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Kinetic Theory of Gases Kinetic Theory of Gases Kinetic Theory of Gases Kinetic Theory of Gases The kinetic theory of gases attempts to explain the microscopic properties of a gas in terms of the motion of its molecules. The gas is assumed to consist of a large number of identical, discrete particles called molecules, a molecule being the smallest unit having the same chemical properties as the substance. Elements of kinetic theory were developed by Maxwell, Boltzmann and Clausius between 1860-1880’s. Kinetic theories are available for gas, solid as well as liquid. However this chapter deals with kinetic theory of gases only. Postulates of kinetic theory of gases 1) Any gas consist large number of molecules. These molecules are identical, perfectly elastic and hard sphere. 2) Gas molecules do not have preferred direction of motion, their motion is completely random. 3) Gas molecules travels in straight line. 4) The time interval of collision between any two gas molecules is very small. 5) The collision between gas molecules and the walls of container is perfectly elastic. It means kinetic energy and momentum in such collision is conserved. 6) The motion of gas molecules is governed by Newton's laws of motion. 7) The effect of gravity on the motion of gas molecules is negligible. Prof. Avadhut Surekha Prakash Manage D.M.S. Mandal’s Bhaurao kakatkar College, Belgaum Kinetic Theory of Gases Maxwell’s law of distribution of velocities This law is to find the number of molecules which have a velocity within small interval (ie. 푐 to 푐 + 푑푐). For deriving this equation Maxwell did following assumptions. a) Speed of gas molecules ranges from zero to infinity. b) In the steady state, the density of gas remains constant. c) Though the speed of the individual molecule changes, definite number of gas molecules have speed between definite range. Consider a gas containing N number of molecules having velocity c and its X, Y and Z components are u, v and w respectively. From the probability theory, the probability of molecules having velocity component 푢 to 푢 + 푑푢 is 푓(푢)푑푢 similarly the probability of molecules having velocity component 푣 to 푣 + 푑푣 is 푓(푣)푑푣 and the probability of molecules having velocity component w to 푤 + 푑푤 is 푓(푤)푑푤. Thus the number of molecules whose velocity lies between 푢 to +푑푢 , 푣 to 푣 + 푑푣 and w to 푤 + 푑푤 is given by 푑푁 = 푁 푓(푢)푑푢 푓(푣)푑푣 푓(푤)푑푤 푑푁 = 푁 푓(푢) 푓(푣) 푓(푤) 푑푢 푑푣 푑푤 … … . (1) If velocity components 푢, 푣 and 푤 along the three axis. The space formed is called as the velocity space. A molecule having velocity components 푢, 푣 , 푤 be represented by a point 푝. Let the small volume 푑푢 푑푣 푑푤 around 푝 Prof. Avadhut Surekha Prakash Manage