Atomic Models & Quantum Numbers

Summary Of Atomic Models

After John Dalton's postulations of the Atomic Theory, several scientists carried out investigations on the structure of the atom to either validate or disapprove the theory. This eventually led to its modifications, as only one out of the five postulations made by Dalton was and is still valid, i.e., all chemical changes result from the combination or the separation of atoms.

The results of the major studies of are summarized below:

J. J. Thompson
Using his discharge tube and cathode rays experiments, he discovered that an atom is made up of positively charged particles called protons, and negatively charged particles called electrons.
He went on to propose that the atom is a sphere of positively charged matter in which negatively charged electrons are embedded. His model looks like an orange filled with seeds.

Ernest Rutherford
Through his gold-foil experiment, he proposed that the atom is made up of a positive core called the nucleus, where the mass of the atom is embedded, and electrons that revolve round the nucleus in their orbits.
He likened his model of an atom to the solar system, with electrons moving round the nucleus, the same way the planets revolve round the sun.

Neil Bohr & The Wave Mechanics
While studying the hydrogen atom, Neil Bohr postulated that the circular orbits of the electrons were quantized, i.e, the electrons revolve round the nucleus at certain and definite energy levels within the atom; and this threw more light in explaining how electrons are arranged in an atom.

As wonderful as his postulation was, like other models; it had its limitations because it could only be used to predict the behavior of electrons in a simple atom like hydrogen (with only one electron), but failed when it came to complicated atoms.

To take care for this shortcoming, the Wave Mechanics model was introduced. It assumes that electrons possess both particles and wave-like properties, which makes them very elusive (difficult to describe), as confirmed by the Heisenberg Uncertainty Principle. This principle suggests that if the momentum of an electron is accurately measured, it will be difficult to know its position and vice-versa. In other words, electrons cannot be localized or restricted to definite or specific energy levels in an atom.

The Wave Mechanics model has now replaced the Bohr's model of the atom.

Quantum Numbers: Overview

The Wave Mechanics model of the atom describes a region around the nucleus called orbital, where there is a possibility of finding an electron with a particular amount of energy. The orbital is defined by a set of parameters known as quantum numbers. It went further to reveal that the energy levels (orbits) are made up of one or more orbitals and the way electrons are distributed around the nucleus is determined by the number and kind of energy levels that are occupied. Examining these energy levels will enable us to predict the behaviour of those electrons in space. We can achieve this by looking at the four quantum numbers in details.

The Four Quantum Numbers

Studies have shown that the energy of an electron in an atom can be defined by four quantum numbers, namely: the principal, subsidiary or azimuthal, magnetic and spin.

1) The Principal Quantum Number, n
This represents the main energy levels or shells or orbits in an atom, and has integral values ranging from, n = 1, 2, 3, 4 to infinity. The shell with the lowest energy level, i.e., the shell closest to the nucleus has a value of n = 1(the first shell). The next has a value of n = 2 (the second shell), and so on. The shells apart from having integral values, also have letters associated with them.

 For example, the shell with n = 1, is known as the K shell, the shell with n = 2, is the L shell, the orbit with n = 3, is the M shell, etc.

Within a particular shell, there is a maximum number of electrons expected to occupy it, and this is determined by the formula:  2n^2 (2 multiplied by 'n squared'), where n is the principal quantum number.

For example,
In K Shell, n = 1; the maximum number of electrons in that shell will be:
                          2 x 1^2 = 2 x 1 = 2 electrons
Similarly, in L Shell, n = 2; the maximum number of electrons will be:
                          2 x 2^2 = 2 x 4 = 8 electrons
Also, in M Shell, n = 3; and the maximum number of electrons in that shell will be:
                          2 x 3^2 = 2 x 9 = 18 electrons
In N Shell, n = 4; the maximum number of electrons will be:
                         2 x 4^2 = 2 x 16 = 32 electrons; and so on

2) The Subsidiary or Azimuthal Quantum Number, l
This indicates the number of energy sublevels present in a particular energy level or electron shell. It has integral values that range from l = 0 to (n-1). In other words, its values depend on the values of the principal quantum number, n.

