QUANTUM MECHANICAL MODEL OF ATOM

The atomic model which is based on the particle and wave nature of the electron is known as wave or quantum mechanical model of the atom. This was developed by Erwin Schrodinger in 1926. This model describes the electron as a three dimensional wave in the electronic field of positively charged nucleus. Schrodinger derived an equation which describes wave motion of an electron. The differential equation is

2ψ /∂x2 + ∂2ψ/∂y2 + ∂2ψ/∂z2 + 8π2m/h2 ( E - v ) ψ = o 

where x, y, z are certain coordinates of the electron, m = mass of the electron E = total energy of the electron. V = potential energy of the electron; h = Planck’s constant and ψ (psi) = wave function of the electron. 

Significance of ψ: The wave function may be regarded as the amplitude function expressed in terms of coordinates x, y and z. The wave function may have positive or negative values depending upon the value of coordinates. The main aim of Schrodinger equation is to give solution for probability approach. When the equation is solved, it is observed that for some regions of space the value of ψ is negative. But the probability must be always positive and cannot be negative, it is thus, proper to use ψ2 in favour of ψ. 

Significance of ψ2: ψ2 is a probability factor. It describes the probability of finding an electron within a small space. The space in which there is maximum probability of finding an electron is termed as orbital. The important point of the solution of the wave equation is that it provides a set of numbers called quantum numbers which describe energies of the electron in atoms, information about the shapes and orientations of the most probable distribution of electrons around nucleus. 

Nodal Points and Planes:

The point where there is zero probability of finding the electron is called nodal point. There are two types of nodes: Radial nodes and angular nodes. The former is concerned with distance from the nucleus while latter is concerned with direction.

No. of radial nodes = n – l – 1
No. of angular nodes = l

Total number of nodes = n – 1

Nodal planes are the planes of zero probability of finding the electron. The number of such planes is also equal to l. 

Illustration:

Calculate radial nodes and angular nodes for the following type of orbitals.

(a)    1s                             (b)    2p

(c)    3p                             (d)    3d

(e)    4s                             (f)     4d
 
Solution:

(a)    0, 0                           (b)    0, 1

(c)    1, 1                           (d)    0, 2

(e)    3, 0                           (f)     1, 2
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