Electron Orbitals and Radial Distribution Function

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This is an advanced Chemistry Tutorial, aimed at university students. The video relates to the content below. More tutorials will be uploaded in time to further describe the various aspects of electron configuration and spacial occupation, however we hope that you find this brief introduction useful in your studies.

What is the Radial Distribution Function?

Electrons are often described as orbiting the nucleus in distinct orbits or shells. More advanced descriptions depict suborbital shells including s, p, d and f sub-shells. It gets a little more complicated than that because electrons are both attracted to the nucleus, as well as repelled away by electrons in other suborbitals. This means that electron density is not uniform in the sub-shells.

The radial distribution function of an electron is the likelihood of finding an electron at different distances (radii) from the nucleus. The short video above constructs and deconstructs an atom, with:

• the simplistic orbital shell model on the bottom right hand corner;
• the sub-orbitals listed in the top right;
• the radial distribution function in the bottom left hand corner; and with
• it all combined into the atomic model in the middle.
  
  

What does the Radial Distribution Function tell us?

It tells us that the electron density varies considerably across the sub-orbitals, especially as the energy levels increase. The x-axis reports radial distance from the nucleus (which is at the origin of the graph). The y-axis gives us an indication of the probability of finding an electron at any given radius. The graph below corresponds to the 2S sub-orbital alone (i.e. it is not an amalgamation like in the video).

There are three pieces of information which the graph gives us:

1. The two peaks tells us that there are two radii where 2S electrons are more likely to be found.
2. The position of the two peaks along the x-axis tell us what these distances are from the nucleus.
3. The height of each peak (measured on the y-axis) tell us how likely we are to find an electron at one radius (peak), compared to the other radius (the other peak). The taller peak tells us we're more likely to find a 2S electron at the radius further away from the nucleus.

Strictly speaking, an electron could exist metres away from its nucleus. The likelihood of this occurring is very small, but it is possible; this is why the graph trails off into infinity. Another way to depict the changes in electron density is in the atomic model below. This image also depicts the 2S orbital by itself. It is clearer without the axis - move your mouse over the image to see this.

This picture gives you the same information as the graph above does, but may be more useful for visual learners to interpret:

1. The colour saturation corresponds to electron density - the more dense the colour, the higher the probability of finding a 2S electron there.
2. The number of rings tells us that there are two radii where the 2S electrons are more likely to be found.
3. Their position tells us how far away these rings of increased electron density are from the nucleus.

Correlating the Two Depictions

The inner ring in the image corresponds to the first peak in the top graph. You know this because

• position: it is closer to the nucleus, thereby corresponding to a peak closer to the origin on the x-axis in the graph.
• colour saturation: it is less saturated in colour, meaning that the electron density is lower than the outer ring (hence a lower peak).

The outer ring in the image corresponds to the second peak. You know this because

• position: it is further away from the nucleus (it has a larger radii), thereby corresponding to a peak further away from the origin on the x-axis in the graph.
• colour: its more saturated colour corresponds to a higher peak than the other ring.

Images

Below are the images used in the video, demonstrating how the shells come together to create an atom.