For example,
In the 1st shell, where n =1, the possible value(s) of l will be:
                            l = 0 to (1-1)
                              = 0 to 0
                              = 0
The above number, 0, should not be seen as a number on its face value, that means 'nothing' or 'naught'; instead, it should be seen as a symbol or item used to represent a particular energy sublevel called s-orbital or subshell.

In the light of the above, it means that there is only one energy sublevel in the 1st shell, and that is the s-orbital or subshell (1s).

Also, in the 2nd shell, where n = 2, the possible values of l will be:
                          l = 0 to (2-1)
                            = 0 to 1 i.e.;
                          l = 0 and 1

Similarly, the number, 1, represents another energy sublevel called p-orbital or subshell. Therefore, there are two energy sublevels in the 2nd shell of an atom, and they are the s-orbital  and the p-orbital (2s and 2p).

In the 3rd shell, where n =3, the possible values of l will be:
                          l = 0 to (3-1)
                            = 0 to 2, i.e.,
                          l = 0, 1 and 2

The number, 2, represents another energy sublevel called d-orbital or subshell. From the above, it implies that three energy sublevels are present in the 3rd shell, and they include: the s-orbital, the p-orbital, and the d-orbital (3s, 3p and 3d)

Similarly, in the 4th shell, where n = 4, the values of l will be:
(work it out yourself and scroll down to confirm your answers)
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                            l = 0 to (4-1)
                              = 0 to 3, i.e.,
                            l = 0, 1, 2 and 3

And the number, 3, stands for an energy sublevel known as f-orbital or subshell. This brings the number of energy sublevels available in the 4th shell of an atom to four, and they include: the s-orbital, the p-orbital, the d-orbital and the f-orbital (4s, 4p, 4d and 4f)

3) The Magnetic Quantum Number, m
This denotes the number of sub-orbitals present in a given subshell or energy sublevel. It has integral values ranging from -l……0……+l (read as -l through 0 to +l). Its values depend on the values of the subsidiary quantum number, l, whose values in turn, depend on those of the principal quantum number, n.

For example,
In the 1st shell, where n = 1, and l = 0, then, the value of m will be:
                          m = -0……0……+0, which gives us 0.
Again the 0 is not 'nothing'  or 'naught', but an item. In other words, there is only one sub-orbital (or compartment) in the 1s subshell, and the maximum number of electrons that can be found in this shell is 2 x 1 = 2

For the 2nd shell, where n = 2, and l = 0, 1
                        when l = 0 (s-orbital),
                                m = 0
                         when l = 1 (p-orbital),
                        then, m = -1, 0, +1

The above indicates that while there is only one sub-orbital in the s-orbital, there are three sub-orbitals (-1,0,+1) in the p-orbital, namely: Px, Py and Pz. This implies that there are altogether four sub-orbitals in the 2nd energy level, and they include: 2S, 2Px, 2Py and 2Pz; and if each of these can accommodate a maximum of two electrons, then, the maximum number of electrons expected in the L shell is 2 x 4 = 8

Looking at the 3rd shell, where n = 3, and l = 0, 1, 2
                                 when l = 0,
                                         m = 0
                                 when l = 1,
                                         m = -1, 0, +1
                                 when l = 2 (d-orbital),
                                then, m = -2, -1, 0, +1, +2

Hence there are five sub-orbitals (-2, -1, 0, +1, +2) in the d-orbital, and altogether, there are nine sub-orbitals in the 3rd energy level; and if each takes two electrons, the maximum number of electrons expected in the M shell is 2 x 9 = 18

Similarly, in the 4th shell, where n = 4, and l = 0, 1, 2, 3
                                 when l = 0,
                                         m = 0
                                  when l = 1,
                                          m = -1, 0, +1
                                  when l = 2,
                                          m = -2, -1, 0, +1, +2
                                  when l = 3 (f-orbital),
                                         m =?
(work it out and scroll down to confirm your answers)
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                                       m = -3, -2, -1, 0, +1, +2, +3