D.M.S. Mandal’s Bhaurao kakatkar College, Belgaum Kinetic Theory of Gases contains 푑푁 number of molecules. Thus molecules have a resultant velocity c is given by 푐2 = 푢2 + 푣2 + 푤2 Differentiating on both side, we get 0 = 2푢 푑푢 + 2푣 푑푣 + 2푤 푑푤 ∵ 푐 = 푐표푛푠푡푎푛푡 ∴ 푢 푑푢 + 푣 푑푣 + 푤 푑푤 = 0 … … … … … (2) The probability density 푓(푢) 푓(푣) 푓(푤) should depend only on the magnitude but not the direction of 푐. ∴ 푓(푢) 푓(푣) 푓(푤) = 푓(푐2) (푤ℎ푒푟푒 푓(푐2) 푠 푓푢푛푐푡표푛 표푓 푐2) Differentiating on both side, we get 푑[푓(푢) 푓(푣) 푓(푤)] = 0 ∵ 푐2 = 푐표푛푠푡푎푛푡 푓′(푢)푑푢푓(푣)푓(푤) + 푓′(푣) 푑푣푓(푢)푓(푤) + 푓′(푤)푑푤푓(푣)푓(푣) = 0 Dividing above equation by 푓(푢) 푓(푣) 푓(푤), we get 푓′(푢) 푓′(푣) 푓′(푤) 푑푢 + 푑푣 + 푑푤 = 0 . (3) 푓(푢) 푓(푣) 푓(푤) Multiplying the equation (2) by an arbitrary constant (β) and adding to equation (3), we get 푓′(푢) 푓′(푣) 푓′(푤) [ + 훽푢] 푑푢 + [ + 훽푣] 푑푣 + [ + 훽푤] 푑푤 = 0 푓(푢) 푓(푣) 푓(푤) As 푑푢, 푑푣 & 푑푤 can not be zero independently, so we get 푓′(푢) 푓′(푣) 푓′(푤) + 훽푢 = 0 + 훽푣 = 0 + 훽푤 = 0 푓(푢) 푓(푣) 푓(푤) 푓′(푢) = −훽푢 푓(푢) Prof. Avadhut Surekha Prakash Manage D.M.S. Mandal’s Bhaurao kakatkar College, Belgaum Kinetic Theory of Gases Integrating wrt u, we get 1 log 푓(푢) = − 훽푢2 + log 푎 2 (푊ℎ푒푟푒 푙표푔 푎 푠 푛푡푒푔푟푎푡표푛 푐표푛푠푡푎푛푡 ) Taking antilog on both side, we get 1 − 훽푢2 2 1 푓(푢) = 푎푒 2 = 푎푒−푏푢 (Where, 푏 = − 훽 = 푐표푛푠푡푎푛푡) 2 2 2 Similarly, 푓(푣) = 푎푒−푏푣 & 푓(푤) = 푎푒−푏푤 Substituting 푓(푢), 푓(푣) & 푓(푤) in eqn (1), we get 2 2 2 푑푁 = 푁 (푎푒−푏푢 ) (푎푒−푏푣 ) (푎푒−푏푤 ) 푑푢 푑푣 푑푤 2 2 2 푑푁 = 푁푎3 푒−푏(푢 +푣 +푤 )푑푢 푑푣 푑푤 ∵ 푐2 = 푢2 + 푣2 + 푤2 2 푑푁 = 푁푎3 푒−푏푐 푑푢 푑푣 푑푤 … … … (4) In this equation constant 푎 and 푏 is found to be 푚 푚 푎 = √ and 푏 = , where 푚 is the mass of the gas particle, 2휋푘푇 2푘푇 푘 is Boltzmann constant and 푇 is temperature of gas Then equation (4) becomes 3 푚푐2 푚 2 − 푑푁 = 푁 ( ) 푒 2푘푇 푑푢 푑푣 푑푤 … … (5) 2휋푘푇 This is one form of Maxwell’s law of distribution of velocities. Prof. Avadhut Surekha Prakash Manage D.M.S. Mandal’s Bhaurao kakatkar College, Belgaum Kinetic Theory of Gases The number of molecules lies within box of volume 푑푢 푑푣 푑푤 whose velocity lies within 푐 to 푐 + 푑푐 is the same as number of molecules within the spherical shell of inner radius 푐 and outer radius 푐 + 푑푐. Hence volume 푑푢 푑푣 푑푤 can be replaced volume of the shell 4휋푐2푑푐. Therefore equation (5) becomes 3 푚푐2 푚 2 − 푑푁 = 푁 ( ) 푒 2푘푇 4휋푐2푑푐 2휋푘푇 3 푚푐2 푚 2 − 푑푁 = 4휋푁 ( ) 푒 2푘푇 푐2푑푐 2휋푘푇 The above equation is called as Maxwell’s equation of distribution of velocity which gives the value of number molecules 푑푁 having velocity 푐 to 푐 + 푑푐. Mean or Average velocity (푪풂풗) The average speed is the sum of all the velocities ranging from 0 to ∞ divided by total number of molecules N. ∞ ∫ 푑푁 푋 푐 ∴ 퐶 = 0 푎푣 푁 Substituting the value of 푑푁 in in above equation 3 푚푐2 ∞ 푚 2 − ∫ (4휋푁 ( ) 푒 2푘푇 푐2푑푐) 푋 푐 0 2휋푘푇 퐶 = 