Quantum Numbers and Atomic Orbitals

By solving the Schrödinger equation (Hy = Ey), we obtain a set of mathematical equations, called wave functions (y), which describe the probability of finding electrons at certain energy levels within an atom.
A wave function for an electron in an atom is called an atomic orbital; this atomic orbital describes a region of space in which there is a high probability of finding the electron. Energy changes within an atom are the result of an electron changing from a wave pattern with one energy to a wave pattern with a different energy (usually accompanied by the absorption or emission of a photon of light).
Each electron in an atom is described by four different quantum numbers. The first three (n, l, m_l) specify the particular orbital of interest, and the fourth (m_s) specifies how many electrons can occupy that orbital.

Principal Quantum Number (n): n = 1, 2, 3, …, ∞
Specifies the energy of an electron and the size of the orbital (the distance from the nucleus of the peak in a radial probability distribution plot). All orbitals that have the same value of n are said to be in the same shell (level). For a hydrogen atom with n=1, the electron is in its ground state; if the electron is in the n=2 orbital, it is in an excited state. The total number of orbitals for a given n value is n².

Angular Momentum (Secondary, Azimunthal) Quantum Number (l): l = 0, ..., n-1.
Specifies the shape of an orbital with a particular principal quantum number. The secondary quantum number divides the shells into smaller groups of orbitals called subshells (sublevels). Usually, a letter code is used to identify l to avoid confusion with n:

l	0	1	2	3	4	5	...
Letter	s	p	d	f	g	h	...

The subshell with n=2 and l=1 is the 2p subshell; if n=3 and l=0, it is the 3s subshell, and so on. The value of l also has a slight effect on the energy of the subshell; the energy of the subshell increases with l (s < p < d < f).

Magnetic Quantum Number (m_l): m_l = -l, ..., 0, ..., +l.
Specifies the orientation in space of an orbital of a given energy (n) and shape (l). This number divides the subshell into individual orbitals which hold the electrons; there are 2l+1 orbitals in each subshell. Thus the s subshell has only one orbital, the p subshell has three orbitals, and so on.

Spin Quantum Number (m_s): m_s = +½ or -½.
Specifies the orientation of the spin axis of an electron. An electron can spin in only one of two directions (sometimes called up and down).
The Pauli exclusion principle (Wolfgang Pauli, Nobel Prize 1945) states that no two electrons in the same atom can have identical values for all four of their quantum numbers. What this means is that no more than two electrons can occupy the same orbital, and that two electrons in the same orbital must have opposite spins.
Because an electron spins, it creates a magnetic field, which can be oriented in one of two directions. For two electrons in the same orbital, the spins must be opposite to each other; the spins are said to be paired. These substances are not attracted to magnets and are said to be diamagnetic. Atoms with more electrons that spin in one direction than another contain unpaired electrons. These substances are weakly attracted to magnets and are said to be paramagnetic.

Table of Allowed Quantum Numbers

n	l	*m_l*	Number of orbitals	Orbital Name	Number of electrons
1	0	0	1	1s	2
2	0	0	1	2s	2
	1	-1, 0, +1	3	2p	6
3	0	0	1	3s	2
	1	-1, 0, +1	3	3p	6
	2	-2, -1, 0, +1, +2	5	3d	10
4	0	0	1	4s	2
	1	-1, 0, +1	3	4p	6
	2	-2, -1, 0, +1, +2	5	4d	10
	3	-3, -2, -1, 0, +1, +2, +3	7	4f	14

Writing Electron Configurations

The distribution of electrons among the orbitals of an atom is called the electron configuration. The electrons are filled in according to a scheme known as the Aufbau principle ("building-up"), which corresponds (for the most part) to increasing energy of the subshells:

1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s, 4d, 5p, 6s, 4f, 5d, 6p, 7s, 5f

It is not necessary to memorize this listing, because the order in which the electrons are filled in can be read from the periodic table in the following fashion:

Or, to summarize:

Periodic Table with Quantum Number scheme

In electron configurations, write in the orbitals that are occupied by electrons, followed by a superscript to indicate how many electrons are in the set of orbitals (e.g., H 1s¹)
Another way to indicate the placement of electrons is an orbital diagram, in which each orbital is represented by a square (or circle), and the electrons as arrows pointing up or down (indicating the electron spin). When electrons are placed in a set of orbitals of equal energy, they are spread out as much as possible to give as few paired electrons as possible (Hund's rule).

examples will be added at a later date

In a ground state configuration, all of the electrons are in as low an energy level as it is possible for them to be. When an electron absorbs energy, it occupies a higher energy orbital, and is said to be in an excited state.

