Schrodingers Time Independent Wave Equation

 

Debroglie hypothesis

As the Newtonian classical approach to deal with problems involving particles at atomic level failed, the concept of wave mechanics was introduced by Louis Debroglie in 1924 and has been successfully applied to atomic and molecular structures.

Louis Debroglie, a French scientist extended the idea of dual nature of radiation to matter. 

According to Debroglie, a materialistic particle posessess both wave and particle characteristics.  

It is to be known that dual nature of light was explained by combining Planck's expression for energy of photon 

E = h𝜈  --------------------(1)

Einsteins Mass energy relation 

E = mc² ------------------(2)

h𝜈 = mc²------------------(3)

as we know,

𝜈 = C/ƛ  -------------------(4)

From (3) & (4)

ƛ = h/P   --------------------(5)

Just in analogy with equation 5, Debroglie proposed concept of matter waves according to which a material particles of mass M moving with a velocity 'v' should have an associated wave length  ƛ called Debroglie's wavelength given by 

ƛ = h/mv = h/P ----------(6)

'm' - according to theory of relativity is not an invariable quantity as in classical physics.

m = m₀/√ 1-(v²/c²) ------ (7)

If particles like electrons are accelerated to various velocities, we can produce waves of various wavelengths. The higher the velocity of electron, the smaller the debroglie's wavelength.

If electron is accelerated by applying potential V, then work done on electron is eV.

mv²/2 = eV ----------------(8)

mv = P = √ 2meV --------(9)

substituting (9) in (6), we have

ƛ = h/√ 2meV  ------------(10)

 

Kinetic Theory of Gases - Important Points

1. The cohesive force (which binds molecules together) between particles of matter which constitutes a gas is extremely small.

2. Mathematical basis of Kinetic Theory of gases was established by Maxwell and Clausius.

3. Kinetic theory of gases relates macroscopic properties with microscopic properties of its molecules. 

 According to this theory,

4.  Molecules are rigid, perfectly elastic and identical in all respects.

5. The average distance traveled by a molecule between two collisions is known as Mean free path.

6.  On the basis of kinetic theory of gases, the pressure exerted by a gas is given by 

                                              P = (1/3)*(m*n*c²/)/V

where, 'n' is no. of moles, 'm' is mass of each molecule and 'M' is total mass of gas, V is volume of gas, c is average speed of molecules, c² is Mean square speed of molecules

c² = (c₁² + c₂² + .......+ cn²)/n

7.  Pressure exerted by a gas 'P'  is 2/3rd of total translational Kinetic energy of molecules per unit volume. 

8. Mean kinetic energy of a molecule is 

(1/2)mc² = (3/2)KT

where 'K' is Boltzmann Constant, K=R/NA  ; R is universal gas constant & NA  is Avagadro's number. 

9. Kinetic energy of a gram molecule of a gas = (3/2)RT

10. Avogadro's Number  

The number of atoms or molecules in a mole of substance is called Avogadro's number. 

Its value is 6.023 x 10²³ mol⁻¹ .

11. Kinetic energy of a molecule depends upon absolute temperature 'T' and it is quite independent of its mass. This fact is known as Kinetic interpretation of temperature.

12. According to kinetic theory of gases, at absolute zero of temperature, the Kinetic Energy of gas becomes zero i.e. molecular motion ceases.

13. Above point is strictly not true because at T=0, the molecules do have some energy known as Zero Point energy.

14. Deduction of gas laws from kinetic theory:

i) Boyles law

  P = (1/3)*Mc² ; PV=constant; 

At constant temperature, pressure of a gas is inversely proportional to volume of a gas.

ii) Charles Law

PV ∝T ; The volume of gas at constant pressure is directly proportional to the temperature (or) pressure of a gas at consatnt volume is directly proportional to temperature.

iii) Avogadro's law

N1=N2; Equal volume of ideal gases existing under same conditions of temperature and pressure contain equal number of molecules. This is called as Avogadro's law.

iv) Dalton's law or partial pressure

P=P1+P2+ ....

the total pressure exerted by gaseous mixture is sum of individual pressures that would be exerted if several gases occupied space in turn, alone.

v) Grahams law of diffusion

The rate of diffusion of a gas through a porous portion is inversely proportional to square root of its density.

