Stefan's -Boltzmann Law

This law states that the total amount of radiant energy emitted by a black body per second per unit area is directly proportional to the fourth power of its absolute temperature i.e. E∝T⁴ or E=𝜎T⁴ where 𝜎 is called Stefan's constant. It has a value of 5.67 x 10⁻⁸ Wm⁻²K⁻⁴. This law is strictly true only when the medium surrounding the black body is vacuum.

The same law was established later by Boltzmann theoretically from thermodynamical considerations. Hence, this law is known as Stefan-Boltzmann law.

Consider the case of black body 'A' at absolute temperature 'T1' which is surrounded by another black body at absolute temperature 'T2'.

Now, 

Heat lost by black body 'A' is 𝜎T1⁴ 
Amount of heat absorbed by black body 'A' from black body 'B' is  𝜎T2⁴.

Therefore, Net amount of heat emitted by body 'A' per second per unit area is 𝜎(T1⁴ - T2⁴).

This is the form of "Stefans Boltzman Law". 

Note: This law is true only when medium surrounding the body is vacuum.

What is Statistical Mechanics?

When we consider bodies at macroscopic level they consist of uncountable atoms or molecules i.e. about 10²³ atoms/gm.mole. In such cases we cannot predict the result of interactions between atoms with the help of ordinary classical laws of motion.

Statistical Mechanics is the branch of Science which establishes the interpretation of macroscopic behaviour of system in terms of its microscopic properties.

The main theme is that it doesn't deal with motion of each particle but it takes into account the average or most probable properties of system without going into interior details of characteristics of its constituents.

The larger is the number of particles in the physical system considered, the more nearly correct are the statistical predictions. The smaller is the no. of particles (no. of degrees of freedom) in a mechanical system, the methods of mechanical system cease to have meaning.

Before the advent of quantum theory Maxwell, Boltzmann, Gibbs etc applied statistical methods making the use of classical physics. These statistical methods are known as Maxwell Boltzmann Statistics.

These statics explained successfully many observed physical phenomenon like temperature, pressure, energy etc; but couldn't explain adequately several other experimentally observed phenomenon like black body radiation, specific heat at low temperature etc.

In order to explain such phenomenon "quantum statistics" was introduced and developed by Fermi, Dirac, Bose, Einstein with new approach by using new quantum idea of discrete exchange of energy between system.

i) Bose-Einstein Statistics
ii) Fermi-Dirac Statistics

Atomic Structure - Important Points for competetive exams

1. Distance of closest approach: It is the distance from which the nucleus of an atom, the alpha particle comes to rest and its kinetic energy is totally converted  into electrostatic potential energy. It is denoted by ro.

ro = (1/4πεₒ)*[(2ze²)/(1/2)*(V²m)]

2. Diameter of atom: 10⁻¹⁰ meter

3. Diameter of Nucleus: 10⁻¹⁴ meter 

4. Impact Parameter(b):  (1/4πεₒ)*[(ze²tan𝜃)/(1/2)*(U²m)]; U is velocity of alpha particle

5. Impact Parameter(b) is inversely proportional to the angle of scattering(𝜃).

6. The equation mvr =n*(h/2π) is called "Bohrs quantisation condition".

7. The equation h𝝂=Ei-Ef is called "Bohr's Frequency condition".

8. Bohr's Radius r = (n²h²εₒ)/πme²

9. Velocity of electron (V) = e²/2nhεₒ 

10. If 'C' is velocity of light; V = [(1/4πεₒ)*(2πe²/Ch)]*(C/n)

11. The factor [(1/4πεₒ)*(2πe²/Ch)] is called fine structure constant. It is denoted by '𝛼'

12. The value of 𝛼=1/137; V = (1/137)*(C/n)

13. Energy of electron En = -(1/4πεₒ)²*(2π²me⁴/n²h²)

14. An electron can have only some definite values of energy while revolving in the orbits n=1,2,3,..... It is called energy quantization.

