ATOMIC STRUCTURE (QUANTUM MECHANICAL MODEL)

by clever academyJuly 08, 20173 Comments


Quantum numbers:


In the bohr theory of spectra, only one quantum number was postulated to describe the electron orbit. Experimental observation on fine structure and the splitting of lines in a magnetic field however showed that, for a complete description of the state of an electron in an atom, four quantum numbers are required. The four quantum numbers are describe below:

  1. Principal Quantum Number (n): This defines the main energy level the electron occupies, and can have any integral value (but not zero), thus, "n" can be 1, 2, 3, 4, 5,.....7. The size of an orbital depends on  "n". The larger the value of "n" is, the larger the orbits. 

  2. Subshell (or Azimuthal or Angular) Quantum number (l): This determines the angular momentum of the electron in its orbital motion around the nucleus. It also governs the shape of the orbital. This can have values l = 0, 1, 2, up to (n-1). The first four Subshell are identified by letters s, p, d, f and g which corresponding to l values of 0, 1, 2, 3 and 4 respectively.

  3. Magnetic Quantum Number (Ml): In isolated atoms of the same elements with the same values of n and l have the same energy. However, within the Subshell, it is possible for these electron to occupy different region of space when the substance is exposed to a magnetic field. The number of allowed values of ml is limited and depends upon l (and therefore on n); ml can have values of l, l-1, ......0, -1,.....-l. For example, if l=2; ml can have 2, 1, 0, -1, -2. The number of orbitals in each Subshell equals the number of values if ml which is given by (2l +1).

For examples,

         if l=0, this is an s-orbitals, ml=2l+1=1
Therefore, there is one s-orbitals

         if l=1, this is a p-orbital, ml=2l +1=3
Therefore, there are three p-orbital

         if l=2, this is a d - orbital,  ml=2l+1=5
Therefore, there are five d - orbitals

         if l=3, this is an f - orbital,  ml=2l+1=7
Therefore,  there are seven f-orbitals.

An orbital can take a maximum number of 2 electrons, this implies that s, p, d and f orbitals can take maximum of 2, 6, 10 and 14 electrons into their one, three, five and seven orbitals respectively.

        4. Spin Quantum Number (Ms): An electron within an orbital has angular momentum. We can imagine an electron as spinning on its own axis as the earth orbits the sun. This spinning causes each electron to behave as a tiny magnet. Electron spin has two possible orientation and the two values of ms are +1/2 or  -1/2, corresponding to the two electrons. This simply explains that two electron cannot move in the same direction , that is if one move 180 degree clockwise the other will move 180 degree counterclockwise. 


                    PAULI EXCLUSION PRINCIPLE


This states that "no two electrons can have a me set of four quantum numbers". By this we mean that no two electrons in any atom behave in an identical manner. Consider Beryllium atom in its ground stare,
                            Be= 1s2 2s2
The two outermost electron are contained in the n=2 energy level (2s2). Since they are in s-orbitals, it implies l=0, ml=0. Therefore, the spin quantum number cannot be the same, that is if one electron moves clockwise , the other moves counterclockwise.
Using this quantum number it is possible to determine the number of electrons each that the main energy level can accommodate.

 In addition, pauli is telling us that two electrons cannot have all four quantum number (that is if two electrons have the same principal, azimuthal and magnetic quantum number, they can not have the same spin quantum number).

NOTE: The total number of electron in an 'n' energy level is given by 2n2.



                            WORKED EXAMPLES

Question 1

Write the designation for the orbital occupied by an electron described by the following quantum numbers. What values of ml and ms could each electron have

(i) n=3, l=1              (ii) n=4, l=0                     (iii) n=3, l=2


Solution:
(i)       n=3, l =1
         l=1, this is a p-orbital
Thereford,  the orbital is 3p
Ml=-1 to +1
      =-1, 0, +1
Ms= +- 1/2

(ii)  n=4, l=0
l=0, this is an s-orbitals
The orbital is 4s
Ml= 0; ms=+-1/2

(iii) n=3, l=2
l=2, this is a d - orbital
The orbital is 3d
Ml=-2 to +2
      =-2, -1, 0, +1, +2
Ms=+-1/2

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