Quantum mechanics Definition:- Quantum mechanics is a

                     Quantum mechanics

Definition:-

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Quantum
mechanics is a fundamental theory which tells us about the nature of smallest
scale of energy of atoms and subatomic particles.

Mechanics:-

                    
The
branch of physics in which we study about the things under the different
forces. Motion is the primary characteristic of the object and other properties
we can explain with the help of mechanics.

History:-

               
In 1900 quantum theory developed for the study of microscopic particles.
Scientist develops the theory which called classical mechanics. It has two main
branches Newtonian mechanics and Maxwell electromagnetic theory. These theories
completely explain the structure of atom  
but for the study of subatomic particles these theories are limited.
After the failure of classical mechanics a new phenomena put forward by
Heisenberg in 1925 which is called matrix mechanics. In 1926 Schrodinger give
us the Idea of wave mechanics which depend upon the dual nature of matter.  So in 1925, 1926 a new theory was purposed
named as a quantum mechanics.

Plank’s quantum theory:-

                                                         
Max plank’s was German physicist 
who experiment on the spectral line emitting from the hot bodies at
various  temperature  in 1900 put forward  and gave name this plank’s quantum  theory of radiation. After that Einstein in
1905 extended more.

Postulate of quantum theory:-

                                                     
1. A hot body emits radiation that is not continuously but
discontinuously. In other words you can say that emission of radiation is not a
continues phenomena but it travel in small packets or bundles unit of wave.
These bundle can exist independently are called photon or quantum of light.

2.  The energy
contain each photon is directly proportional to the frequency of emitted
radiation.

             
E = h?

This equation is applied on all type of radiation
and this is called plank’s equation.

 

 3. The energy
absorbed or emitted by anybody it will be equal to one quantum of energy (=h?)
or the multiple of the whole number. It can never be the fractional value. 

Application of quantum theory:-

 

                                                               
In
1905 Elbert Einstein explains the phenomena of photo electric effect or
radiation. “When the beam of light is falls on the surface of alkali metals
electrons are emitted from the surface of metal this phenomenon is called photo
electric effect.”

Photoelectric emission of electron is only possible
when the frequency of light is ??
. This
is called threshold frequency .All metal has its different value below this
electron will not emit. Kinetic energy is directly proportional to the
frequency of light but does not change on the intensity of light .Increase the
number of electron as the intensity of light increase per unit time. This
phenomenon of photoelectric emission could be understood in terms of plank’s
quantum theory. He said that the electromagnetic radiation (light) has both
wave and particle like properties .each photon of light has energy equal to

                                                        
E = h?   

When a
photon of frequency  ?
strikes on the surface of metal it gives energy to the electrons which are
present on the surface of metal .A part of energy is utilize to liberate the
electron other stored as a kinetic energy so total energy equal to

                                                   
h? =  work function + K.E

                  
                     h? = ? + K.E

Importance of quantum mechanics:-

                                                                
These
days all the branches of science are correlate with the quantum mechanics.
Quantum mechanics was developed early in the 20th century when
scientist started search about smaller particle like atom or electron etc.
Classical mechanics explains the motion of large bodies. It could not explain
the properties of sub atomic particles. Classical mechanics explain that the
energy emitted or absorbed in the body in continues form. But according to the
quantum mechanics this phenomenon occurs in discontinues form. Classical
mechanics failed to explain the phenomena of heat capacity, BBR and
photoelectric effect etc.

Application of quantum mechanics:-

Ø  It
enables scientist to calculate the energy level and other properties of
molecules.

Ø  It
tells us about the periodic variation in the properties like ionization energy,
electron affinity and atomic size etc.

Ø  It
enables scientist to understand the nature of bond between the atoms and
molecules.

Ø  It
helps to measure bond angle, bond length and energy shell.

Ø  It
helps us to know about whether molecule is dipole or not. 

Ø  It
also provides the basis for understanding the result of spectroscopic measurements.

Ø  Quantum
mechanics also helps to determine the thermodynamics properties of bulk matter.

                                   Classical
mechanics

Definition:-

         In classical mechanics
we studied about the motion of large bodies or particles.