15. Energy Quantization: 

      E1 = -(1/4πεₒ)²*(2π²me⁴/n²h²) ;
      E2 = (1/4)*E1
      E3 = (1/9)*E1   ............................E∞ = 0

      E = -13.6/n²

 16. Rydberg's constant for Hydrogen (RH) is (1/4πεₒ)²*(2π²me⁴/ch³). Its value is 1.09678 x 10⁷m⁻¹

 17. Value of (1/4πεₒ) is 9x10⁹ C²N⁻¹m⁻².

 18. The charge 'e' of the electron is measured by Millikan's Oil drop method.

 19. The ratio of charge to mass(e/m) for an electron is measured by "Thomson".  

 20. Mass of electron(m) = 9.1 x 10⁻³¹ Kg

 21. Mass of Proton is 1835 times that of mass of electron.

 22. Canal Rays or Positive Rays are discovered by "E. Goldstein". Wien observed that these rays can be deflected in magnetic field and hence they are called Positive Rays.   

23. Rest mass energy of electron is 931 MeV

24. Orbital frequency of electron is (1/T) = (V/2πr)

25. Ionization energy of a Hydrogen atom is 13.6 eV

26. The excitation energy required by the electron to excite from state n1 to state n2 is En₂-En₁

Spectral series of Hydrogen atom

27. Lyman series lie in Ultravoilet region.

28. Balmer series lie in near UV region and visible region.

29. Paschen series lie in infrared region.

30. Brackett series also lie in infrared region. 

31. Pfund series also lie in far infrared region.

                                                          (1/) = R[(1/nf²)-(1/ni²)]

In addition to the above, nf=6, Humprey series results.

32. Velocity of an electron is independent of its mass.

33. Velocity of an electron is inversely proportional to the orbit.

34. The electron in the inner most orbit has highest velocity.

35. Velocity of a electron is independent of its mass.   

36. Orbital frequency is inversely proportional to the cube of 'n' i.e. 𝜈∝(1/n³).

37. If Ep & Ek represents Potential & Kinetic energies of the orbital electron, then Ek = -Ep/2.

38. When a Hydrogen atom is raised from the ground state to an excited state both kinetic energy and potential energy decrease.E∝(1/n²).

39. The energy difference between two consecutive energy levels decreases as the quantum number 'n' increases.

40. Bohr used conservation of angular momentum to explain his theory.

41. The velocity of an electron in the ground state is e²/2hεₒ = 2 x 10⁶ m/sec 

42. The ground state energy of Hydrogen atom is -13.6 eV. The energy needed to ionise the Hydrogen atom from its second excited state is 1.51 eV.

43. According to Bohr's principle, the relation between principle quantum number(n) and radius of orbit is  r ∝ n²

Heat Transfer due to convection

In this type of heat transfer, molecules of fluids move up bodily due to heating. Such heat transfer occurs when a fluid such as air or water comes in contact with an object whose temperature is higher than that of fluid. As temperature of fluid in contact with hot body increases, the fluid expands and thus becomes less denser and due to buoyant forces it rises & the position is occupied by cooler surrounding fluid and the process continues.

"Convection" is part of many natural process. Atmospheric convection plays an important role in determining global climate patterns & daily weather changes.

The rate of heat transfer by convection depends on the temperature difference between the surfaces and also on their areas. 

Heat Conduction

In this mechanism, heat transfer is due to vibration amplitudes of molecules & atoms present in solids.

Consider a cubicle of solid. Let us maintain one face of cube at high temperature (TH) and other opposite face at low temperature(Tc).

Due to temperature difference an amount of heat energy (Q) passes from  hot face to cold face in time 't'.

Conduction rate  Pcond (amount of energy transferred for uni time) is

Pcond = Q/t = K*A*(TH-Tc)/d;

where 'K' is coefficient of thermal conductivity, a constant for given material.
           'd' is thickness of slab
           'A' Area of slab
            't' is time of conduction

Therefore, Q= K*A(TH-Tc)*t/d

Note: i) 'K' depend on nature of material of which slab is made
         ii) A good thermal conductor has 'K' greater value.

Thermal Resistance to conduction(R-value):

This explains resisting of thermal conductivity. The R-value(thermal resistance) of a slab of thickness 'd' is defined as R=d/k. Thus material having less value of 'K' will have higher R-value and thus acts as a good thermal insulator.

