PHYSICS CLASS XII Chapter 3 - Kinetic theory of gases and radiation

PHYSICS
CLASS XII
CHAPTER - 3
KINETIC THEORY OF GASES AND RADIATION
MAHARASHTRA STATE BOARD

1. CHAPTER 3 Kinetic Theory of Gases and Radiation CLASS XII PHYSICS MAHARASHTRA STATE BOARD

2. Can you recall? 1. What are different states of matter? 2. How do you distinguish between solid, liquid and gaseous states? 3. What are gas laws? 4. What is absolute zero temperature? 5. What is Avogadro number? What is a mole? 6. How do you get ideal gas equation from the gas laws? 7. How is ideal gas different from real gases? 8. What is elastic collision of particles? 9. What is Dalton's law of partial pressures?

3. SOLID LIQUID GAS MOLECULES ATOMS Matter is made up of tiny particles called molecules. All types of matter are made up of extremely small particles. STATES OF MATTER

4. Scanning Tunnelling Microscope (STM) Molecules are always in a state of motion and even when inside matter, they never stop moving.

5. Closely Packed Very loosely Packed Loosely Packed

6. Gas law’s Boyle’s Law: V at constant T Charles Law: V T at constant P Avogadro’s Law: V Gay-Lussac’s Law: P T at constant V Combining these all law’s, we get V n T PV = nRT Where, R is proportionality constant OR Universal Gas Constant R = 8.314 J  

7. Mole’s Unit: mol Formula: Mole = Concept: 1 mole of substance Contain 6.022 X No. of mole (n) =  

8. ATOM Weight of one atom of an element = Weight of one molecule of a compound = For Gas, n = Where, n mole N Number of molecules in a gas Avogadro's number OR Number of molecule in 1 mole of gas  

9. IDEAL GAS Equation, PV = nRT PV = RT …………. [n = ] Here, R = Where, is the Boltzmann Constant So, PV = PV = N Where, N is the Number of Particles NOTE: A gas obeying the equation of state i.e., PV = nRT at all pressure, and temperature is an ideal gas.  

10. Behaviour of gas Gas contained in container is characterized by its pressure, volume and temperature. Particle of gas are always in constant motion. It is very hard to evaluate the motion of single particle of gas. We find average of physical quantities and then relate to the macroscopic view of gas.

11. Fig.(a): A gas with molecules dispersed in the container: A stop action photograph. Fig.(b): A typical molecule in a gas executing random motion.

12. IDEAL gaS A gas obeying ideal gas equation is nothing but an ideal gas. Intermolecular interaction is different or absent. REAL GAS Compose of atom/molecule which do interact with each other. If there is atom/molecule are so apart then we can say that the real gases are ideal. This can be achieved by lowering pressure and increasing temperature.

13. MEAN FREE PATH Represented by ‘ ’ Defined as average distance travelled by a molecule with constant velocity between two successive collision. Mean free path is inversely vary with density. Mean free path also inversely proportional to size of the particle of gases. Conclusion: ……….[ ]  

14. EAXMPLE Obtain the mean free path of nitrogen molecule at 0 °C and 1.0 atm pressure. The molecular diameter of nitrogen is 324 pm (assume that the gas is ideal). Solution: Given T = 0 °C = 273 K, P = 1.0 atm = 1.01× Pa and d = 324 pm = 324 × m. For ideal gas PV = N T, ∴ Using Eq., mean free path = = 0.8 X Note that this is about 247 times molecular diameter.  

