February 3

Why is the s orbital spherical: its special place

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Why is the s orbital spherical: its special place

By Juman Hijab

atoms-electrons

Updated on: October 1, 2021 Original date: February 3, 2021


Tetrahedron

It's special place

The s-orbital has the most advantages of all the orbitals: 


  1. It is spherical which means the electron can "live" in a huge expanse of space.
  2. There are at most 2 electrons within the orbital. Thus, it is relatively easy for the electrons to stay out of each other's way.
  3. If we place the electrons on the vertices of a tetrahedron, the electrons can move from position to position but always stay far apart from its sister electron.  
  4. An electron in an ns-orbital can fluidly move into an (n+1)s-orbital or an (n-1)s-orbital. This gives rise to the hydrogen spectral lines (how it does this will be discussed in a future article).
  5. And, last but not least, the tetrahedral geometry for the s electrons allows them to venture closest to the nucleus. This is something all electrons want to do; they do it best in a tetrahedral structure.


The ability of the s-orbital electron to court the nucleus

An s-orbital electron living in a tetrahedral Platonic solid can get real close to the nucleus.

When you look at the faces of the tetrahedron above, it is clear that the each of the four faces is quite close to the center. In fact, for a unit sphere, the volume of an inscribed tetrahedron is only 12.3% of the sphere's volume. This small volume allows the tetrahedral faces to almost kiss the center point of the sphere.


In contrast, look at the table below: the volumes of all the other Platonic solids (and the non-Platonic rhombic dodecahedron) take up significantly more space within a sphere. This means that the electrons of non-tetrahedral solids will have more difficulty veering towards the nucleus. This is born out by  the distances from the mid-face of the solids to the center. those are quite a bit longer than that of a tetrahedron.


Platonic solidS (and rhombic  Dodecahedron)

Distance from center of sphere to mid-face of solid

Tetrahedron

12.3%

Octahedron

31.8%

Cube

36.8%

Icosahedron

60.5%

Dodecahedron

66.5%

Rhombic Dodecahedron

47.7%

Another way of looking at this is that an electron moving from the mid-face of a tetrahedron to the center is  2/3 closer to the nucleus than if it were coming from the vertex of the tetrahedron.


The diagram below shows a white central nucleus. The long red arrow is the distance from the vertex to the nucleus. The short red arrow is the distance from a tetrahedral mid-face to the nucleus; notice how much closer that is to the nucleus.

s orbital 2/3 closer to nucleus

s orbital 2/3 closer to nucleus

The spherical shape of the s-orbital

Having only two electrons sharing 4 vertices and the ability to get closer to the nucleus from the mid-face of the tetrahedron is the basis for the spherical shape of the s-orbital. Here's how it works. 

  • The electron tries to stay apart from its sister by staying at the vertex of the tetrahedron
  • However, it is pulled in to the nucleus; the most effective way to get closer to the nucleus is to veer towards one of the mid-faces of the tetrahedron.
  • Because electrons are such fast moving objects, their movement creates 2-D and 3-D reality (being in two places at the same time).  Electrons moving from the vertices and the sides to reach the mid-face of the tetrahedron creates a "plane" of electrons on each of the tetrahedral faces.
A tetrahedral plane of electrons

A tetrahedral "plane" of electrons veering towards the nucleus

  • However, at the same time, the internal energy of the electron pulls this "plane" of electrons outwards towards the sphere's rim. It is as if you are pushing the tetrahedral face inwards towards the nucleus with your fist; but then the tetrahedral face boings back into place, overshooting into a curved surface.
  • The net result is a spherical tetrahedron, as shown in this image. Each of the tetrahedral faces creates a curved surface that is continuously moving inwards towards the nucleus and then outwards towards the sphere's rim. This is nicely shown in drum wave motion analogy (the drum membrane is shown moving in and out in a continuous fashion).
Spherical tetrahedron

Spherical tetrahedron

Picture credits: 

  1. By extender_01. Atomic orbitals basic vector illustration. Types s,p,d. Shutterstock.com, ID: 176273831. 
  2. By sciencepics. One tetrahedral void showing the geometry. Shutterstock.com, ID: 255673555.
  3. By Tomruen at English Wikipedia - Own work, Spherical Tetrahedron. Public Domain, https://commons.wikimedia.org/w/index.php?curid=17032995.

Juman Hijab

About the author

Juman is a retired physician after having been in clinical practice for more than four decades. Her lifelong interest has been in the chemistry of life.

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