In this theory matter made up of
particle having certain mass. This theory is based on the Newton’s law of
motion. According to these law ;

1.      A
body will stay at rest or a body will move with constant velocity until a
unbalanced force act on that body.

2.      If
any force on the body acts in the direction of motion , an acceleration will
produce in the direction of motion  it
will directly proportional to the force acting on the body and inversely
proportional to the mass of the body

                                      F = ma

3.      Action
and reaction are equal in force but their direction is opposite.

Limitations
in classical mechanics:-

·       
It could not explain the motion of those
bodies whose velocity equal to the velocity of light.

·       
It could not explain the radiation emitted
from black bodies or hot mass.

·       
It could not explains the dual nature of
light e.g. wave and particles.

·       
It could not explain the atomic spectra
observed with different atoms.

 Difference between
classical and quantum mechanics

Classical mechanics:-

                               
I.           
It is apply on the macromolecules   like plants, stars etc.

                            
II.           
It obeys the Newton’s law of motion.

                          
III.           
It is based on the particle nature of
the object.

                         
IV.           
 It can measure 100% accurate.

                            
V.           
Two properties can measure
simultaneously e.g. position and momentum.

                         
VI.           
It tells us that energy emit or absorb
continuously.

                       
VII.           
It has large unit for the measurement
like km, kg etc.

                    
VIII.           
Classical mechanics fails to describe
the relation between matter and energy.

                         
IX.           
It cannot explain the new scientific
problem.

Quantum mechanics:-

                                
I.           
It deals with the study of micro
particles like atom, electrons, molecules etc.

                              
II.           
It obeys the Schrodinger wave equation.

                           
III.           
It is based on the dual nature of light.

                           
IV.           
It cannot measure 100% accurate.

                             
V.           
It cannot measure two properties at the
same time.

                           
VI.           
It tells us that energy is no longer
continuous.

                        
VII.           
It has small units like mm, nm, cm etc.

                      
VIII.           
It explains the relation between matte
and energy.

                           
IX.           
It is the science of future and explains
both physical and medical science.

                             Quantum
numbers

The four numbers which completely describe the
characteristics of an atom e. g.  Energy,
shape, orientation of an orbitals and spin of electron are called quantum
numbers. There are:

1. Principal Quantum number (n)                      2. Azimuthal Quantum
number (l)

3. Magnetic Quantum numbers (m)                 4. Spin Quantum number (ms)

1. Principal Quantum number (n):-

                                                                          
This is the quantum number which is denoted by ‘n’. It tells us distance
of the shell from the nucleus and its energy. It has any integral value 1,2,3,4
with respect to K, L, M, N, O etc respectively. The value of n tells the energy
level in which electron circulate around the nucleus. IF the value of n is
greater the energy will be high and vice versa. ‘n’ also gives us the energy of
electron in the shell.

2. Azimuthal Quantum number (l):-

                                                        
This
describes the shape of orbital. It has a value 0, 1, 2 ….n-1. The values cannot
exceed from 3 due to the limited of electrons. The sell is called s shell if
the value of l=0 and its shape is spherical. The shell is called p orbital if
the value of l=1 and it has a dumbbell shape. If the value of l=2 this orbital
is called d orbital. The orbital is called f orbital if the value of l=3.It is
used to represent the orbital or sub-shells. The value of l depends upon the
n-1.

3. Magnetic quantum number (m):-

                                                        
This
quantum number usually describes the magnetic characteristics of an electron
and the orientation of the orbital. Its value depend upon the values of l its
range from-l to +l zero is also included. For s sub-shell l=0and m=0 because s
shell has only one orientation in space and can b arranged along any axis. S
shell is not divided in further sub-shell. In case of p shell l=1 and m=0, ±1.
P sub shell has three space orientations along all the three axes X, Y, and Z
axis. They all are perpendicular to each other. For d sub shell l=2 and m=0,
±1, ±2.It has five orientation in the space. All of these are
not same or identical in shape.

