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Saturday, July 30, 2016

Applications of Molecular Orbit Theory

Electronic configuration and Molecular Behavior:
  • The distribution of electrons among various molecular orbitals is called the electronic configuration of the molecule.
Important information of configuration:
1. Stability of Molecules:
  1. The molecule is stable if  NB is greater than NA
  2. The molecule is unstable if NB is less than NA
    Where, NB → Bonding e- s NA → Anti-Bonding e- s
2. Bond order:
  • Bond order is defined as one half the difference between the number of electrons present in the bonding and the anti-bonding orbitals.Bond order = ½ [NB - NA]
  • The rules discussed in the above regarding the stability of the molecule can be restated in terms of Bond order as follows.
    • A positive Bond order means a stable molecule while negative (or) zero bond order means an unstable molecule.
    • Nature of Bond: Integral bond order values of 1, 2 (or) 3 correspond to single, double (or) triple bonds respectively.

3. Bond Length:
  • The bond order between two atoms in a molecule may be taken as an approximate measure of the Bond length.
  • Bond length decreases bond order increases.

4. Magnetic Nature:
  • If all the molecular orbitals in a molecule are doubly occupied the substance is diagrammatic.
  • If one (or) more molecular orbitals are singly occupied it is paramagnetic.
    EX: O2 Molecule
Bonding in some Homonuclear diatomic molecules
1. Hydrogen Molecule (H2):
  • There are 2 electrons in hydrogen molecule which are present in σ1S molecular orbital so electronic configuration of hydrogen molecule is H2 : (σ1S)2
    The bond order of H2 molecule can be calculated as given below
    Bond order = [NB - NA] / 2
  • Since there is no unpaired electron in hydrogen molecule, therefore it is diamagnetic.
2. Helium molecule (He2):
  • In (He2) molecule there are 4 electrons. These electrons will be accommodated in σ1S and  σ*1S molecular orbital leading to electronic configuration.
    He2 : (σ1S)2 (σ*1S)2
  • Bond order He2  is ½ (2-2) = 0
  • He2  Molecule is therefore unstable and does not exist.
3. Lithium Molecule (Li2):
  • The electronic configuration of lithium is 1S2,2S1. There are six electrons in Li2 the electron configuration of Li2 molecule therefore is Li2 : (σ1S)2 (σ*1S)2 (σ2S)2
  • The above configuration is also written as KK (σ2S)2 where KK represents the closed K shell structure (σ1S)2 (σ*1S)2  
  • There are four electrons present in bonding molecular orbitals and two electrons in anti-bonding molecular orbitals.
    Bond order ½ [4 - 1] = 1
  • It has no unpaired electrons so it is diamagnetic.
4. Oxygen molecule (O2):
  • The electronic configuration of oxygen atom is 1S2 2S2 2P4 each oxygen atom has 8 electrons hence, in O2 molecule there are 16 electrons. The electronic configuration of O2 molecule therefore
    O2 : (σ1S) (σ*1S)2 (σ2S)2 (σ*2S)2 (σ2PZ)2 (π2P2x= π2P2y) (π*2P1x = π*2P1y)
  • From the electronic configuration of O2 molecule, t is clear that ten electrons are present in anti-bonding molecular orbitals and six electrons are present in anti-bonding molecular orbitals.
    Its bond order = ½ [NB - NA] = ½ [10 - 6] = 4
  • It may be noted that it contains two unpaired electrons π*2Px and π*2Py molecular orbitals.
  • O2 Molecule should be paramagnetic.
5. Nitrogen molecule (N2):
  • The electronic configuration of Nitrogen is 1S2 2S2 2P3, there are 14 electrons in N2. The electronic configuration of
    N2 : σ1S2 < σ*1S2 < σ2S2 < σ*2S2 < (π2P2x=π2P2y) < σ2P2z
  • Bond order of N2 = ½ [10 -4] = 3
  • There is no unpaired electron in the electronic configuration of N2
  • It is diamagnetic in nature

Friday, July 29, 2016

Law of conservation of energy

Energy is either created or destroyed but converts from one form to another.
At point A:
A body of mass ‘m’ is thrown up with a velocity ‘u’ from the ground level. Let us take the point as A. therefore it has only kinetic energy and potential energy is equal to ‘0’ at A.
Energy at A
K.E = ½ mu2
P.E = 0.
Total Energy = ½ mu2

At point B:
Now let the velocity of a body after reaching a height ‘x’ from the ground be ‘u’. Let us take that point as B.
So it is
K.E at x is ½ m (u’)2
P.E is mgx
Total Energy = ½ m (u’)2  + mgx.

