Discussion about various quantum numbers and the concept of probability distribution curves

1. Quantum numbers & Probability
distribution curves

2. Quantum numbers
• The mathematical solution of Schrodinger’s equation introduced four numbers that
were called quantum numbers , to determine the energy of an electron in multi-
electron atoms
• Used to describe completely the movement and trajectories of each electron within an
atom.
• Principal Quantum Number (n)
• Azimuthal Quantum number (l)
• Magnetic Quantum number (m)
• Spin Quantum number (s)

3. Principal Quantum Number
• describes the electron shell, or energy level, of an atom.
• describes tells the most probable distance of the electrons from the nucleus
• has whole number values 1 , 2 , 3 , 4 , ……. etc
• Also be designated as K,L,M,N, …… etc
• larger the number n , the larger the size of the orbit
• The maximum number of electrons in n is 2n2
• Also called as orbit.

4. Azimuthal Quantum Number
• Describes the shape of a given orbital.
• Each value of l indicates a specific subshell
• Subshell has whole number value starting from 0,1,2,3,4,……….
• The value of ℓ ranges from 0 to n − 1
• The maximum number of electrons in ℓ is 4ℓ +2
• Symbols of sub-levels , s = 0 , p = 1 , d = 2 , f = 3
• denotes the angular momentum and is given by the formula

5. Magnetic Quantum Number
• Determines the number of orbitals and their orientation within a subshell
• Depends on the azimuthal quantum number l
• For a given value of l, the value of ml ranges between the interval -l to +l.

7. Spin Quantum Number
• independent of the values of n, l, and m
• gives insight into the direction in which the electron is spinning
• The possible values are +½ and -½.
• Helps in the determination of an atom's ability to
generate a magnetic field or not.

8. Summary

9. Radial Probability function
• Ψ2 gives the probability of finding and electron at specific coordinate in 3D space.
• Multiplying this probability by the area (4πr2)available at that distance will give us the Radial
Distribution Function for the given electron.
• Probability of finding electron from nucleus
without reference to its direction.
• Prob. at origin = zero.
• The distance for finding electron in
an orbital increases with ‘n’
• For 1s electron, the max. prob of finding
electron = 0.529AO (Bohr’s radius)
• Radial node (nodal surface) – probability of
Finding electron is zero.

10. Radial Probability function
The number of radial nodes for an orbital = n-l-1.

11. Radial probability function

12. Angular probability function
• The square of the angular distribution function describes the probability of
finding the electron at angles θ and φ.
• The Angular distribution function describes the basic shape of the orbital, or the
number of lobes in an orbital.
• The angular distribution functions depend only on the quantum number l.
Total no. of angular nodes for any orbital = l
S – orbital is independent
of θ and φ.
Spherical

13. Shape of p - orbitals
• Dumb- bell shape
• Two lobes touching each other at the origin
• They are directional
• Two lobes are separated by a plane (nodal plane) where the
probability of finding electron is zero.
One
Nodal plane

14. Shape of d- orbitals
• Double dumb-bell shape
• Four lobed passing through the origin
• Two orbitals through the passing x,y,z axes
• Three lobes passing between the axes
• Two nodal planes are present.