The spacings between the levels are not to scale. The first five energy levels are shown. The energy only depends on the principal quantum number, n. The orbitals and the number of each type are also shown. For n = 4, there is a single s orbital, three different p orbitals, five different d orbitals, and seven different f orbitals. The diagram would continue with the n = 6 level. The different levels are sometimes referred to as shells.
HYDROGEN ATOM s ORBITALS
Although the hydrogen energies only depend on the principal quantum number n, l and m still play an important role. These quantum numbers determine the shapes of the orbitals, and they determine other aspects of the hydrogen atom’s properties. For example, the m quantum number is called the magnetic quantum number. The three 2p orbitals, 2p1, 2p0, and 2p-1, differ by their m quantum number. When the hydrogen atom is put in a magnetic field, the energies of these three orbitals are no longer the same.
From the energy levels calculated with the Schrödinger equation (see Figure 10.1) it is clear how the empirical diagram in Figure 9.3 arises. The optical transitions seen in the line spectrum of the hydrogen atom and described by the Rydberg formula are transitions between the energy levels of the hydrogen atom, energy levels that are calculated using quantum theory with no adjustable parameters.
As mentioned above, the n, l, and m quantum numbers all go into determining the shapes of the wavefunctions. The s orbitals have l = 0. l = 0 means that the electron has no angular momentum in its motion relative to the nucleus of the atom. All directions look the same, so s orbitals are spherically symmetric three-dimensional probability amplitude waves. Figure 10.2 shows schematic representations of the 1s, 2s, and 3s orbitals (probability amplitude waves). The darker shading indicates a greater probability of finding the electron that distance from the center. The distances at which the probabilities have maxima are shown by the solid circles. The centers of the white regions in the 2s and 3s orbitals (dotted circles) are nodes, that is, regions where the probability of finding the electron goes to zero. In going from the 1s to the 2s to the 3s, the size of the orbital becomes much larger. The electron has a greater probability of being found further away from the nucleus as the n quantum number increases.
FIGURE 10.2. The 1s, 2s, and 3s orbitals shown