Note:

i) 'R' is properly assigned to specified thickness of slab but not to material of slab.
ii) In steady state, conducting rates thru any no. of materials must be equal.
     Therefore, Pcond = A*(TH-Tc) / Σ(d/K)
iii) Heat is transferred from molecule to molecule by conduction. In this case molecules do not bodily move but simply vibrate.

Black body and its Radiation

A perfectly black body is the one which absorbs all the radiations of all wavelengths incident on it. Since it neither reflects not transmits any radiation it appears black in color what may be the color of incident radiation.

According to Kirchoffs law, a body which is capable of absorbing radiation must also be capable of emitting all possible wavelengths. So a perfectly black body is a good absorber as well as good radiator. When it is heated to a suitable high temperature, it emits radiations of all wavelengths (continuous spectrum). As the radiations emitted by black body is rich in maximum possible wavelengths and hence such Radiations are known as full Radiation or Total Radiation.     

The wavelength of emitted Radiation by a black body depends only on its temperature and is independent of the material of the body.

There is no body acting as perfect black body. The nearest approach is lamp black or platinum black. These are capable of absorbing the visible and a part near infrared but far infrared (heat Radiation) are reflected. So perfectly black body is just an ideal concept. For all practical purposes a lamp blacked surface can be considered as perfectly black body.

Energy Distribution in Black body Radiation

The distribution of energy in black body radiation for different wavelengths and at various temperatures was determined experimentally by Lummer and Pringsheim in 1899. They used the black body as an electrically heated chamber with narrow aperture.

The temperature of heated enclosure is measured by thermocouple.

The parallel beam of Radiation is allowed to incident on a "fluorspar prism" instead of a glass prism. The reason behind not using glass prism is that it absorbs some heat radaition.

The radiation is detected by means of Bolometer. Bolometer is an instrument to detect Thermal Radiation. The Bolometer is a linear type due to Lummer and Kurlabaum and is fitted with galvanometer 'G'. The deflection produced in the Galvanometer gives the intensity of Radiation, Eƛ. This is defined such that quantity Eƛ.dEƛ is the energy, for wavelengths lying between ƛ and ƛ+dƛ emitted per second per unit surface area of black body.   

The wavelengths at different parts of the spectrum was calculated by "Prism Dispersion Formula".

The experiment results are as follows:

1) The emission from a Black body at any temperature is composed of Radiation from all wavelengths.

2) At a given temperature, the energy is not uniformly distributed. As the temperature of the black body increases, the intensity of radiation for each wavelength increases. This shows that the total amount of energy is radiated per unit area per unit time increases with rise of temperature.

i.e., T 𝛼 Eƛ

3) The total energy of radiation at any temperature is given by the area between the curve corresponding to that temperature and horizontal axis. The increase in area found in accordance with Stefans law.

4) The amount of radiant energy emitted is small at very short and very long wavelengths. At a particular temperature, the spectral radiance Eƛ is maximum at particular wavelength ƛm. Most of the energy is emitted at wavelengths not very  different from ƛm.

5) The wavelength corresponding to the maximum energy represented by the peak of the curve shifts towards shorter wavelengths as the temperature increases. This is called Wiens Displacement. According to this law ƛm x T = constant.

This shows that as the temperature is increased, the black body emits the radiation of shorter wavelengths such that the product of temperature 'T' and maximum wavelength ƛm is a constant. 

The constant is called Wiens Displacement constant and has value 0.2896 x 10⁻² mK. 

There is change in wavelength due to Doppler effect.

Laws Related to Black Body:

a) Kirchoffs Law
b) Stefan-Boltzmann law
c) Wiens Law
d) Rayleigh-Jeans Law
e) Plancks Law

Important points to be noted:

i)  Wiens formula agrees in short wavelength region.
ii)  Rayleigh-Jeans formula agrees for long wavelength region.
iii) Plancks formula covers the entire region.
iv) When radiation is passed through a black body is passed through a prism, acontinuous spectrum is obtained. The energy is distributed in various wavelengths varying from 0 to infinity.
v) The law that connects the intensity with the wavelength is known as law of distribution of intensity of black body radiation.
vi) According to Stefans law, u=𝝈T⁴ where 𝝈 is Stefans constant and 'u' is energy density.