Kinetic Theory of Gases and Radiation PDF

KINETIC THEORY OF GASES AND RADIATION  Assumption Kinetic theory of Gases 1)A gas consists of a large number of tiny particles called molecules. 2)The molecules are perfectly elastic sphere of very small diameter. 3)All the molecules of same gas are identical in shape, size and mass. 4)The m...

KINETIC THEORY OF GASES AND RADIATION  Assumption Kinetic theory of Gases 1)A gas consists of a large number of tiny particles called molecules. 2)The molecules are perfectly elastic sphere of very small diameter. 3)All the molecules of same gas are identical in shape, size and mass. 4)The molecules are always in state of random motion. 5)Due to there elastic collision, they collide with each other & also wall of container with no loss of K.E. during collision. 6)Actual volume occupied by a gas molecule is very small compared to total volume occupied by the gas. 7)Between two successive collision, molecule travel in straight line with constant velocity called free path. 8)The time taken for collision is very small as compare to the time required to cover free path between two successive collision. 9)Average K.E. of gas molecules is directly proportional to the absolute temperature Free path- “The distance travelled in straight line by the molecule between two successive collisions is called free path.” Mean free path- “The average distance covered by the molecule between two successive collisions is called mean free path.”  If r = 0 and a = 0 , then tr = 1 , which means all the incident radiation transmitted through the object. The object is perfect transmitter and is completely transparent to the radiation. The substance through which the radiation can pass called diathermanous substance. tr ≠ 0 A diathermanous substance neither good absorber nor good reflector. Example: glass, quartz, sodium chloride, hydrogen, oxygen, dry air. If tr = 0 , a + r = 1 , the object does not transmit any radiation. It is said to be opaque to the radiation. This type of substance called Athermanous substance. Example: water, wood, iron, copper, moist air , benzene etc. If tr = 0 and a = 0 , then r = 1, all the incident energy is reflected by object. It is a perfect reflector. A good reflector is a poor absorber and poor transmitter. If r = 0 , and tr = 0 , then a = 1 , all the incident energy absorbed by the body. Such an object is called perfectly black body. Explain Ferry’s perfectly black body  It is a double walled hollow metallic sphere having a small aperture through which heat can enter. The space between the walls is evacuated. The inner surface has a small conical projection in front of aperture. it’s interior is coated with lamp black. The hole is directed towards the source of radiation. Any heat which is incident along the axis of the sphere is not allowed to be reflected back by providing a conical projection exactly in front of opening. The cone scatters the radiation as it undergoes multiple reflection and it continues till the entire heat entering the sphere is absorbed (nearly 98%). The effective area of perfectly black body, is the area of aperture. Emission of Heat Radiation The amount of heat (Q)radiated by a body depends upon: 1.The Absolute temperature of the body (T). 2.The nature of the body – material, nature of surface – polished or not etc. 3.Surface area of the body (A) 4.Time duration for which the body emits radiation(t) Qα A Q α t Hence Q α A t Q = RAt where R is the Radiant power or emissive power. 𝑄 𝑅= 𝐴𝑡 ∴ Emissive power is defined as the quantity of heat radiated per unit area per unit time is defined as emissive power of the body at given temperature Dimension of emissive power are [L 0 M 1 T -3] SI unit: J m -2 s -1 or W / m 2 Coefficient emission or Emissivity ( e ) : The coefficient of emission or emissivity of a given body is the ratio of emissive power of a body (R) to the emissive power of the black body (R B) at the same temperature. ∴ e = R/RB For a perfectly Black body e = 1 For a perfect Reflector e = 0 For Ordinary body 0 < e < 1 Note : Emissivity is larger for rough surfaces and smaller for smooth and polished surface. Emissivity depends upon temperature and wavelength of radiation. Kirchhoff’s law of heat radiation and its theoretical proof Consider an ordinary body A and a perfectly Black Body B suspended in a constant temperature enclosure. The bodies A and B radiate heat to the enclosure and enclosure also radiate heat to the bodies, after some time, the temperature of both the bodies will become same as that of temperature of enclosure. Q = amount of heat radiation incident on each body per unit area per unit time. Let , a = coefficient of absorption of body A , e =coefficient of emission of body A,R = emissive power of body A , R B = emissive power of black body B For Ordinary Body A: Quantity of radiant heat absorbed by body A= quantity of heat emitted by body A Qa=R (1) For Black Body B: Q =RB (2) Dividing equation (1) by (2), we get 𝑅 𝑎= (3) 𝑅𝐵 𝑅 We know , 𝑒 = (4) 𝑅𝐵 From equation (3) and (4) a=e Spectral Distribution of Blackbody Radiations At each temperature, the Blackbody emits continuous heat radiation spectrum. ii) The energy associated with the radiation of a particular wavelength increases with increase in temperature of Blackbody. iii) At a given temperature of the Blackbody, the amount of energy associated with radiation initially increases with wavelength and after becoming maximum corresponding to a wavelength λ max, it decreases. The wavelength λ max is called wavelength of maximum emission. iv) The area under each curve represents the total energy emitted by the perfect Blackbody per second per unit area over the complete wavelength range at that temperature. v) The energy distribution is not uniform. The peak of the curve shifts towards the left – shorter wavelengths, i.e. the value of λ max decreases with increase in temperature Wien’s Displacement Law Wien’s displacement law states that the wavelength for which the emissive power of the blackbody is maximum is inversely proportional to the absolute temperature of black body. This law is called displacement law because as the temperature increases the maximum intensity of radiation emitted by it gets shifted or displaced towards the shorter wavelength side. 1 λ max α 𝑇 𝑏 ∴ λ max = 𝑇 ∴ λ max T = b where b is called Wien’s constant b = 2.897 × 10 – 3 m K Stefan – Boltzmann Law of Radiation: According to this law, “The rate of emission of radiant energy per unit area or the power radiated per unit area of a perfect black body is directly proportional to the fourth power of its absolute temperature. R α T4 or R = σ T 4 Where σ is Stefan’s constant. σ = 5.67 × 10 – 8 J m -2 K -4 or Wm-2K-4and dimension of σ are [ L0 M1 T-3 K-4] Hence the power radiated by Black body depends only on temperature and not on any other characteristics such as colour, materials, nature of surface. If Q is the amount of radiant energy emitted in time t by a perfect black body of surface area A at temperature T, then 𝑄 𝑄 =σT4 [∴ 𝑅 = ] 𝐴𝑡 𝐴𝑡 For a body , which is not blackbody, the energy radiated per unit area per unit time is still proportional the fourth power of temperature but is less than that of blackbody. For ordinary body R = e σ T4 Where e is emissivity of the surface. If the perfect blackbody having absolute temperature T is kept in a surrounding which is at a lower absolute temperature T0 of surrounding. Then Energy radiated by blackbody at temperature T per unit area per unit time = σ T 4 The energy absorbed from surrounding per unit area per unit time = σ T0 4 ∴ The net energy loss by perfect blackbody per unit area per unit time = σ T 4 - σ T0 4 = σ (T 4 - T04) For an ordinary body, net loss in energy per unit area per unit time = e σ (T 4 – T0 4) If T < T0, then net gain in thermal energy of body per unit area per unit time is e σ (T04 – T 4 )