Absolutely Small - Michael D. Fayer [61]
Figures 10.5 and 10.6 show plots of the wavefunctions (top panels) and the radial distribution functions (bottom panels) for the 2s and 3s orbitals. The wavefunction of the 2s orbital has a node, that is, a place where the wavefunction is zero. Nodes were discussed in connection with the particle in a box wavefunctions (see Figure 8.4). At a node the probability of finding a particle, in this case the electron, is zero. The 2s wavefunction begins positive, crosses zero at the node located at twice the Bohr radius, 2a0, and then is nega tive. The wavefunction then decays to zero. By 8 Å the value of the wavefunction is very small. As we have discussed in detail, the wavefunctions are probability amplitude waves. Like other waves, they can be positive or negative. The bottom panel of Figure 10.5 displays the 2s radial distribution function. This is the probability of finding the electron a distance r from the nucleus. Probabilities are always positive because they are the square of the wavefunction, which is always positive. A wave can be positive or negative, but it makes sense that a probability is always a positive number or zero. The radial distribution function shows that most of the probability is between about 2 and 4 Å, which also can be seen in Figure 10.2 but not as quantitatively. The peak of the probability is located at ∼2.8 Å.
FIGURE 10.5. A plot of the 2s hydrogen atom wavefunction (top panel) and the radial distribution function (bottom panel), as functions of r, the distance from the proton. The wavefunction begins positive, goes through a node at slightly more than 1 Å (2a0), and then decays to zero. The radial distribution function shows that the maximum probability of finding the electron peaks at about 2.8 Å, with most of the probability between 2 and 4 Å (see Figure 10.2). The distance r is in Å, which is 10-10m.
FIGURE 10.6. A plot of the 3s hydrogen atom wavefunction (top panel) and the radial distribution function (bottom panel), as functions of r, the distance from the proton. The wavefunction begins positive, goes through a node, becomes negative, goes through a second node, and becomes positive again. It then decays to zero. The radial distribution function shows that the maximum probability of finding the electron peaks at about 7 Å, with most of the probability between 5 and 11 Å (see Figure 10.2). The distance r is in Å, which is 10-10m.
In Figure 10.6, it can be seen that the 3s wavefunction has two nodes, that is, the wavefunction crosses zero twice. The hydrogen atom wavefunctions have this in common with the particle in a box wavefunctions (see Figure 8.4). For n = 1, there is no node. For n = 2, there is one node. For n = 3, there are two nodes. The number of nodes for the s orbitals is n-1. The 3s wavefunction begins positive, goes negative, and then becomes positive again. It finally de cays to zero, and is very small by about 16 Å. The 3s radial distribution function shows that most of the probability is relatively far from the nucleus. The peak probability is at ∼7 Å. Most of the probability is between 5 and 11 Å. The three radial distribution functions shown in Figures 10.4, 10.5, and 10.6 are quantitative plots of the information shown schematically in Figure 10.2. As the principal quantum number, n, gets larger, the s orbitals become larger and have more nodes.
THE SHAPES OF THE p ORBITALS
For the 2s orbital, n = 2, l = 0, and m = 0. However, for n = 2, l can also equal 1 with the associated three values of m, m = 1, 0, -1. The three different m values give rise to the three different 2p orbitals. These are shown in the energy level diagram, Figure 10.1. The three different 2p orbitals are represented schematically in Figure 10.7. As mentioned above, because of their shapes, the 2p orbitals are usually referred to as 2pz, 2py, and 2px. Each orbital has two lobes, a positive lobe and a negative lobe. Which lobe is assigned positive and negative is arbitrary, but the sign must change because there is an angular nodal plane. The