Properties of Monatomic Ions

The electrons in the outermost shell (the ones with the highest value of n) are the most energetic, and are the ones which are exposed to other atoms. This shell is known as the valence shell. The inner, core electrons (inner shell) do not usually play a role in chemical bonding.
Elements with similar properties generally have similar outer shell configurations. For instance, we already know that the alkali metals (Group I) always form ions with a +1 charge; the "extra" s¹ electron is the one that's lost:

IA	Li	1s²2s¹	Li+	1s²
	Na	1s²2s²2p⁶3s¹	Na+	1s²2s²2p⁶
	K	1s²2s²2p⁶3s²3p⁶4s¹	K+	1s²2s²2p⁶3s²3p⁶

The next shell down is now the outermost shell, which is now full — meaning there is very little tendency to gain or lose more electrons. The ion's electron configuration is the same as the nearest noble gas — the ion is said to be isoelectronic with the nearest noble gas. Atoms "prefer" to have a filled outermost shell because this is more electronically stable.

The Group IIA and IIIA metals also tend to lose all of their valence electrons to form cations.

IIA	Be	1s²2s²	Be²⁺	1s²
	Mg	1s²2s²2p⁶3s²	Mg²⁺	1s²2s²2p⁶
IIIA	Al	1s²2s²2p⁶3s²3p¹	Al3+	1s²2s²2p⁶

The Group IV and V metals can lose either the electrons from the p subshell, or from both the s and p subshells, thus attaining a pseudo-noble gas configuration.

IVA	Sn	[Kr]4d¹⁰5s²5p²	Sn²⁺	[Kr]4d¹⁰5s²
			Sn⁴⁺	[Kr]4d¹⁰
	Pb	[Xe]4f¹⁴5d¹⁰6s²6p²	Pb²⁺	[Xe]4f¹⁴5d¹⁰6s²
			Pb⁴⁺	[Xe]4f¹⁴5d¹⁰
VA	Bi	[Xe]4f¹⁴5d¹⁰6s²6p³	Bi³⁺	[Xe]4f¹⁴5d¹⁰6s²
			Bi⁵⁺	[Xe]4f¹⁴5d¹⁰

The Group IV - VII non-metals gain electrons until their valence shells are full (8 electrons).

IVA	C	1s²2s²2p²	C4-	1s²2s²2p⁶
VA	N	1s²2s²2p³	N3-	1s²2s²2p⁶
VIA	O	1s²2s²2p⁴	O2-	1s²2s²2p⁶
VIIA	F	1s²2s²2p⁵	F-	1s²2s²2p⁶

The Group VIII noble gases already possess a full outer shell, so they have no tendency to form ions.

VIIIA	Ne	1s²2s²2p⁶
	Ar	1s²2s²2p⁶3s²3p⁶

Transition metals (B-group) usually form +2 charges from losing the valence s electrons, but can also lose electrons from the highest d level to form other charges.

B-group	Fe	1s²2s²2p⁶3s²3p⁶3d⁶4s²	Fe²⁺	1s²2s²2p⁶3s²3p⁶3d⁶
			Fe³⁺	1s²2s²2p⁶3s²3p⁶3d⁵

Atomic Sub Shells

Atoms have electrons arranged in definite energy levels. The electrons within the first quantum shell are closest to the nucleus and have the lowest energies.
Subsequent energy levels further away from the nucleus have higher energies.
The shells are labelled by assigning each one a principal number, n, where n = 1, 2, 3…
Evidence for electrons being arranged in energy levels is given by atomic spectroscopy and successive ionisation energy values.

The energy levels are evident due to the large increase in ionisation energies after the removal of certain electrons.
The number of electrons allowed in one energy level can be summarised by the formula 2n² where n = energy level quantum number.
Within energy levels in atoms, there are also sub energy levels, or orbitalss.
The orbitals are labelled s, p, d and f. In the first energy level, there is only one orbital (s). The second shell has two orbitals (s and p), the third three (s, p, d) and the fourth has four (s, p, d, f).
The different orbitals can hold a different number of electrons:
- 1 s orbital can hold 2
- 3 p orbitals can hold 6
- 5 d orbitals can hold 10
- 7 f orbitals can hold 14

This explains why the total number of electrons in each energy level is 2n².