15. Pressure of ideal gas A cubical box of side L. It contains n moles of an ideal gas. Volume of that cube is, V = , Where L We have to find relation between pressure P of gas with molecular speed Here, change in momentum in –component = - m = - 2 m So, total momentum applied on the wall is 2 m The total distance travelled by particle of gas during collision of molecule and wall is 2L     m  

16. ……[Speed = ] For 1-molecule, Average force = Average rate of change of momentum = = = = x Average Force ( ) = Similarly, for particle 2,3,4, ……with respect to x-component velocities are , , Average Force ( ) =  

17. Pressure(P) = = = ………….( i ) Average Velocity ( ) = N = Put it in equation ( i ) Pressure(P) = Pressure(P) = ……..[V= ] ……………(ii)  

18. Now, as there is perfectly collision in molecule = = = = …………………….. (a) For average velocity of all component (x, y, z) + + + + ……… From equation (a) = Put in equation (ii) P = = =  

19. ROOT MEAN SQUARE (rms) SPEED P = = We know that, P V = n R T = …………[n = ] = = = …….[ = ]  

20. INTERPRETATION OF TEMPERATURE IN KTG We know that, P = P V = = ) ……………( i ) In ideal gas there is no interaction molecule so P.E. = 0. Here, Total Energy (E) = K.E. + P.E. = N ( ) + 0 = N ( ) …………….(ii) Put in equation ( i ) P V = E ……….(iii) Using Ideal Gas, P V = N E ……….(iv) N E E = N  

21. Example At 300 K, what is the rms speed of Helium atom? [mass of He atom is 4u, 1u = 1.66 × kg; = 1.38 × J/K] Solution: Given T = 300 K, m = 4 × 1.66 × kg Average K. E = = = = = 187.05 X = 13.68 X = 1368 m/s  

22. Energy of single molecule is K.E = + + For gas at temperature T, The average K.E per molecule is <K.E.> <K.E> = < > + < > + < > But the mean distance is is the molecular energy in single x or y or z component.  

23. DEGREE OF FREEDOM The molecule is free to move in whole space. Molecule can move in 3-D space. “Degree of freedom of a system are defined as the total number of co-ordinate required to describe the position and configuration of the system completely.” DIATOMIC MOLECULE Molecules having 3-translational degrees (x, y, z) Diatomic molecule rotate around axis Suppose, a molecule moving along axis z with with K.E = For axis y , K.E = So, Total energy = E (Translational) + E (Rotational) = [ + + ] + [ + ]   Fig.: The two independent axes z and y of rotation of a diatomic molecule lying along the x-axis.

24. From equipartition of energy: Each and every molecule separated with molecule. For, Translational motion 3 ( dof ) [ 3 X ] Rotational motion 2 ( dof ) [ 2 X ] Vibrational motion 2 ( dof ) [ 2 X ] [ m + k ] The energy of molecule [ ] is valid for high temperature and not valid for extremely low temperature where quantum effects become important.  

25. SPECIFIC HEAT CAPACITY If temperature of gas increase (Cause considerable change in volume and pressure) Specific heat at constant volume Specific heat at constant pressure MAYER’S RELATON Consider, 1 mole of gas (Ideal) enclosed in cylinder Let P, V, T be the pressure, volume and temperature dT = Change in temperature (Volume is constant d Increase in the internal energy dE Where,  

26. If gas is heated with same given temperature at constant pressure then, dV Change (increase) in volume Here, work is done to pull backward the piston dw = p dv dE + dw = ………(ii) Where, Molar specific heat at constant pressure From equation ( i ) - - ……………… (iii)  

27. For, 1 mole of gas, PV = RT …………(n = 1) But, when pressure is constant P dV = R dT - = R dT - ………(Mayer’s relation) Now, - Where, J – mechanical equivalent of heat Here, = Molar mass of gas , = Respective principle of heat - = ( - =  

28. a) MONOATOMIC GAS For translational motion 3 X = = ……….( = N = = R + = =  

29. b) diaTOMIC GAS Gases like Each molecule have 3-Translational and 2-Roational dof . E = [3 X [ 2 X ] = [ [ ] = = = ……….( = = = R + = = ……….(For rigid rotator body)  

30. For non rigid vibrating molecule E = Translational Energy + Rotational Energy + Vibrational Energy E = [3 X [ 2 X ] [ 2 X ] = = = ……….( = = = R + = =  

31. b) POLYaTOMIC GASES E = Translational Energy + Rotational Energy + (f x Vibrational Energy) Where, f Number of vibrational dof E = [3 X [ 3 X ] [ f X 2 X ] = (3+3+2f) = (3 + f) = = ……….( = (3+f) = (3+f) = R + (3+f) =  

32. HEAT Heat is the form of energy which can be formed from one object to another object. MODES OF CONDUCTION Conduction Convection Radiation Does not required material medium for transfer Emission of heat radiant energy Electromagnetic wave Can travel through both material and vacuum medium. Fastest way of heat transfer.   Requires material medium for transfer.