At point c:
Now let us assume the body has reached the maximum height of ‘Hmax’. Let this point be ‘C’.
Here P.E is mgHmax
K.E is 0
At C i.e., at maximum Height
K.E = 0
P.E = mgHmax
According to law of conservation of energy at A, B & C must be equal. As, the gravitational force is a conservative force.
TA = TB = TC
½ mu2 = ½ mu2 + mgx = mgHmax

Thursday, July 28, 2016


Evaluation adjoint and inverse Matrix using determinants and elementary transformations
MINOR: Let A = [aij] be a square matrix of order n. Then the minor Mij of aij in A is the determinant of the square sub-matrix of order (n-1) obtained by leaving ith row and jth column of A
  1. If , we have
M11 = Minor of A11 = 2, M12 = Minor of A12 = —3,
M21 = Minor of A21 = —7, M22 = Minor of A22 = 4
COFACTORS: Let A = [aij] be a square matrix of order n. Then the cofactor Cij of aij in A is equal to (—1) i+j. times the determinant of the sub-matrix of order (n-1) obtained by leaving ith row and jth column of A.
If follows from this definition that
Cij =Cofactor of aij in A
= (-1) i+j Mij, where Mij is minor of aij in A
Thus, Cij = Mij if i+j is even
And, Cij = - Mij if i+j is odd.
  1. If , then we have
C11 = (-1)1+1 M11= M11 = = 2,
C12 = (-) 1+2M12 = - M12 = = 7
C13 = (-1)1+3 M13 = M13 = = 8,
C23 = (-1)2+3 M23 = - M23 = = 8 etc.
Let A = [aij] be a square matrix of order n and let Cij be cofactor of aij in A. Then the transpose of the matrix of cofactors of elements of A is called the adjoint of A and is denoted by adj A.
Thus, adj A = [Cij] T.
(adj A) ij = Cij = Cofactor of aji in A.
Find the adjoint of matrix.
We have,
Cofactor of A11 = s,
Cofactor of A12 = —r,
Cofactor of A21 = -q
And, Cofactor of A22 = p.
It is evident from this example that the adjoint of a square matrix of order 2 can be easily obtained by interchanging the diagonal elements and changing signs of off-diagonal elements.
If , then by the above rule, we have,
Find the adjoint of matrix
Let Cij be cofactor of aji in A. Then cofactors of elements of A are given by
PROPERTIES OF ADJOINT The following are some properties of adjoint of a square matrix which are states as theorems.
Let A be a square matrix of order n. Then
A (adj A) = |A|In = (adj A) A
Let A be a non-singular square matrix of order n. Then
|adj A| = |A|n-1
If A and B are non-singular square matrices of the same order, then
adj AB = (adj B) (adj A)
If A is an invertible square matrix, then
adj AT = (adj A) T.
If A is a non-singular square matrix, then
adj ( adj A) = |A|n-2 A
If A is a symmetric matrix, then adj A is also a symmetric matrix.

DEFINITION A square matrix of order n is invertible if there exists a square matrix B of the same order such that
AB = I„ = BA
In such a case, we say that the inverse of A is B and we write,
A-1= B
The inverse of a matrix has the following properties
Every invertible matrix possesses a unique inverse.
A square matrix is invertible iff it is non-singular.
The inverse of A is given by
  1. AB=AC => B=C
  2. BA=CA => B=C
Clearly, AB = BC but B ≠ C.
(Reversal law) if A and B are invertible matrices of the same order, then AB is invertible and
(AB)-1 = B-1 A-1
If A is an invertible square matrix, then
AT is also invertible and (AT)-1 = (A-1) T.
If the non-singular matrix A is symmetric, then A-1 is also symmetric.

If A is a non-singular matrix, then