This gives a total of seven sub-orbitals in the f-orbital; and in all, there are 16 sub-orbitals (one 4s, three 4p, five 4d and seven 4f) in the 4th energy level. If each accommodates two electrons, the total number of electrons in the level will be 2 X 16 = 32

4) The Spin Quantum Number, s
This is the last in the series, and describes the spin of electrons in a  sub-orbital, which can only hold a maximum of two electrons that spin on their axes in opposite directions. The spin quantum number has two values, which are independent of the quantum numbers:  +1/2 and -1/2, where +1/2 shows the upward spin, and -1/2 shows the downward spin.

Principles And Rules Guiding The Filling Of Electrons Into Orbitals

Having understood the concept of the four quantum numbers, it is time to discuss the three principles that guide the way electrons are filled into orbitals. They are:

1) Pauli's Exclusion Principle: This states that no two electrons in an atom can have the same values for all the four quantum numbers. If they have the same values for the principal, subsidiary and magnetic quantum numbers, their spins will be different.

Let's consider the two electrons in a helium atom, (which has one shell). They are both found in the 1st and only shell of the atom, which is n = 1. Then, they both occupy the s-orbital (l = 0), which is the only orbital in the 1st shell. Of course, when l = 0, m = 0; which implies that the two electrons both share the same sub-orbital. However, when entering the sub-orbital, they do so with different spins, +1/2 (upward) and -1/2 ( downward). This opposite spinning of the electrons is the only arrangement that can enhance the stability of the orbitals. The above explains why the electron configuration of He is 1s^2 (the sign ^ stands for power).

Similarly, if we consider the electron configuration of lithium atom, which has three electrons (1s^2, 2s^1); we will see that the third electron goes into the s-orbital of the 2nd shell with an upward spin. If we compare this with the upwardly spined 1s electron, we will observe that they have the same values for their spin (+1/2), magnetic (0) and subsidiary (s-orbital, i.e, l = 0) quantum numbers, but their principal quantum numbers, n are different. While one is 1(1st shell), the other is 2 (2nd shell).
In summary, according to this principle, no two electrons in an atom can exhibit the same behaviour.

2) Hund's Rule of Maximum Multiplicity:
This rule states that electrons fill degenerate orbitals singly first, before pairing starts. Degenerate orbitals are orbitals with the same energy level. They include  the three sub-orbitals of the p-orbital, the five sub-orbitals of the d-orbital, and the seven sub-orbitals of the f-orbital.

So, if we consider a nitrogen atom with seven electrons, its electron configuration is 1s^2, 2s^2, 2(Px)^1, 2(Py)^1, 2(Pz)^1 and not 1s^2, 2s^2, 2(Px)^2, 2(Py)^1. The 2Px, 2Py and 2Pz are degenerate orbitals, which take up the 5th, 6th and 7th electrons respectively.

3) Aufbau's Principle:
This principle states that electrons fill orbitals with lower energy first, before filling those with higher energy. In other words, electrons will go into an orbital with a lower (n+l) value before going into the one with a higher (n+l) value. This is why electrons will fill 4s ((n+l) value of 4+0 = 4), before filling 3d ((n+l) value of 3+2 = 5).

In cases where the (n+l) values are the same, e.g., 3d (3+2=5) and 4p (4+1=5); electrons will fill the orbital with a lower n value first, 3d, before filling the one with a higher n value, 4p.

