Orbitals and Quantum Numbers

ORBITALS AND QUANTUM NUMBERS

ORBIT VS. ORBITAL

 PHYSICAL SIGNIFICANCE OF THE QUANTUM NUMBERS

1s and 2s orbitals

 2s and 2p Orbitals

d orbital

 ELECTRONIC CONFIGURATION OF ATOMS

Aufbau principle Electrons are placed in the lowest energetically available subshell. Thus orbitals are filled in the order of increasing energy, using two general rules  to help predict electronic configurations:-

1. Electrons are assigned to orbitals in order of increasing value of (n+ℓ). Orbitals with lower value of (n+l) are filled first as they have lower energy.
2. For subshells with the same value of (n+ℓ), electrons are assigned first to the sub shell with lower n.
3. The order in which the orbitals are filled is as follows:
4. 1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s, 4d, 5p, 4f, 5d, 6p, 7s...
5. It is based on (n+ l) rule. It states that the orbital with lower value of (n +l) has lower energy.

 AUFBAU DIAGRAM

PAULI'S EXCLUSION PRINCIPLE

It is impossible for any two electrons of an atom to have the same values of the all four quantum numbers. If even three quantum numbers are same for any two electrons of the atom then fourth quantum number will definitely be different.

In other words an orbital can have maximum of two electrons

Hund’s Rule of Maximum Multiplicity

This rule deals filling of electrons in the orbitals belonging to the same subshells of equal energy called degenerate orbitals.

5.  It states that pairing of electrons in the orbitals belonging to the same sub shell (p, d or f) does not take place until each orbital of that sub shell gets one electron that is singly occupied.
5.  Some of the orbitals acquire extra stability due to their symmetry.

 EXCEPTION: HALF-FILLED AND FILLED D ORBITALS

•  When a subshell is half-filled or completely filled, it lowers the overall energy of the atom.
•  In other words, this is favoured.
•  This usually doesn't affect electron configurations, except in the d and f subshells.
•  Example: Copper (29 electrons)
•  Normal electron configuration:

      1s²2s²2p⁶3s²3p⁶4s²3d⁹

•  Added stability if 3 d is full (electron gets promoted to 3 d from 4s)
•  1s²2s²2p⁶3s²3p⁶4s¹3d¹⁰

The region where this probability density function reduces to zero is called nodal surfaces or simply nodes.

Boundary surface diagram: In this representation, a boundary surface or contour surface is drawn in space for an orbital on which the value of probability density ψ ²(r) is constant. However, for a given orbital, only that boundary surface diagram of constant probability density is taken to be good representation of the shape of the orbital which encloses a region or volume in which the probability of finding the electron is very high, say, 90%.

Radial nodes: Radial nodes occur when the probability density wave function for the electron is zero on a spherical surface of a particular radius. Number of radial nodes = n – l – 1

Angular nodes: Angular nodes occur when the probability density wave function for the electron is zero along the directions specified by a particular angle. Number of angular nodes = l

Total number of nodes = n – 1

Degenerate orbitals: Orbitals having the same energy are called degenerate orbitals

IMPORTANT FORMULAE

1.         
2. Frequency of electromagnetic wave  
3. Energy of photon E = hn =  
4. 
5. Bohr’s postulate mvr =  (Angular momentum of ‘e’ in an orbit)            
6. 
7. r_n = r₁ x n²  same atom different orbit
8. r_Z  = r_H / Z   same orbit different atom
9. 
10. u_n = u₁ / n        same atom different orbit
11. u_Z = u_H x Z      same orbit different atom
12.   E₁ for H= -21.72X10^-12 erg  = -13.6 eV
13. E_n = E₁/n²             same atom different orbit
14. E_Z = E_H x Z²    same orbit different atom
15. 
16. A^o
17. Ön=a (Z – b), where n is frequency of X-rays given out by metal of at. No. Z
18. Average atomic weight =  
19. KE of photoelectric electron (½ mv²) = hn - hn_o (hn_o = work function) & n_o threshold frequency
20. deBroglie equation l =  
21. Uncertainty
22. (Schrodinger wave equation)  
23. Angular momentum of ‘e’ in an orbital=  
24. Magnetic moment of an atom=.  where n is no. of unpaired electrons
25. Nodes
    1. Radial (spherical) nodes=(n – l – 1)
    2. Angular (non spherical) nodes=l
    3. Total nodes= (n - 1) 
26. No of spectral lines =