The sub shells can be summarised in the diagram below, notice that the n=3 and n=4 sub shells overlap.

Atomic Orbitals

The s, p, d and f s orbitals compose of regions of space where the electrons can be found.

Electrons within the same orbitals have the same amount of energy.
The electrons do not lie within a specific place within an orbital, but can be found anywhere around the orbital.
Each atomic orbital can contain 2 electrons, which can spin in two directions , the electrons within the same orbital spin in opposite directions.
To describe accurately the state of an electron, you need to be able to state:
1. The electron shell it is in (1, 2, 3, 4….)
2. The orbital it is in within the electron shell (s, p, d, f)
3. Its spin.

Filling the orbitals

The arrangement of electrons within an atom, is referred to as the electronic configuration of an atom.
The orbitals are filled within a definite order to produce the lowest energy arrangement available.
The lowest energy levels are always filled first.
Where there is more than one orbital within the same energy, the separate orbitals are filled separately with the same spin (think of bus analogy, i.e. people will aim to sit on their own seat on the bus, but when the seats are full, they will be forced to sit on a seat with someone else!). Then the electrons pair up in orbitals.

Difference between shells, subshells and orbitals

Here's a graphic I use to explain the difference :
Shells, sub-shells, and orbitals

All electrons that have the same value for n (the principle quantum number) are in the same shell
Within a shell (same n), all electrons that share the same l (the angular momentum quantum number, or orbital shape) are in the same sub-shell

When electrons share the same n, l, and ml, we say they are in the same orbital (they have the same energy level, shape, and orientation)

So to summarize:

same n - shell
same n and l - sub-shell
same n, l, and ml - orbital

Now, in the other answer, there is some discussion about spin-orbitals, meaning that each electron would exist in its own orbital. For practical purposes, you don't need to worry about that - by the time those sorts of distinctions matter to you, there won't be any confusion about what people mean by "shells" and "sub-shells." For you, for now, orbital means "place where up to two electrons can exist," and they will both share the same

l, and

ml values, but have opposite spins (

ms).

Orbitals that have the same value of the principal quantum number n form a shell. Orbitals within a shell are divided into subshells that have the same value of the angular quantum number l. Chemists describe the shell and subshell in which an orbital belongs with a two-character code such as 2p or 4f. The first character indicates the shell (n = 2 or n = 4). The second character identifies the subshell. By convention, the following lowercase letters are used to indicate different subshells.

s: l = 0

p: l = 1

d: l = 2

f: l = 3

What is called an orbital might differ according to the context. With orbitals in the context of shells and subshells one usually means atomic orbitals, i.e. two-electron eigenstates of an atom's Hamilton operator which are characterized by the three quantum numbers: the principal quantum number

n, the angular quantum number

l and the magnetic quantum number

m. But often the word orbital is also used for spin-orbitals, i.e. one-electron eigenstates of the system's one-electron Hamilton operator which are characterized not only by

l and

m but also by the spin quantum number

ms which can be either

+12 or

−12.

Nucleons Calculation

Nucleons

The nuclei of atoms are made up of protons and neutrons. These two components of the nucleus are referred to as nucleons. The electrons occupy the space outside the nucleus. Since an atom is electrically neutral, the number of protons in the nucleus is exactly equal to the number of electrons. This number is the atomic number given by the symbol Z.
Atomic number represents the number of protons in an atom. As atoms are electrically neutral, an atom contains as many electrons as it has protons. The total number of protons and neutrons present in one atom of an element is known as its mass number.Mass number = number of protons + number of neutrons
It can also be said that:Mass number = atomic number + number of neutrons
Atomic number and mass number are represented on the symbol of an element as follows:

atomic number mass number nucleon number

Example:

Represent the atom of sodium whose atomic mass is 23 and atomic number is 11. Calculate the number of protons, electrons and neutrons.

Atomic number Z -11Atomic mass A - 23No. of protons = Z = 11 No. of electrons = 11No. of neutrons = A - Z23 - 11 = 12