Physics Supplement to my class. Kinetic Theory

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1 Physics Supplemet to my class Leaers should ote that I have used symbols for geometrical figures ad abbreviatios through out the documet. Kietic Theory 1 Most Probable, Mea ad RMS Speed of Gas Molecules 1.1 Most Probable Speed Most probable speed of the molecules of a gas is that speed which is possessed by maximum fractio of total umber of molecules of the gas. FOR EXAMPLE: If the speed of 10 molecules of a gas are 2,3,3,4,4,4,5,6,7,7,m/s, the the most probable speed is 4m/s, as maximum fractio of total molecules possess this speed ad importat to ote here that it s ot the maximum speed of the molecules. I the case of Ideal Gases v mp = 2k B T m (1.1) Where m is the mass of the molecule, k B is Boltzma costat ad T is temperature of the gas. For a give gas, value of most probable speed icreases with rise i temperature. 1.2 Mea Speed or Average Speed It is the Average speed with which a molecule of a gas moves. It is equal to the sum of the idividual speeds of the molecules divided by the umber of molecules. If v 1, v 2, v 3,...v are the speeds of idividual molecules, the their mea or average speed is v av = v 1 + v 2 + v v (1.2) 1.3 Root Mea Square Speed It is defied as the square root of the mea of the squares of the radom velocities of the idividual molecules of a gas. If v 1, v 2, v 3,...v are radom velocities of molecules of a gas, the r ms speed of molecules is v1 2 v r ms = + v v v 2 (1.3) Vaisesika Learig 1

2 For Ideal gases v r ms = Where, the symbols have their usual meaig. 3k B T NOTE:- From equatios (1.1),(1.2), (1.3), we fid that v mp : v av : v r ms = 8 2 : π : 3 v r ms is maximum ad v mp is miimum. Here is a example to clear the above cocepts EXAMPLE : You are give five umbers: 7, 13, 34, 69, ad What is the average value av of these umbers? 2. What is the r ms value r ms of these umbers? ANSWER: (1) The average value ca be foud from av = = 42.8 (Aswer ) 5 (2) The r ms value ca be foud from r ms = = 53.7 (Aswer ) 5 The rms value is greater tha the average value because the larger umbers beig squared are relatively more importat i formig the rms value. m (1.4) 2 Degree of Freedom(DoF) The umber degree of freedom of dyamical system is defied as the total umber of co-ordiates or idepedet quatities required to describe completely the positio ad cofiguratio of the system. For example: (Cosiderig a poit mass of the particle without Rotatio.) 1. Whe a particle is movig alog a straight lie, say alog X- axis, its positio ca be specified by its displacemet alog the X-axis. Therefore, such a particle has oe traslatioal degree of freedom. e.g. A bob of a oscillatig simple pedulum. 2. Whe a particle is movig i a plae, its positio ca be determied by its displacemets alog the X-axis ad Y-axis. Therefore, it has two traslatioal degree of freedom. e.g. A At movig o the floor. Vaisesika Learig 2

3 3. Whe a particle is movig i space, its positio ca be determied by its displacemets alog the X-axis, Y-axis ad Z-axis. Therefore, it has three traslatioal degree of freedom. e.g. A buzzig bee, flyig birds etc. Cosider a system of two particles, each havig three degree of freedom so that the umber of DoF of both particles is six. If the two particles remai at a fixed distace (a defiite relatio)from each other the the umber of coordiates required to describe the cofiguratio of the system reduces by oe.therefore, the system has 5 DoF. Geeralisig the above observatio The umber of DoF (N) of a dyamical system is obtaied by subtractig the umber of idepedet relatios from the total umber of coordiates required to specify the positio of costituet particles of the system. i.e. N = 3A R (2.1) Where, A = umber of particles of the system, R = umber of idepedet relatios amog particles For Moo atomic gases(e.g. Neo, Argo, Helium etc.) put A = 1, R = 0 i eq.(2.1) therefore N = = 3 For Di atomic gases(e.g. H 2, O 2, N 2 etc.) The molecule is capable of traslatory motio of its cetre of mass. Therefore it has three traslatioal DoF. But i additio, the molecule ca rotate about its cetre of mass i horizotal ad vertical plaes. Thus a di atomic molecule also has two rotatioal DoF. Also, Put A = 2 ad R = 1 i eq.(2.1), we get For Tri atomic gases(e.g. H 2 S, SO 2 etc.) N = = 5 Liear Molecule ( ) I this case, A = 3 ad R = 2(as two distaces betwee the atoms are fixed). from (2.1), N = = 7 No-Liear Molecule (atoms at the vertices of a ) I this case, A = 3 ad R = 3(as three distaces betwee the atoms are fixed). from (2.1), N = = 6 Vaisesika Learig 3