4. Spin quantum number (ms):-

                                                                    
This quantum number is denoted by ms . It describes the
direction of the spin. Alkali metals have only single electron in their
outermost shell. Spectrum is emitted when electron jump from ground state to
exited state. This spectrum is consisting of pair of lines that is why it is
called doublet line structure. The electron while moving around the nucleus or
spin around their axis .The direction of spin may be clockwise or
anti-clockwise. It has value of +1/2 or -1/2. The possibility in every case
will be 50% clockwise and 50% anti clockwise.

                                         Chemical
bonding

Chemical bond:-

                            A
force of attraction which holds two atom or molecules to form a compound is
called chemical bond. The chemical combination of elements depends upon the electronic
configuration of the elements. The noble gases has very stable electronic
configuration that is why these gases show very less bonding.

                                  Valence
bond theory (VBT)

Definition:-

                  
A
covalent bond is formed by the overlapping of half filled orbitals in the
valence shell of the two atoms formed the pair of electron one from the each
atom. As a result of overlap the electron with opposite become paired to
stabilize them if the overlapping is large bonding will stronger.

     VSPER
theory tells us about the shape of molecules but does not give reasons for the
bond formation. But VBT explain both of these bond formation and shapes of
molecules.

 Postulate
of VBT:-

Ø The
atom which unites them through the bonding in the molecule should retain their
identity.

Ø  The formation of
covalent bond is due to the overlap of atomic orbital of two different atoms,
each having one unpaired electron .These electrons overlap with opposite spin.
This bond is localized between two atoms. If atomic orbitals are in same spin
then no bond formation takes place.

Ø If
the atomic orbitals have more than one unpaired electron the possibility of
formation of more than one bond.

Ø If
the valence electros are already paired then they cannot take part in bonding
un pairing electron required energy which release from the bond formation.

Ø The
strength of covalent bond is depending upon the overlapping of atomic orbitals.
Greater overlap greater will be the b0nd strength.

Explanation:-

                      The formation of
different types of overlapping has been discussed in VBT like s and so orbital,
s and p orbital and p and p orbital.

       In the
formation of H2 molecule from H atom 
the two identical s orbital of two different atoms show linear
overlapping  and the density if the
electrons become concentrated due to sigma bond which is formed between two H
atoms.

         A
bond formed when two half filled atomic overlap in such way that the
possibility of presence of electrons is between the centers of the both atoms
nuclei is called sigma (?) bond. 

                                           OR

A bond which is formed due to the head to head
overlapping of atomic orbitals.

When there is the bond formation both of these atoms
come interact with each other they are not distinguishable. There are two
possibilities

                                            H1A.
H2B? case A

                                            H2A
.H1B? case B

                                            ?1
= ?A (1). ?B (2)

                                                                
?2 =?A (2). ?B (1)

Here ? is a wave function.

True wave function is ?1+?2

                                           ? =
C1?1 + C2?2

C1 and C2 are the mixing
coefficients.

For H2 molecule C1 = ±C2

It means that if C1 = 1 and C2
= ± 1

Therefore the two possibilities for the wave
function. One corresponding to the C1 = 1 and C2  = 1 and the second is C1 = 1
and C2 = -1.

                     
? = ?A (1).?B (2) + ?A (2).?B (1)

                     
? = ?A (1).?B (2) _ ?A (2).?B (1)

Pauling has shown that the calculation of Es
suggested by Hitter and london can be improved by considering that the
additional ionic character in which both the electrons are attached to the same
atom. So new equation will be;

                       ?3 = ?A (1).
?A (2)

                       ?4 = ?B (1).
?B (2)

By adding these four equations we will get final
equation;

                  
? = ?A (1). ?B (2) +?A (2). ?B
(1) + ?A (1). ?A (2)+?B (1).?B (2)

                   
? = ? covalent + ? ionic

When the value of c governs the extent of ionic
character mixed with the covalent valence bond structure. This approvement of
wave function is called ionic covalent resonance and the approvement in energy
can be traced to the extra freedom it gives to the distribution of electrons.
The real values of c is about 0.25 which indicate that ionic structure
contribute

 of
the overall structure of

. In hetero-nuclei the molecule ionic
character are more important and in species like NaCl they dominate the
covalent character.

