1.7     Bond Strength

In Figure 1.42, the energy of the electrons in the two atomic orbitals is greater than their energy in the bonding molecular orbital Φ_B. As noted earlier, lower energy means greater stability, but just how much energy is the stabilization apparent in Figure 1.42 worth? The difference between the energies of the two electrons in Φ_B and the energies of the two separate electrons in atomic 1s orbitals is 104 kcal/mol (435 kJ/mol). This amount of energy is very large, and consequently the H₂ molecule is held together (bound) by a substantial amount of energy. This same amount of energy is released into the environment if H₂ molecules are formed from separated hydrogen atoms. It would require the application of exactly this amount of energy to generate two hydrogen atoms from a molecule of H₂. The 104 kcal/mol represents the stabilization of two electrons in Φ_B (Fig. 1.44).

Actually, if we could measure it, the temperature would rise ever so slightly as two hydrogen atoms combined to make a hydrogen molecule because 104 kcal/mol would be released as heat energy into the vessel. The environment in the vessel would warm up as the process proceeded. A reaction that liberates heat, one where the products are more stable than the starting materials, is called an exothermic reaction. The opposite situation, in which the products are less stable than the starting materials, requires the application of heat and is called an endothermic reaction.

The enthalpy change (ΔH°) of any chemical reaction is estimated as the difference between the total bond energy of the products and the total bond energy of the reactants. The value of ΔH° for a reaction between hydrogen atoms to give H₂ is −104 kcal/mol (−435 kJ/mol).

The sometimes troublesome sign convention shows ΔH° for an exothermic reaction as a negative quantity. For an endothermic reaction, ΔH° is positive. For example,

TABLE 1.10 Some Average Bond Dissociation Energies

Table 1.10 gives averages over a range of compounds. A few compounds may lie outside these values. In later chapters, more precise values of specific bonds will appear.

Let’s consider the simplest molecule, H₂⁺, which is H₂ with only one electron and a positive charge. Even this molecule is bound quite strongly.¹⁰ The small amount of information already in hand—a molecular orbital picture of H₂ and the bond strength of the H―H bond (104 kcal/mol)—enables us to construct a picture of this exotic molecule and to estimate its bond strength. We can imagine making H₂⁺ by allowing a hydrogen atom, H·, to combine with H⁺, a bare proton. All the work necessary to create an orbital interaction diagram for this reaction was done in the construction of Figure 1.41. We are still looking at the combination of a pair of hydrogen 1s atomic orbitals, so the building of a diagram for H₂⁺ produces precisely the same orbital diagram, reproduced in Figure 1.47. The only difference comes when we put in the electrons. Instead of having two electrons, as does H₂, with one coming from each hydrogen atom, H₂⁺ has only one electron. Naturally, it goes into the lower-energy, bonding molecular orbital, Φ_B. The electron spin is shown down, but it could equally well be shown up. There is no difference in energy between the two spins, and we have no way of knowing which way a single spin is oriented.

What might we guess about the bond energy of this molecule H₂⁺? If two electrons in Φ_B result in a bond energy of 104 kcal/mol, it seems reasonable to guess first that stabilization of a single electron would be worth half the amount, or about 52 kcal/mol (218 kJ/mol). That is, when an H· atom and an H⁺ ion combine to form the species H₂⁺, we are estimating that 52 kcal/mol of heat is given off. And we are amazingly close to being correct! It has been experimentally determined that the H₂⁺ molecule is bound by 64 kcal/mol (Fig. 1.47), which means that the system H₂⁺ is 64 kcal/mol (268 kJ/mol) more stable than the system of a separated H· and H⁺.

It may seem somewhat counterintuitive that a low-energy (strong) bond is associated with a large number for the BDE. The lower the energy of a bond, the higher the number representing bond strength! The low-energy, strong bond in H₂ has a BDE of 104 kcal/mol, for example, whereas the higher-energy, weaker bond in H₂⁺ has a BDE of only 64 kcal/mol. A strong bond means a low-energy, stable species. The diagram in Figure 1.44 should help keep this point straight. A strong bond means a more stable bond and it means a lower position on an energy diagram.

Remember the sign convention: 2 H· → H―H, ΔH° = −104 kcal/mol. The minus sign tells us that this reaction is exothermic as read from left to right. An exothermic reaction will have products that are more stable than the starting materials. The products will be lower than the reactants on the energy diagram.

The antibonding molecular orbital (Φ_A) has been empty in all of the examples we have considered. Why the emphasis on Φ_A? What is an empty orbital, anyway? The easiest, nonmathematical way to think of Φ_A in H₂ is as the place the next electron would go. If the bonding molecular orbital Φ_B is filled, as it is in H₂, for example (Fig. 1.44), another electron cannot occupy Φ_B and must go instead into the antibonding orbital, Φ_A. This would create H₂⁻ (H₂ + e⁻ → H₂⁻).

Dihelium, He₂, a strictly hypothetical species, is an example of a molecule in which electrons must occupy antibonding orbitals. Each He, like each H in H₂, brings only a 1s orbital to the mixing. Just as in the molecules H₂ and H₂⁺, the molecular orbitals for He₂ are created by the combination of two 1s atomic orbitals (Fig. 1.48). Each helium atom brings two electrons to the molecule, though, so the electronic occupancy of the orbitals is different from that for H₂ or H₂⁺. The bonding orbital (Φ_B) can hold only two electrons (Pauli principle) so the next two must occupy Φ_A. The two electrons in this antibonding orbital are destabilizing to the molecule, just as the two electrons in Φ_B are stabilizing. As a result, no net bonding occurs in the hypothetical molecule He₂. The stabilization afforded by the two electrons in the bonding molecular orbital is canceled by the destabilization resulting from the two electrons in the antibonding molecular orbital. Molecular helium He₂ is, in fact, unknown.

 

⁹To describe a bond as strong or weak is an arbitrary function of human experience. A bond that requires 100 kcal/mol (418 kJ/mol) to break is “strong” in a world where room temperature supplies much less thermal energy. It would not be strong on the sunlit side of Mercury (where it’s hot: the average temperature is about 377°C, or 710°F). Similarly, if you were a life form that evolved on Pluto (where it’s cold: the average temperature is about −220°C, or −361°F), you would regard as strong all sorts of “weak” (on Earth) interactions, and your study of chemistry would be very different indeed. One area of research in organic chemistry focuses on extremely unstable molecules held together by weak bonds. The idea is that by understanding extreme forms of weak bonding, we can learn more about the forces that hold together more conventional molecules. Chemists who work in this area deliberately devise conditions under which species that are normally most unstable can be isolated. They create very low temperature “worlds” in which other reactive (predatory) molecules are absent. In such a world, exotic species may be stable, as they are insulated from both the ravages of heat (thermodynamic stability) and the predations of other molecules (kinetic stability).

¹⁰Here is the answer to Problem 1.23. The simplest molecule is hydrogen (H₂) minus one electron. Another electron cannot be removed to give something even simpler because H₂²⁺ is not a molecule―there are no electrons to bind the two nuclei.