푎푣 푁 Prof. Avadhut Surekha Prakash Manage D.M.S. Mandal’s Bhaurao kakatkar College, Belgaum Kinetic Theory of Gases 3 ∞ 푚푐2 푚 2 − 퐶 = 4휋 ( ) ∫ 푒 2푘푇 푐3푑푐 푎푣 2휋푘푇 0 3 ∞ 푏 2 2 퐶 = 4휋 ( ) ∫ 푒−푏푐 푐3푑푐 푎푣 휋 0 ∞ ퟏ 2 1 ∵ ∫ 푒−푏푐 푐3푑푐 = 2푏2 0 푚 ( Where 푏 = ) 2푘푇 3 푏 2 1 퐶 = 4휋 ( ) 푎푣 휋 2푏2 4 퐶 = √ 푎푣 휋푏 4 ∴ 퐶푎푣 = √ 푚 휋 ( ) 2푘푇 8푘푇 ∴ 퐶 = √ 푎푣 휋푚 Where 푚 is the mass of gas molecule. If 푀 is molecular 푀 weight which can be given by 푀 = 푚푁퐴. Then 푚 = Therefore 푁퐴 the above equation becomes. 8푘푇 퐶 = 푎푣 √ 푀 휋 ( ) 푁퐴 Prof. Avadhut Surekha Prakash Manage D.M.S. Mandal’s Bhaurao kakatkar College, Belgaum Kinetic Theory of Gases 8푘푁 푇 ∴ 퐶 = √ 퐴 푎푣 휋푀 8푅푇 ∴ 퐶 = √ ∵ 푘푁 = 푅 푎푣 휋푀 퐴 Root Mean Square Velocity (푪풓풎풔) The rms speed is the square root of the sum of all the squared velocities ranging from 0 to ∞ divided by total number of molecules N. ∞ ∫ 푑푁 푋 푐2 ∴ 퐶 = √ 0 푟푚푠 푁 Substituting the value of 푑푁 in in above equation 3 푚푐2 ∞ 푚 2 − ∫ (4휋푁 ( ) 푒 2푘푇 푐2푑푐) 푋 푐2 √ 0 2휋푘푇 퐶 = 푟푚푠 푁 3 ∞ 푚푐2 푚 2 − 퐶 = √4휋 ( ) ∫ 푒 2푘푇 푐4푑푐 푟푚푠 2휋푘푇 0 3 ∞ 푏 2 = √4휋 ( ) ∫ 푒−푏푐2 푐4푑푐 휋 0 푚 ( Where 푏 = ) 2푘푇 Prof. Avadhut Surekha Prakash Manage D.M.S. Mandal’s Bhaurao kakatkar College, Belgaum Kinetic Theory of Gases 3 1 푏 2 3 휋 2 = √4휋 ( ) 푋 ( ) 휋 8 푏5 3 = √ 2푏 3 = √ 푚 2 ( ) 2푘푇 3푘푇 퐶 = √ 푟푚푠 푚 푀 Substituting 푚 = in above equation. We get 푁퐴 3푘푇 퐶 = 푎푣 √ 푀 ( ) 푁퐴 3푘푁 푇 ∴ 퐶 = √ 퐴 푎푣 푀 3푅푇 ∴ 퐶 = √ ∵ 푘푁 = 푅 푎푣 푀 퐴 Most Probable Velocity (푪풑풓) The most probable velocity is the velocity possessed by maximum number of molecules. For this condition is, Prof. Avadhut Surekha Prakash Manage D.M.S. Mandal’s Bhaurao kakatkar College, Belgaum Kinetic Theory of Gases 푑[푑푁] = 0 푑푐 3 푚푐2 푑 푚 2 − ∴ (4휋푁 ( ) 푒 2푘푇 푐2푑푐) = 0 푑푐 2휋푘푇 3 푚푐2 푚 2 푑 − 4휋푁 ( ) ( 푒 2푘푇 푐2푑푐) = 0 2휋푘푇 푑푐 3 푚푐2 푚푐2 푚 2 − 2푚푐 − 4휋푁 ( ) ( 푒 2푘푇 [2푐] + [− ] 푒 2푘푇 푐2) = 0 2휋푘푇 2푘푇 3 푚푐2 푚 2 − 푚푐 4휋푁 ( ) 푒 2푘푇 ( 2푐 − [ ] 푐2) = 0 2휋푘푇 푘푇 푚푐 2푐 − [ ] 푐2 = 0 푘푇 푚푐3 ∴ = 2푐 푘푇 2푘푇 ∴ 푐2 = 푚 2푘푇 ∴ 푐 = √ 푚 Thus most probable velocity, 2푘푇 퐶 = √ 푝푟 푚 푀 Substituting 푚 = in above equation. We get 푁퐴 Prof. Avadhut Surekha Prakash Manage D.M.S. Mandal’s Bhaurao kakatkar College, Belgaum Kinetic Theory of Gases 2푘푇 퐶 = 푎푣 √ 푀 ( ) 푁퐴 2푘푁 푇 ∴ 퐶 = √ 퐴 푎푣 푀 2푅푇 ∴ 퐶 = √ ∵ 푘푁 = 푅 푎푣 푀 퐴 Relation between 푪풑풓 , 푪풂풗 & 푪풓풎풔 2푅푇 8푅푇 3푅푇 퐶 ∶ 퐶 ∶ 퐶 = √ : √ ∶ √ 푝푟 푎푣 푟푚푠 푀 휋푀 푀 8 = √2: √ ∶ √3 휋 2 3 = 1 ∶ ∶ √ √휋 2 = 1: 1.128 ∶ 1.228 Mean free path Mean free path of gas molecules is defined as the average distance travelled by a molecule between two successive collisions. Prof. Avadhut Surekha Prakash Manage D.M.S. Mandal’s Bhaurao kakatkar College, Belgaum Kinetic Theory of Gases Expression for mean free path Consider a gas in container having n molecules per unit volume.
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