4 3 Determiatio of γ from the DoF Suppose a poly atomic gas molecule has DoF. Therefore Iteral eergy of oe gram mole of the gas is U = 1 2 k B T N A = 2 RT C v = du dt = d dt C p = C v + R ( 2 RT ) = 2 R C p = 2 R + R = ( )R ( γ = ) γ = C p γ = C v ( 2 + 1) R 2 R = 2 ( ) (3.1) This is the relatio betwee the γ ad DoF. QUESTIONS 1. Whe alcohol or acetoe (ail polish remover) is rubbed o your body, your body temperature decreases. Explai this effect. 2. A liquid partially fills a cotaier. Explai why the temperature of the liquid decreases if the cotaier is the partially evacuated. (Usig this techique, oe ca freeze water at temperatures above 0 C.) 3. Whe a automobile travels for a log distace the air pressure i the tyres icreases. Why? 4. A gas storage tak has a small leak. The pressure i the tak drop more quickly if the gas is hydroge tha if it is oxyge. Why? 5. Why the lad has a higher temperature tha the ocea durig the day but a lower temperature at ight? 6. Although the velocity of air molecules is early 0.5km/s yet the smell of scet spreads at a much slower rate. Why? 7. Why evaporatio causes coolig? 8. Whe air is pumped ito a cycle tyre the volume ad pressure of the air i the tyre both are icreased. What about Boyle s law i this case? Vaisesika Learig 4

5 9. Explai why there is fall i temperature with altitude. ANSWERS 1. Whe we pour some alcohol or acetoe (ail polish remover) o our palm the particles gai eergy from your palm or surroudigs ad evaporate causig the palm to feel cool. (See also aswer 7) 2. The faster-movig atoms are the oes that escape from the liquid s surface (that is, they are the oes that evaporate). That meas the average speed of the remaiig atoms is decreased ad the temperature decreases as the average speed of the atoms decreases. 3. O drivig a automobile for a log time, the work doe agaist frictio is coverted ito heat. The gas i the tyre gets heated ad hece the pressure of the gas icreases, because P T 4. Rate of diffusio of a gas is iversely proportioal to the square root of the desity. so hydroge leaked out more rapidly. 5. Specific Heat of water is more tha lad(earth). Therefore for give heat chages i temperature of lad is more tha ocea. 6. The air molecules travel alog a zigzag path due to frequet collisio as a result their displacemet per uit time is very small. 7. Durig evaporatio fast movig molecules escape from a liquid surface so the average kietic eergy of the molecules left behid is decreased thus the temperature of the liquid is lowered. 8. Whe air is pumped, more molecules are pumped i. Boyle s law is applicable oly whe umber of molecules remais costat. So Boyle s law is ot obeyed. 9. As the molecules move higher their potetial eergy icreases ad hece kietic eergy decreases ad hece temperature decreases. Also at greater heights, more volume is available. The gas expeds ad hece coolig occurs. This Documet Is The Itellectual Property Of Vaisesika Learig, New Delhi (A No-Profit Orgaizatio for Sciece Popularizatio) All right reserved. This documet is prepared oly to beefit the CBSE 10+2 studets. No part of this publicatio may be reproduced, stored i a retrieval system or trasmitted, i ay form or by ay meas, electroic, mechaical, photocopyig, recordig or otherwise without the prior permissio of the publisher. Vaisesika Learig 5