Atomic Structure

Quantum Mechanical Model of Atom

Dual Behavior of Matter

In 1924, de Broglie proposed that matter (like radiation) should also exhibit dual behavior, i.e. both particle and wave-like properties. So, electrons should also have momentum as well as wavelength. Based on this, de Broglie gave following relation between wavelength (λ) and momentum (p) of a material particle.

`λ=h/(mv)=h/p`

Where, m is mass, v is velocity and p is momentum of the particle.

De Broglie’s prediction was confirmed experimentally when it was found that an electron beam undergoes diffraction. It is important to recall that diffraction is a phenonmenon characteristic of waves. According to de Broglie, every object in motion has a wave character. Because of large masses, wavelengths of ordinary objects are so short that their wave properties cannot be detected. But the wavelenghts associated with subatomic particles (with very small mass) can be detected experimentally.

Heisenberg’s Uncertainty Principle

In 1927, Heisenberg proposed the uncertainty principle. As per this principle, it is impossible to determine simultaneously the exact position and exact momentum (or velocity) of an electron.

It can be shown by following equation.

`Δx×Δp_x≥h/(4π)`

Or, `Δx×Δ(mv_x)≥h/(4π)`

Or, `Δx×Δv_x≥h/(4πm)`

Where, Δx is the uncertainty in position and Δp_x or Δv_x is the uncertainty in momentum or velocity of the particle. If the position of electron is known with high degree of accuracy then velocity of electron will be uncertain. On the other hand, if velocity of electron is known precisely, then position of electron will be uncertain.

Significance of Uncertainty Principle

It rules out existence of definite paths or trajectories of electrons and other similar particles.
The effect of Heisenberg Uncertainty Principle is significant only for motion of microscopic objects and is negligible for that of macroscopic objects.
In dealing with milligram-sized or heavier objects, the associated uncertainties are hardly of any real consequence.
The precise statements of position and momentum of electrons have to be replaced with statements of probability.

Reasons for failure of Bohr’s Model

The wave character of electron is not considered in Bohr’s Model.
Bohr said that electrons move in well defined orbits. But this contradicts the Heisenberg’s uncertainty principle. We cannot be certain about the path of a particle unless we are sure about its position and velocity.
So, Bohr’s model could not be extended to atoms other than hydrogen.

Quantum Mechanics

Classical mechanics successfully describes the motion of all macroscopic objects which have essentially a particle-like behavior. But it fails when applied to microscopic objects because it ignores the concept of dual behavior of matter and the uncertainty principle.

Quantum mechanics is a theoretical science that deals with the study of motions of microscopic objects that have both observable wave-like and particle-like properties. When quantum mechanics is applied to macroscopic objects the results are the same as in case of classical mechanics.

Orbitals and Quantum Numbers

Each orbital is designated by three quantum numbers labelled as n, l and m_l

Principal Quantum Number (n): The principal quantum number determines the size and to large extent the energy of the orbital. It is a positive integer with value of n = 1, 2, 3, ……

The principal quantum number also identifies the shell. The number of allowed orbitals is given by n².

n = 1 2 3 4 …………..

shell = K L M N …………….

Size of an orbital increases with increase of principal quantum number ‘n’. The energy of the orbital will increase with increase of n.

Azimuthal Quantum Number (l): It is also known as orbital angular momentum or subsidiary quantum number. It defines the three-dimensional shape of the orbital. For a given value of n, l can have n value ranging from 0 to n – 1. For example; for n = 3, the possible values of l are 0, 1 and 2.

Each shell consists of one or more sub-shells or sub-levels. The number of sub-shells in a principal shell is equal to the value of n. For example; in the second shell (n = 2) there are two sub-shells (l = 0, 1). Each sub-shell is assigned an azimuthal quantum number.

Value for l: 0 1 2 3 4 5 ………….

Notation for sub-shell: s p d f g h …………………..

Magnetic Orbital Quantum Number (m_l): This quantum number gives information about the spatial orientation of the orbital with respect to standard set of coordinate axis. For any sub-shell (defined by l) 2l + 1 values of m_l are possible and these values are given as follows:

`m_l=-l, -(l-1), -(l-2), …..0, 1 …(l-2), (l-1), l`

So, for l = 0 the permitted value of m_l = 0, [2×0 + 1] = 1

For l = 1, value of m_l = -1, 0 and +1.

The following chart give the relation between the subshell and the number of orbtials associated with it.

Value of l	0	1	2	3	4	5
Subshell notation	s	p	d	f	g	h
Number of orbitals	1	3	5	7	9	11

Electron Spin Quantum Number (m_s): An electron spins around its own axis, the way earth spins on its axis. So, an electron has instrinsic spin angular quantum number. Spin angular momentum of the electron can have two orientations relative to the chosen axis. These two orientations are distinguished by the spin quantum numbers m_s which can take the values of +1/2 or -1/2. The are called the two spin states of the electron and are generally represented by two arrows ↑ (spin up) and ↓ (spin down). Two electrons with different m_s values are said to have opposite spins. An orbital cannot hold more than two electrons and these two electrons should have opposite spins.

Summary of four quantum numbers:

n defines the shell, determines the size of the orbital and also to a large extent the energy of the orbital.
There are n subshells in the n^th shell, and l identifies the subshell and determines the shape of the orbital.
m_l designatyes the orientatiopn of the orbital.
m_s refers to orientation of the spin of the electron.