Summary Of Orbitals With Their (n+l) Values

                           1s = 1 + 0 = 1;   2s = 2 + 0 = 2
                           2p = 2 + 1 = 3;   3s = 3 + 0 = 3
                           3p = 3 + 1 = 4;   3d = 3 + 2 = 5
                           4s = 4 + 0 = 4;   4p = 4 + 1 = 5
                           4d = 4 + 2 = 6;   4f = 4 + 3 = 7
                           5s = 5 + 0 = 5;   5p = 5 + 1 = 6
                           5d = 5 + 2 = 7;   5f = 5 + 3 = 8
                           6s = 6 + 0 = 6;   6p = 6 + 1 = 7
                           6d = 6 + 2 = 8;   6f = 6 + 3 = 9
                            7s = 7 + 0 = 7;   7p = 7 + 1 = 8
                            7d = 7 + 2 = 9;   7f = 7 + 3 = 10

Order Of Filling Electrons Into Orbitals
From the above analysis, we can easily derive the order by which electrons are filled into orbitals, as shown below:
1s 2s 2p 3s 3p 4s 3d 4p 5s 4d 5p 6s 4f 5d 6p 7s 5f 6d 7p 6f 7d 7f
With this, we can write the electron configuration of any known element, taking into consideration all the rules and principles.

Electron Configuration Of The First 30 Elements

Hydrogen, H  - 1: 1s^1

Helium, He - 2: 1s^2

Lithium, Li - 3:  1s^2 2s^1

Beryllium, Be - 4: 1s^2 2s^2

Boron, B - 5: 1s^2 2s^2 2p^1

Carbon, C - 6: 1s^2 2s^2 2p^2

Nitrogen, N - 7: 1s^2 2s^2 2p^3

Oxygen, O - 8: 1s^2 2s^2 2p^4

Fluorine, F - 9:  1s^2 2s^2 2p^5

Neon, Ne - 10: 1s^2 2s^2 2p^6

Sodium, Na - 11: 1s^2 2s^2 2p^6 3s^1

Magnesium, Mg - 12: 1s^2 2s^2 2p^6 3s^2

Aluminium, Al - 13: 1s^2 2s^2 2p^6 3s^2 3p^1

Silicon, Si - 14: 1s^2 2s^2 2p^6 3s^2 3p^2

hosphorus, P - 15: 1s^2 2s^2 2p^6 3s^2 3p^3

Silicon, S - 16: 1s^2 2s^2 2p^6 3s^2 3p^4

Chlorine, Cl - 17: 1s^2 2s^2 2p^6 3s^2 3p^5

Argon, Ar - 18: 1s^2 2s^2 2p^6 3s^2 3p^6

Potassium, K - 19: 1s^2 2s^2 2p^6 3s^2 3p^6 4s^1

Calcium, Ca - 20: 1s^2 2s^2 2p^6 3s^2 3p^6 4s^2

Scandium, Sc - 21: 1s^2 2s^2 2p^6 3s^2 3p^6 4s^2 3d^1

Titanium, Ti - 22: 1s^2 2s^2 2p^6 3s^2 3p^6 4s^2 3d^2

Vanadium, V - 23: 1s^2 2s^2 2p^6 3s^2 3p^6 4s^2 3d^3

Chromium, Cr - 24: 1s^2 2s^2 2p^6 3s^2 3p^6 4s^1 3d^5

Manganese, Mn - 25: 1s^2 2s^2 2p^6 3s^2 3p^6 4s^2 3d^5

Iron, Fe - 26: 1s^2 2s^2 2p^6 3s^2 3p^6 4s^2 3d^6

Cobalt, Co - 27: 1s^2 2s^2 2p^6 3s^2 3p^6 4s^2 3d^7

Nickel, Ni - 28: 1s^2 2s^2 2p^6 3s^2 3p^6 4s^2 3d^8

Copper, Cu - 29: 1s^2 2s^2 2p^6 3s^2 3p^6 4s^1 3d^10

Zinc, Zn - 30: 1s^2 2s^2 2p^6 3s^2 3p^6 4s^2 3d^10

Take note of the discrepancies in the electron configurations of Cr and Cu. This is due to the extra stability conferred on evenly filled (singly or doubly) d-orbitals, bearing in mind that the d-orbitals are inner orbitals.

Do These:

Write the electron configuration of the following species:
i) Cr3+
ii) Fe2+
iii) Fe3+
iv) Mn4+
(Hint: Electron loss starts from the 4s orbital, which is an outer orbital)

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