(Chemistry Ch-2) 7. Quantum Mechanical Model of Atom

• Macroscopic objects have particle character, so their motion can be described in terms of classical mechanics, based on Newton’s laws of motion.
• Microscopic objects, such as electrons, have both wave-like and particle-like behaviour, so they cannot be described in terms of classical mechanics. To do so, a new branch of science called quantum mechanics was developed.
• Quantum mechanics was developed independently by Werner Heisenberg and Erwin Schrodinger in 1926.
• Quantum mechanics takes into account the dual nature (particle and wave) of matter.
• On the basis of quantum mechanics, a new model known as quantum mechanical model was developed.
• In the quantum mechanical model, the behaviour of microscopic particles (electrons) in a system (atom) is described by an equation known as Schrodinger equation, which is given below:

Where,

= Mathematical operator known as Hamiltonian operator

ψ = Wave function (amplitude of the electron wave)

E = Total energy of the system (includes all sub-atomic particles such as electrons, nuclei)

• The solutions of Schrodinger equation are called wave functions.

Hydrogen atom and Schrodinger equation

• After solving Schrodinger equation for hydrogen atom, certain solutions are obtained which are permissible.
• Each permitted solution corresponds to a definite energy state, and each definite energy state is called an orbital. In the case of an atom, it is called atomic orbital, and in the case of a molecule, it is called a molecular orbital.
• Each orbital is characterised by a set of the following three quantum numbers:

• Principal quantum number (n)
• Azimuthal quantum number (l)
• Magnetic quantum number (m_l)

• For a multi-electron atom, Schrodinger equation cannot be solved exactly.

Important Features of the Quantum Mechanical Model of an Atom

• The energy of electrons in an atom is quantised (i.e., electrons can only have certain specific values of energy).
• The existence of quantised electronic energy states is a direct result of the wave-like property of electrons.
• The exact position and the exact velocity of an electron in an atom cannot be determined simultaneously (Heisenberg uncertainty principle).
• An atomic orbital is represented by the wave function ψ, for an electron in an atom, and is associated with a certain amount of energy.
• There can be many orbitals in an atom, but an orbital cannot contain more than two electrons.
• The orbital wave function ψ gives all the information about an electron.
• |ψ|² is known as probability density, and from its value at different points within an atom, the probable region for finding an electron around the nucleus can be predicted.

Orbitals and Quantum Numbers

Smaller the size of an orbital, greater is the chance of finding an electron near the nucleus.

Each orbital is characterised by a set of the following three quantum numbers:

• The principal quantum number (n)
  • Positive integers (n = 1, 2, 3,………)
  • Determines the size and energy of the orbital
  • Identifies the shell
  • n = 1, 2, 3, 4, ……..

Shell = K, L, M, N, ……..

With an increase in the value of n, there is an increase in the number of allowed orbitals (n²), the size of an orbital and the energy of an orbital.

• The Azimuthal quantum number (l)
  • Also known as orbital angular momentum or subsidiary quantum number
  • Defines the three-dimensional shape of an orbital
  • For a given value of n, l can have n values, ranging from 0 to n − 1.

For n = 1, l = 0

For n = 2, l = 0, 1

For n = 3, l = 0, 1, 2

For n = 4, l = 0, 1, 2, 3, ………and so on

• Each shell consists of one or more sub-shells or sub-levels. The number of sub-shells is equal to n, and each sub-shell corresponds to different values of l.

For n = 1, there is only one sub-shell (l = 0)

For n = 2, there are two sub-shells (l = 0, 1)

For n = 3, there are three sub-shells (l = 0, 1, 2)…….. and so on

Value for l	0	1	2	3	4	5 ……
Notation for sub-shell	s	p	d	f	g	h ……

Sub-shell notations corresponding to the given principal quantum numbers and azimuthal quantum numbers are listed in the given table.

Principal quantum number (n)	Azimuthal quantum number (l)	Sub-shell notations
1	0	1s
2	0	2s
2	1	2p
3	0	3s
3	1	3p
3	2	3d
4	0	4s
4	1	4p
4	2	4d
4	3	4f

• The magnetic orbital quantum number (m_l):
  • Gives information about the spatial orientation of the orbital with respect to the standard set of co-ordinate axis
  • For a given value of l (i.e., for a given sub-shell), 2l + 1 values of m_l are possible.

m_l = −l, −(l − 1), − (l − 2), …0, 1,….(l − 2), (l − 1), l

Example:

For l = 0, m_l = 0 (one s-orbital)

For l = 1, m_l = −1, 0 + 1 (three p-orbitals)

For l = 2, m_l = −2, −1, 0, +1, +2 (five d-orbitals)

For l = 3, m_l = −3, −2, −1, 0, + 1, +2, + 3 (seven f-orbitals)

The relation between the sub-shell and the number of orbitals is given in the following table:

Sub-shell notation	Number of orbitals
s	1
p	3
d	5
f	7
g	9
h	11

• There is a fourth quantum number known as the electron spin quantum number (m_s).
  • It designates the orientation of the spin of an electron. There are two orientations of an electron, known as the two spin states of an electron: +and −or ↑(spin up) and ↓(spin down)
  • An orbital cannot hold more than two electrons.