Thursday, March 17, 2011

Quantum Mechanical Model

1. Overview of Theory of Quantum Mechanics
    The theory of quantum mechanics explains the behaviour of particles in the atomic and subatomic realms. These particles include photons (particles of light) and electrons. Since the electrons of an atom determine many of its chemical and physical properties, quantum mechanics is foundational to understanding chemistry.

    The most common way to describe electrons in atoms according to quantum mechanics is to solve the Schrödinger equation for the energy states of the electrons within the atom. When the electron is in these states, its energy is well-defined but its position is not. The position of an electron is described by a probability distribution map called an orbital.

2. Orbital
    A three-dimensional region in space around the nucleus where there is a high probability of finding an electron.
  

3. Quantum numbers
    Four quantum numbers are used to characterise the electron orbitals:
    (a) Principal quantum number (n) determines the energy (or energy level) of the electron and size of the orbital.
      n = 1, 2, 3, ... ... ∞
 
    (b) Angular momentum quantum number (l) (or azimuthal / subsidiary / orbital quantum number) determines the shape of the orbital.
     l = 0, 1, 2, ... ... (n-1)
     [The values of l depend on the value of principle quantum number, n]

    (c) Magnetic quantum number (m) determines the orientation of the orbital in space
     m = -l, ... ..., -2, -1, 0, +1, +2, ... ..., +l
     [The permitted values of m depend on the value of l, number of permitted value of m = 2l + 1]

    (d) Electron spin quantum number (s) determines the direction of spinning motions of an electron on its own axis (either clockwise or anti-clockwise).
     s = +1/2 or -1/2
 

4. Shapes of atomic orbitals
    (a) s orbital: spherical with the nucleus at the center. Electrons move anywhere within the sphere.
       

       
               As n increases, the size of s orbital increases.

    (b) p orbitals: dumb-bell shape. Have a node at the nucleus.
       

    (c) d orbitals:
         (i) Four d-orbitals (dx2-y2, dxy, dyz, dxz) have four lobes extending out perpendicular to each other.     
         (ii) One d-orbital (dz2) has two lobes extending out along the z-axis with a torus (doughnut-shaped ring) around the center on the x-y plane.
        

5. Degenerate orbitals
    All the three p orbitals and five d orbitals are degenerate orbitals. Degenerate orbitals are orbitals with same energy but differ in their orientation.

6. Relationship between quantum numbers
 

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