Absolutely Small - Michael D. Fayer [76]
The Bond Length Is the Distance That Gives the Lowest Energy
At the distance r0, the energy is the minimum. r0 is the separation of the H atoms that is the most stable (lowest energy). This distance is called the bond length. It is the separation between the two protons in a stable hydrogen molecule. The difference between the bottom of the “potential energy well” and the zero of energy is the dissociation energy. The dissociation energy is the amount of energy that would have to be put into a hydrogen molecule to break the chemical bond, which would produce two hydrogen atoms. The potential energy well for the hydrogen molecule is equivalent to a hole in the ground that a ball rolls into. The top of the hole is the zero of energy. The ball falls to the bottom of the hole to minimize the gravitational potential. Gravity pulls the ball down. To lift the ball out of the gravitational well requires energy, as the gravitational pull on the ball must be overcome. The deeper the hole, the more energy it will take to lift the ball out of the hole. With molecules, the deeper the potential energy well, the more energy it will take to get out of the well, that is, to break the chemical bond.
The distance scale along the r axis of Figure 12.1 is not shown. But it is interesting to discuss two distances. At what distance do the hydrogen atoms first really begin to feel each other? Figure 10.3 shows that the 1s orbital probability amplitude wave for a hydrogen atom becomes very small at a distance from the nucleus of approxi mately 3 Å (3 × 10-10 m). So one might expect that when two hydrogen atoms are a little closer than 6 Å they would start to interact. In Figure 12.1, the point where the potential energy curve (solid curve) just stops touching the zero of energy line (dashed line) is approximately 6 Å. So the atoms begin to feel each other when the atomic wavefunctions just begin to overlap significantly. The point r0 is the location of the minimum of the potential energy curve. It is the length of the bond. Experiments and calculations have deter mined this distance to be 0.74 Å. If the atoms are further apart or closer together than this distance, the energy is higher. The potential energy curve shown in Figure 12.1 is from an actual quantum mechanical calculation. It is a relatively low-level calculation that can be done completely with pencil and paper; no computers are necessary. This approximate calculation gives r0 = 0.80 Å, so it is a little off. If you want to see the monumental amount of math that goes into even this relatively simple calculation, see Michael D. Fayer, Elements of Quantum Mechanics, Chapter 17 (New York: Oxford University Press, 2001). Much more complicated quantum theoretical calculations of the H2 molecule can produce all of the properties of the hydrogen molecule with accuracy better than can be obtained through experimental measurements. Such accurate calculations are possible because the hydrogen molecule is so simple. For large molecules, experiments still beat calculations.
FORMING BONDING MOLECULAR ORBITALS
Figure 12.1 shows that a chemical bond will be formed between two hydrogen atoms to yield the H2 molecule, but it doesn’t show why. As mentioned in Chapter 11, a covalent bond involves the sharing of electrons by atoms. When a molecule is formed, the atomic orbitals combine to form molecular orbitals. For the hydrogen molecule, we start with two hydrogen atoms, Ha and Hb. Each has a single electron in an atomic 1s orbital. We will call these orbitals 1sa and 1sb. These two atomic orbitals are represented in the top portion of Figure 12.2 as circles. This is a simple schematic of the delocalized electron probability amplitude wave shown in Figures 10.2 through 10.4.