Root mean square velocity (crms) = √c² = √(3KT)/m = √(3P)/ρ

15. Law of equipartition of energy (deduced by James Clark Maxwell)

i) Average value of the components of velocity 'C' (i.e. u,v & w) along 3 directions should be equal or for a molecule all 3 directions are equivalent i.e. u=v=w

ii) Total mean kinetic energy of molecule is E = (3/2)KT; 

     K is Boltzman constant and T is absolute temperature.

iii) A molecule has "three translational" degrees of freedom.

iv) Total Kinetic Energy of a dynamical system is equally divided among all its degree of freedom and it is equal or (1/2)KT per degree of freedom.This is called Law of eqipartition of energy. 

v) For a "monoatomic molecule", we have only translational motion because they are not capable of rotation. Thus for one molecule of a monoatomic gas total energy E=(3/2)KT. 

vi) For a "diatomic molecule" we can suppose it to be two sphere joined by a rigid rod. Such a molecule can rotate about any one of 3 mutually perpendicular axes. The rotational inertia about an axis along rigid rod is negligible compared to that about an axis perpendicular to rod, so rotational energy consists of to terms such as (1/2)I𝓌y² & (1/2)I𝓌z².

vii) For special description of center of mass of a diatomic gas molecule, 3 coordinates will be required. Thus, for a diatomic gas molecule having both rotational & translational motion; 

E = (3/2)KT + 2(1/2)KT = (5/2) KT

viii) For "tri-atomic gases", each molecule contains 3 spheres joined together by rods so that molecule is capable of rotating energetically 3 mutually perpendicular axes. Hence, for triatomic molecule having both translational & rotational motion energy 'E' will be 

 E = 3(1/2)KT + 3(1/2)KT = 3KT

ix) If a molecule in all has 'f' degrees of freedom, its average total energy would be (1/2)fKT. 

About Plastics

What is Plastic?

Plastic is a polymer in which monomers are arranged in linear or cross linked chains.

Alexander Parkes is the creator of first plastic known as “Parkesine”.

Types of Plastics:

a) Thermoplastic 

This is the plastic which gets deformed easily by heating. Some of the thermoplastics are polythene and PVC. This is a plastic which will soften when heated and harden when cooled. Thermoplastic is a polymer that turns into a liquid when heated and freezes to a glassy state when cooled sufficiently.

b) Thermosetting Plastics 

These are the plastics which moulded once cannot be softened/deformed by heating. Bakelite and Melamine are such plastics.

 Properties of Plastics

1.  Plastics are easily mouldable into different shapes.
2. Plastics could be recycled.
3. Plastics could be melted.
4. Plastics could be rolled into sheets.
5. Plastic could be drawn into wires
6. Plastics are nonreactive to chemicals and atmosphere.
7. Plastics are light in weight
8. Plastics are cheaper than metals.
9. Plastics are poor electric conductors and could be used as insulators.
10. Plastics are poor thermal(heat) conductors and hence could be used for cooking utensils etc.

Biodegradable and non-biodegradable Plastics

Plastics, which are decomposed by natural process through bacteria in the presence of water, sunlight and oxygen are called bio-degradable plastics.   Plastics which cannot be decomposed by natural processes are called non-biodegradable plastics.

Plastics take long time for decomposition which causes pollution of environment. Burning of plastics causes poisonous fumes into air.

SOUND - Important Notes

1. A vibrating body produces sound.
2. Sound has energy
3. Sounds are produced by vibrating bodies and the air that passes through orifices of the instruments.
4. Sound needs a medium to propagate.
5. The to and fro motion of a body from its mean position is known as one vibration.
6. The maximum displacement of vibrating body from its mean position is called as amplitude.
7. The number of vibrations per second is called as frequency.
8. The shrillness of a sound is known as pitch.
9. The pitch of the sound depends upon its frequency.
10. Normal sound consists of mixed frequencies.
11. The sounds which are pleasant to hear are called as music. Music is a combination of sounds that are produced in an order and pleasant to hear.
12. The sounds which are unpleasant to hear are called as noise.
13. The sounds that a normal human being can hear are called as audible sounds.
14. The sounds that a normal human being cannot hear are called inaudible sounds.
15. Frequency of the audible sounds ranges from 20 Hertz (20 vibrations/sec) to 20000 Hertz(20000 vibrations/sec).