Kirchoff's Law for black body

This law states that at a given temperature, the ratio of emission power to the absorption power for a given wavelength is same for all bodies and is equal to the emissive power of a black body at the same temperature.

Energy emitted per square meter is eƛdƛ.

Energy absorbed per unit area is QƛdQ.

eƛ/aƛ = ∂Qƛ /∂ƛ = constant

where eƛ is emission power ; aƛ is absorptive power

∂Qƛ is energy falling on unit area per second within same wavelength range & temperature.

 

(∂Qƛ /∂ƛ) depends only on temperature and for a given temperature it is constant.

For a perfectly black body, aƛ = 1

Note: Kirchoffs law is true only for temperature radiation.

Pyrometers - important points for exams

1. They are instruments used for measuring high temperatures.

2. The Pyrometers based on the principles of radiation are termed as Radiation Pyrometers.

3. The fact that the radiation emitted by a black body depends only upon its temperature provides radiation methods for measuring high temperatures. 

4. There are two types of Radiation Pyrometers

i) Total Radiation Pyrometers
The instruments that measure total radiation emitted by the body under test are called as total Radiation Pyrometers.
The temperature is determined by making use of Stefans law.

ii) Optical or Spectral Pyrometers
The optical pyrometers compare the intensity of radiation of a certain wave length emitted by the body with that of radiation of same wavelength emitted by a standard body at known temperature. The temperature of the body obtained by applying Weins displacement law or Planck's law. 

5. Advantages
a) They can be used to measure any high temperatures even when the hot bodies inaccessible.
b)They need not be put in contact with the hot body not raised to the temperature of the body.
c) There is no difficulty in extrapolation because radiation laws are valid at all temperatures.  

6. Disadvantages
a) Their range roughly begins at 600 degC and so temperatures below this limit cannot be measured.
b) The temperature obtained by these Pyrometers is always less than the actual temperature of source because they use radiation laws which are only true for black bodies. So the measures are accurate for black bodies only.

7. Ferry's total Radiation Pyrometer

The distance of pyrometer from hot body should be such that image formed by concave mirror is larger than opening of mirrors.

Calculation of temperatures

Let T & T0 be the absolute temperatures of the hot body and that of pyrometer case.

Now, according to Stefans Boltzmans law, the millivolt meter deflection 𝛳 will be proportional to T⁴-T0⁴.

As T0 is very small compared to T, 𝛳 ∝ T⁴.

Note: In practice it is found that power lies between 3.8 and 4.2 due to stray reflections from the walls etc.

𝛳 ∝ T^K 

Log𝛳 = K log T

The graph drawn between log𝛳 & logT comes out to be a straight line calibration graph.
To measure a high temperature a rotating sector is used which allows only radiation equal to 𝛳/360 to enter pyrometer.

If a pyrometer measure a temperature T1, then the actual temperature T2 of the body is given by

 T1⁴/T2⁴ = 𝛳/360;   T2 = T1 (360/𝛳)

8. Disappearing filament Optical Pyrometer

First Morse made this device later Holborn and Kurlbaum improved it.

The current through filament is adjusted in such a way that filament just disappears against background. Current in filament is recorded with help of ammeter. Since both filament & image are equally bright, they must be emitting equal amounts of energy per unit area per second and hence, they must be at same temperature.

The temperature of filament 'T' (or unknown temperature of hot body) can be calculated from the formula 

I = a+bT+CT2, where I is strength of current flowing through the circuit and a,b,c are constants

The value of these constants can be determined by calibrating the instrument against known temperatures.

Note: The instrument is suitable for measuring temperatures from 600 to 1500 degC. The range can be raised to 2700 degC by using a rotating sector.

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.