The formation of an electron pair in 

 is
based on the following assumptions,

·       
Each hydrogen atom has half-filled 1s
atomic orbital which contain one electron.

·       
Hitler and London tells us that the
electron are not easily understand and the wave function must explain this
fact.

·       
 The molecular wave function

 is
obtained by the linear combination of wave function of atomic orbital with the
concept of the resonance.

·       
The electrons in the molecule which are
pairing should satisfy the Pauli’s Exclusion principal and these electrons have
different values of spin quantum numbers.

·       
Electron pairing results to the formation
of attractive forces.

·       
The overlapping of orbitals have same
shape and energy, the orbital overlap with each other to specific extent. There
will be stronger bond formation with respect to greater overlap.

The overall concept by the description of 

 molecules is that the covalent bond is formed
by the interconvert of forces between a pair of electrons having different
spin.

Pi, ? Bond:

The bond which is formed due to the sidewise overlapping of two
half-filled atomic orbitals which are parallel to each other and the density of
electron are above and below the center axis is called pi ? bond.

Molecular Orbital Theory (MOT):

In Molecular orbital theory all the valence electrons in the
molecule are linked with all the nuclei. In other words all the valence
electrons have an effect on the stability of the molecule. MOT considered that
when the molecule is formed the valence shell atomic orbital stops to exist.
They are replaced by a new set of energy levels with the related new density
distribution.

“An
electronic energy level in a molecule and the corresponding density in the
space is called Molecular orbital”

For example we take hydrogen molecule which has two identical
atoms each having one electron in their s-orbital. They are named as 

 and 

 and their wave function will be 

 and 

 the effective overlap will occur only when

·       
The orbital have similar energy state

·       
The orbital overlap to specific extent

Orbital have same shape

Where

and

 are mixing coefficient the can be replaced by
single coefficient ?

Two possible molecular orbital wave functions are obtained from
addition and subtraction

By taking square of both of these equations we have

 

 

+

+

+           +

                           +                                                                       

 

 

 

 

+

+

+

                        –

                                                                                                                       

1s

1s

?

?

??

Energy

 

 

 

 

 

 

 

 

 

 Molecular orbital is formed when the addition
of two 1s atomic orbital having the same spin overlap with inter nuclear axis
such overlap leads to reinforcement of the wave function between two nuclei the
probability of finding electron between the two nuclei. The bonding molecular
orbital

 having lower energy than that the atomic
orbitals.

Both bonding and anti-bonding molecular orbital having
cylindrical symmetry about their inter nuclear axis these orbitals are known as
sigma orbitals and these bonds are called as sigma bonds.

Bond Order:

The bond order is
defined as one half difference between the number of electron in the bonding
and anti-bonding molecular orbital in the molecule.

                                                                                                               

Bond order =

The bond order in

=1

Effective combination of atomic orbitals
and its conditions:

       
I.           
The atomic orbitals which are combing
together should have same energy and similar magnitude In case of formation of
a homo-nuclear diatomic of 

 types and the

-AO of atom 

 will not be combined with

-AO of another atom 

 of
same element, where  

 and

 are
two atoms of molecule A2 since their energies are differs from each other so
the energy gap between 2s and 2p atomic orbital is too high They will not
combine. In case of formation of hetero nuclear diatomic molecule of 

 type can be formed.

    
II.           
The density of atomic orbital must be
overlapped with each other if they are going to form molecular orbital this
condition is known as Principal of Maximum overlap. The atomic orbitals should
have same symmetry about their axis this condition is called as symmetric
condition for the combining of atomic orbitals.

On
the basis of symmetric conditions it is stated that the sum of atomic orbitals
which have same energy will overlap but cannot combine to the molecular
orbital. The molecular orbital cannot be formed by the overlapping of s-orbital
of one atom and p-orbital of another atom which is perpendicular to the
molecule axis. It can be 

 and

.

The non-formation of
Molecular orbital is due to the un-symmetry of s and p orbitals it can be said
that + + is neutralized by the + – overlap

 

x

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