## How big is an H atom?

Atomic "size" is not well defined. Electrons determine size, but electrons are spread nonuniformly over ill-defined volumes around an atomic nucleus. While the electron's wavefunction in any given state (i.e., any given set of n, l, and m quantum numbers) shows how that electron is spread, there are several possible ways to turn the wavefunction into a measure of "size."

One of the most common such measures, and one of the most useful and straightforward to interpret, is the average radial distance of the electron from the nucleus. Quantum mechanics beyond the scope of Chem 6 shows that this average distance is given by the following simple expression:

(This is Eq. [15.22] on page 557 in the text, written in a somewhat more compact way and with Z = 1 because we are only concerned with hydrogen at the moment. The "<r>" notation is a common one for averages. It is also much more web-friendly!) In this equation, a0 is the Bohr radius, 0.529 Å, and n and l are the principal and angular momentum quantum numbers, respectively.

The spread in location itself is conveniently given by the radial probability distribution, discussed on pages 553 through 555 in the text. In particular, Figure 15.23(c) graphs these functions for the three lowest energy spherical wavefunctions, 1s, 2s, and 3s. It is instructive to plot these curves along with the value for <r>. While we're at it, let's throw in a plot of the potential energy V(r) (which is just Coulomb's law, shown on page 538) and indicate the total energy of each of these states (which is just En in equation [15.19a] on page 550).

This is a lot of information! Let's walk through such a graph for the 1s state, shown below.

Here we have an x axis which is just the radial electron-proton distance, expressed as multiples of the Bohr radius. A green line locates <r> for this state. The y axis is shown in energy units appropriate for the potential energy V(r) and the total energy E1. The total energy line is drawn over the region of r that corresponds to classical motion: an electron with total energy E1 would oscillate along a line that extends 2a0 in length on each side of the proton. The radial probability distribution is shown in black, and it is scaled in the y direction arbitrarily. Note the gray area on the right of r = 2a0. In this area, the potential energy is greater than the total energy, which means, in classical mechanics, that the kinetic energy would have to be negative. This is impossible in classical mechanics, but it IS possible in quantum mechanics! Note that the radial probability distribution is significant over a large part of the gray area. This means that the electron could be found in regions of space that are forbidden in classical mechanics. This is an example of quantum mechanical tunneling.

Note, however, that the average radius (<r> = 1.5a0 for the 1s state) is in the region of space allowed by classical mechanics.

Now let's draw the same kind of curve for the 2s state. (Note that both the energy and the distance scales have changed. The 2s atom is quite a bit bigger and quite a bit higher in energy than 1s!)

Again, we see a tunneling region, and <r> is within the classical region. We can also see the radial node, just like in figure 15.23. It is the radius (r = 2a0) where the radial probability distribution is zero. Compare this picture to the picture of the electron density on the H atom wavefunction picture page (or the artist's representation of the wavefunction shown in figure 15.23).

Finally, let's look at 3s:

Note the two radial nodes, and note that a greater portion of the radial probability distribution is within the white, classically allowed region of space for 3s than for 2s or 1s. If we were to draw these figures for greater and greater n values, we would find that this trend continues. That's because as n increases, the atom becomes more and more closely described by classical mechanics. This is an example of something first discovered by Bohr and called the Correspondence Principle.