Resizing Matter: Part 2

In Part 1, we considered how resizing matter impacts the relative strength of ionic bonds in simple salts. This was an easy case because we assumed that (1) charge magnitudes remain unchanged, (2) Coulomb’s Law holds, (3) the distance between ion centers is the only variable parameter that affects the bond, and (4) masses of atoms scale proportionally but don’t impact the ionic bonds.

Resizing a living creature (e.g., Engorgio-ed spiders or Ant-Man) is much more complicated. Living systems are made up of a plethora of organic molecules containing covalent bonds. Furthermore, these molecules interact with each other through intermolecular forces.

Let’s get fundamental. What would happen if you magically or technologically resized an atom?

My General Chemistry students would argue that the size of an atom is related to the effective nuclear charge – how strongly the nucleus of an atom pulls on the outermost (valence) electrons. In this view, electrostatics (via Coulomb’s Law) explains why the electron is attracted to the nucleus – we call this the potential energy contribution to the overall nucleus-electron system. But this doesn’t explain why the electron doesn’t collapse into the nucleus. Since electrons are governed by quantum mechanics, the kinetic energy of the electron must also be taken into account. The mass and speed of electrons could affect the system if the atom is resized.

Assuming the average distance of an electron to the nucleus doubled, the (electrostatic) potential energy would decrease two-fold (although the attractive force decreases four-fold from Coulomb’s Law). The increase in mass of the nucleus should not impact the system because the electrons move much faster than the nucleus. To electrons, the sluggish nuclei resemble stationary objects (P-Chem students should recognize the Born-Oppenheimer approximation). If the mass of an electron doubles, but its average speed does not change, then (as a simple approximation) the kinetic energy also doubles. Thus, the potential and kinetic energies balance each other out.

If you doubled the radius of a sphere, it’s volume increases eightfold. Assuming a spherical atom, if you wanted to keep the density of the atom equivalent, you would also need particle masses to increase eightfold, and not merely double. But maybe it’s okay for the density of the atom to decrease as the size increases. After all, atoms don’t merge into each other (they do squish into each other a bit, but not a whole lot) because of the Pauli Exclusion Principle. (P-Chem does come in useful!) The Pauli principle isn’t affected by changes in mass of the particles.

Before we get to covalent bonds (likely Part 3), let’s consider interatomic interactions in an atomic gas. The noble gases are a good example here. Imagine a sample of helium where all atoms doubled in size and mass as described above.

For the most part, helium atoms bounce off each other when they collide. However, there is a very slight attraction between atoms due to the polarizability of each atom. The much larger xenon atom, for example, has a higher polarizability and so xenon-xenon attractions are stronger than helium-helium attractions. In xenon, although the nucleus is much more positive, the outermost electrons are very effectively shielded by shells of inner electrons. Overall xenon’s effective nuclear charge is smaller, and hence the atom sizes are larger.

But would polarizability change if we simply resized helium? Okay, at this point I cracked open a few of my P-Chem books and started reading. Instead of inundating you with equations, let’s simplify it to the following: Atom polarizability is a measure of how much the atom responds to an applied electric field. It is roughly proportional to the volume of the atom. There are a bunch of niggly details, but let’s keep it simple for now. Since volume increases eight-fold with a doubling of atomic radius, polarizability is expected to increase even if the mass of the electron doubled. This means that the interatomic attractions increase with an increase in size. This will be important when we consider a more complex system consisting of a collection of molecules interacting with each other in, say, a biological cell. The intermolecular forces should increase with an increase in size.

Given the density (pun intended) of these ideas, it seems prudent to stop for now. We’ll continue this analysis in Part 3.