Absolutely Small - Michael D. Fayer [59]
The increased size of the orbitals is the reason that the energy increases as the quantum number n increases. The formula for the energy levels of the hydrogen atom has a negative sign in front of it, En = -RH/n2. We use a sign convention that a lower energy is a more negative energy. The hydrogen atom is composed of a proton and an electron attracted to each other through a Coulomb interaction, that is, an electrostatic interaction. Opposite charges attract. The proton is positively charged and the electron is negatively charged. When a proton and electron are infinitely far apart, they do not feel each other. There is no attraction because they are so far apart. The system has its zero of energy when the particles are separated at infinity. The electron and proton attraction increases as they get closer to each other. The energy of the system decreases, becoming increasingly negative. The 2s orbital has the electron further away from the proton on average than the 1s, and the 3s orbital has the electron still further away from the proton on average. This is clear from Figure 10.2. As the quantum number increases, the energy is a smaller negative number. For larger values of n, it takes less energy to separate the electron and proton, that is, to ionize the atom. Ionization is the process of pulling the electron out of an atom so that they are no longer bound together. For n = 1, it takes an energy of RH to ionize the atom. This is the amount of energy that needs to be put into the atom to overcome the binding of -RH. When n = 2, it only takes RH/4 to ionize a hydrogen atom. When n = 3, even less energy, RH/9, is needed to ionize the atom.
SPATIAL DISTRIBUTION OF s ORBITALS
To get a better feel for the spatial distribution of the probability of finding the electron in some position, it is useful to make two types of plots of the wavefunctions. One is just to plot the wavefunction as a function of distance from the nucleus. This type of plot is useful but somewhat misleading. The second type of plot is called a radial distribution function, which will be described shortly. Figure 10.3 is a plot of the wavefunction Ψ(r) as a function of the distance from the proton, which is at the center of the atom. This type of plot is the probability amplitude of finding the electron along a single line radially outward from the center. In Figure 10.2, r is along a horizontal line starting at the center of the shaded electron distribution and moving outward to the right. Figure 10.3 shows that the probability of finding the electron along a single line decreases rapidly, and is close to zero by a distance from the nucleus of 3 Å.
The problem with the type of plot shown in Figure 10.3 is it does not account for the three-dimensional nature of the atom. Looking at the 1s orbital in Figure 10.2, you can see that you can find the electron at some distance from the center by moving along a line to the right, but also along a line to the left, or up, or down. You can also move in any diagonal direction a distance r and have the same probability of finding the electron. Since the atom is three dimensional, you can also move in or out of the page and find the electron. If you want to know the probability of finding the electron a certain distance r from the proton, you need to sum all of these different radial directions.
FIGURE 10.3. A plot of the 1s wavefunction Ψ(r) as a function of r, the distance from the proton. Ψ(r) is proportional to the probability of finding the electron along a line radially outward from the center of the atom. The distance r is in Å, which is