Resizing Matter: Part 3

After considering two simpler situations, we are now ready to tackle a more difficult problem. How would the strength of covalent bonds change if we magically resized matter? We care about this because living creatures are predominantly made up of molecules containing covalent bonds. So if you were engorging a spider to scare Ron Weasley, or you have obtained the technology of Ant-Man’s suit, you might care to know the side-effects – which may turn out not to be secondary but instead devastatingly problematic.

The challenge here is that the covalent bond is not as easy to describe because it is inherently quantum mechanical. Back in Part 1, we used a simple model to describe ionic bonds relying on the classical Coulomb’s Law, which works satisfactorily for the most part. Then in Part 2, we discussed weak attractive inter-particle forces using an intuitive (also classical) polarizability model. There is no classical analog to describe covalent bonding, and the intuitive model found in many general chemistry textbooks is misleading. Here’s one from a former textbook I’ve used before.

I show students this figure because it is familiar to them if they have taken chemistry in high school. The main idea: By conveniently locating the negatively charged electrons between the positively charged nuclei, this shared pair of electrons attracts both atomic centers and thereby holds the molecule together. A wave of the hand then concludes that this attractive force outweighs the repulsive forces. The figure implies these repulsive forces are electrostatic, and while that might be true for H₂, it’s certainly not true for anything larger when Pauli repulsion kicks in. I tell the students that the figure is partly true but partly misleading.

Why is it partly true: The potential energy term due to electrostatic attraction between the electrons and the nuclei does give rise to the attractive force in the covalent bond. And it is sufficiently strong to outweigh the repulsive forces at the appropriate equilibrium bond distance. Why it is misleading: A chemical bond is defined with reference to the infinitely separated atoms. As the atoms approach each other to form a bond, the energy-stabilizing contribution due to the potential energy is actually reduced. Egad! Shouldn’t that cause the bond not to form? It turns out that the repulsive contribution of the kinetic energy is reduced even more. Hence, there is a net stabilization of energy due to forming the covalent bond. (Quantum Mechanics folks: We’ve assumed that the covalent bond is represented by the Heitler-London wavefunction.)

All this is a bit too complicated for the general chemistry student, and so in recent years I’ve moved away from this approach of introducing the topic of chemical bonding. My most recent approach is to use the General Bond Energy Curve early on, and briefly discuss different attractive and repulsive contributions. Then we get into the details of ionic bonds, covalent bonds, and intermolecular forces. The equilibrium distances and equilibrium bond energies change but the overall curve still has a steep repulsive part on the left, an optimum well in the middle, and a shallower attractive part on the right that asymptotically approaches zero energy – the reference state for infinitely separated particles, be they atoms, ions or molecules.

Now back to resizing matter. How would the strength of the covalent bond change if atoms magically or technologically changed in size? Using the same assumptions laid out in Part 2, we would expect that the outermost electrons are held more weakly by the nucleus and therefore the energy-stabilizing attractive contribution due to potential energy should be weakened. We see this if we compared the covalent bonds of the Cl₂ versus I₂ molecules. The larger iodine atoms lead to a longer and weaker I–I bond compared to the smaller chlorine atoms which have a shorter and stronger Cl–Cl bond. Br₂ lies in between, as befits the intermediate atomic size of bromine. (F₂ is anomalous because of the Pauli Principle!)

How would the kinetic energy change? That’s a bit more difficult to picture. The simple notion of using 0.5 mv² likely fails us here. That’s because to some extent the electrons are “localized” in the mid-point region, but in a different sense the kinetic energy drops because they have more “space” to move around in per the overlap of the two orbitals that contribute to the bond. Technically the decrease in kinetic energy can be approximated by the square of the gradient of the molecule’s wavefunction compared to the separated atoms. Does the decrease in kinetic energy decrease proportionally to the decrease in potential energy?

Honestly, I’m not sure what the answer should be. (Sorry to disappoint you!) If there is a relationship between the two, it’s likely not to be linear. To solve this problem, I would actually need to do is set up a quantum mechanical calculation involving two “mutant” atoms with approximate properties I would expect for a magically resized atom. The answers from this calculation would still be approximate, but I would have a better idea how both equilibrium bond energies change as a function of the resized atoms. I’m too lazy to do the calculation at the moment so I will make a guess. (I imagine this is how my students feel sometimes when they encounter my exam questions.) I will guess that bigger atoms lead to longer bonds with weaker attractive potential energy, and although the repulsive kinetic energy term is also lower in magnitude, the fact that a chemical bond exists suggests that the potential energy term is overall larger in magnitude than the kinetic energy term, and therefore more impacted as the equilibrium bond distances change.

While I’m dissatisfied with my answer, and assuming I’m in the Advanced Arithmancy course, I submit it to the Hogwarts professor grading my exam. Hopefully it will be graded soon and an answer key provided. That way I don’t have to actually do the hard work and be willing to accept a not-so-good grade. I suppose sometimes school is about trade-offs, and I’m reminded in a tiny way what it’s like to be in the shoes of my students.