Energy Levels: UNIZOR.COM - Physics4Teens - Atoms - Electronic Structure...

Notes to a video lecture on http://www.unizor.com

Energy Levels

In the previous lecture we have addressed how electrons are positioned around nucleus.
We mentioned that, according to Bohr's model, they are only located on stationary orbits within shells and subshells.

We had a nice formula for the maximum number of electrons that can be located on each subshell, depending on its order number, traditionally designated by a letter (subshell #1 is designated letter s, the next #2 is p, then d, f, g etc.) For subshell #m the maximum number of electrons on it is 4m−2.
The number of subshells within shell #N is N.
Consequently, the maximum number of electrons held in each shell is 2·N².

This mathematically perfect picture gets more complicated, when we consider the energy level associated with each subshell.
The main principle of distribution of energy among subshells, that is, the greater radius of a subshell - the greater energy electrons within this subshell possess, is absolutely true.
The problem is, the further we go from the nucleus - the closer to each other are the shells and, subshells of greater radius of a shell #N might overlap with subshells of smaller radius of shell #N+1.

Let's consider a few elements in a sequence of their atomic numbers, that is the number of protons in their nucleus or electrons within shells on different levels.

Hydrogen has only one proton in the nucleus and one electron within the first subshell (designated by a letter s) of the only shell #1.
So, the electron structure of the atom of hydrogen, where the number specifies the shell number and a letter specifies the subshell, where the superscript at the subshell letter indicates the number of electrons within the same subshell, is:
1s¹

Helium has two protons in the nucleus and two electrons within the first subshell (designated by a letter s) of the only shell #1.
The subshell s (the first one) can hold maximum of 2 electrons, so everything is fine.
The electron structure of the atom of helium is:
1s²

Lithium has three protons in the nucleus and three electrons.
The first (and only) subshell s of the shell #1 can hold only 2 electrons. Therefore, a new shell #2 should exist and one electron should be in its first subshell s.
The subshell s of the second shell can hold maximum of 2 electrons, so everything is fine.
The electron structure of the atom of lithium is:
1s² 2s¹

Let's skip a few elements and consider carbon with atomic number 6.
Its 6 electrons should fill the shell #1 with its only subshell s holding 2 electrons, then the shell #2 with its subshell s holding another 2 electrons, then, considering shell #2 has two subshells, s and p, the second subshell p will hold the remaining 2 electrons.
The subshell p, as the second subshell, can hold maximum of 6 electrons, so we still have room.
The electronic structure of an atom of carbon is, therefore,
1s² 2s² 2p²

Skip a few more elements to silicon with atomic number 14.
We already know that the first shell can hold 2 electrons in one and only subshell s.
The second shell can hold 2 electrons in subshell s and 6 electrons in subshell p.
That totals 10 electrons. To accommodate 4 other electrons the third shell should be used.
Its subshell s will hold 2 electron and subshell p will hold remaining 2 ones.
The electronic structure of silicon is
1s² 2s² 2p⁶ 3s² 3p²

So far, increasing in atomic number is synchronous with sequential filling of shells and subshells within shells.
But let's consider element argon. Its atomic number is 18 and it completely fills one subshell s of the first shell, both subshells of the second shell and two (out of three) subshells of the third shell.
Its electronic structure is
1s² 2s² 2p⁶ 3s² 3p⁶

The next element is potassium with atomic number 19.
It would be reasonable to expect that, since the second shell's last subshell p is already filled as well as two first subshells of shell #3, and the third shell allows three subshells, the extra electron #19 should be in the third subshell d (that is the third subshell's letter) of the shell #3.
However, the experiments showed that the last electron #19 goes to the fourth shell's first subshell s instead. Why?

Here we face the fact that shells overlap, and higher energy level of the third subshell d of shell #3 exceeds the lower energy level of the first subshell s of shell #4.
That's why the location of electron #19 of potassium is not 3d¹ but 4s¹.
That is the reason why the electronic structure of potassium is
1s² 2s² 2p⁶ 3s² 3p⁶ 4s¹

The order of filling the electronic subshells, as the number of electrons in the atom grows, under normal conditions corresponds to an order of increasing their energy levels. This is known as the Aufbau principle or Madelung or Klechkovsky rule.

According to this principle, and taking into consideration that shells get closer and closer to each other as we move away from a nucleus, while the number of subshells is increasing linearly with the shell number, the order of filling the electronic structure of an atom is not the numerical order of shell/subshell numbers, but as follows: 1s→2s→2p→3s→3p→4s→
→3d→4p→5s→4d→5p→6s→
→4f→5d→6p→7s→5f→
→6d→7p→...

In the lecture "Orbiting Electron" of the chapter "Building Blocks of Matter" of this course we have derived a formula for a total energy of an orbiting electron:
E = −k·e²/(2·r)
where
k is Coulomb constant
e is the electric charge of an electron
r is the radius of an orbit.

From the above formula we see that absolute value of the energy is decreasing with increasing of a radius of an orbit, but, since it's negative, the energy itself is increasing.
As a radius of an orbit of an electron is increasing, its (negative) energy is increasing, getting asymptotically closer to zero. At the same time the number of subshells of each shell is increasing with the shell number.
That explains the overlapping energy characteristic of the shells.