33. INTERACTION OF THERMAL RADIATION AND MATTER When any thermal radiation falls on object then some part is reflected, some part is absorbed and some part transmitted. Consider, Q Total amount of heat energy incident Total energy (Q) = Where, a = , r = , = Coefficient of absorption, reflection, transmission  

34. COEFFICIENT OF HEAT READIATION Coefficient of absorption/absorptive power/absorptivity “The ratio of amount of heat absorbed to total amount of heat incident.” a = Coefficient of reflection/reflectance “The ratio of amount of heat reflected to total amount of heat incident.” r = Coefficient of transmission or transmittance “The ratio of amount of heat is transmitted to total amount of heat incident.” =  

35. Cases Perfect transmitter: r = 0, a = 0, 1 Object is said to be completely transparent. b) Di-athermanous substance: “Substance through which heat radiation can pass is called di-athermanous” Neither a good absorber nor good reflector E.G., Quartz, Sodium Chloride, Hydrogen, Oxygen c) Athermanous substance: 0, a + r = 1 “Substance which are largely opaque to thermal radiation i.e. do not transmit heat.”  

36. d) Good reflector/poor absorber/poor transmitter: 0, a = 0, r = 1   e) Perfect blackbody: 0, r = 0 and a = 1 “All the incident energy is absorbed by the object such an object is called a perfect blackbody.”   NOTE : All the a (absorption), r (reflection) and t (transmission) depends upon wavelength of incident radiation. “A body which absorb entire radiant energy incident on it, is called an ideal or perfect blackbody.” Platinum black absorb nearly 97 % of incident radiant heat on it. Good absorber are good emitter and poor absorber are poor emitter. Fig.: Ferry’s blackbody.

37. EMISSION OF HEAT RADIATION Pierre prevost published a theory of radiation known as theory of exchange of heat. At temperature 0 K or above it. Body radiate thermal energy and at same time they absorb radiation received from surrounding. Emission per unit time depends upon surface of emission, its area and size. Hotter body radiate at higher rate than the cooler body

38. AMOUNT OF HEAT RADIATED BY THE BODY DEPENDS ON – The absolute temperature of body The nature of body (Polished or not, size, colour) Surface area of body (A) Time duration Amount of heat radiated Surface area of body and time duration Q At Q = Rat Where, R Emissive power/Radiant power R = SI Unit – J Dimension - [ ]  

39. EMISSIVITY “The coefficient of emission or emissivity (e) of a given surface is the ratio of the emissive power R of the surface to emissive power of perfect black body at same temperature.” e = For perfect blackbody, e = 1, Perfect reflector, e = 0 KIRCHOFF’S LAW OF HEAT RADIATION “At a given temperature, the ratio of emissive power to coefficient of absorption of a body is equal to the emissive power of a perfect blackbody at the same temperature for all wavelength.” At thermal equilibrium, emissivity is equal to absorptivity.  

40. WEIN’S DISPLACEMENT LAW The wavelength for which emissive power of a blackbody is maximum is inversely proportional to the absolute temperature of the blackbody. Where, b Wein’s constant b = 2.897 X mk Law is useful to determine distant length.  

41. SREFAN-BOLTZMANN LAW OF RADIATION “The rate of emission of radiant energy per unit area or the power radiated per unit area of a perfect blackbody is directly proportional to fourth power of its temperature.” R R = Where, Unit J Dimension ] We know that, R = So, R = = …………(For black body) For ordinary body, R =  

42. Let, T Energy radiated per unit time = Energy radiated per unit time in surrounding = Net loss of energy by perfect blackbody per unit time = - = - ) For ordinary body = - )