{"id":254,"date":"2023-03-02T13:26:59","date_gmt":"2023-03-02T13:26:59","guid":{"rendered":"https:\/\/www.vartmaaninstitutesirsa.com\/?page_id=254"},"modified":"2023-04-01T05:22:12","modified_gmt":"2023-04-01T05:22:12","slug":"unit-ix-behaviour-of-perfect-gases-and-kinetic-theory-of-gases","status":"publish","type":"page","link":"https:\/\/vartmaaninstitutesirsa.com\/index.php\/unit-ix-behaviour-of-perfect-gases-and-kinetic-theory-of-gases\/","title":{"rendered":"Unit IX: Behaviour of Perfect Gases and Kinetic Theory of Gases"},"content":{"rendered":"

\u00a0 \u00a0 \u00a0Chapter\u201313: Kinetic Theory of gases<\/strong><\/h1>\n

Kinetic theory of gases \u2013 assumptions, concept of pressure. Kinetic interpretation of temperature; rms speed of gas molecules; degrees of freedom, law of equi-partition of energy (statement only) and application to specific heat capacities of gases; concept of mean free path, Avogadro\u2019s number. Equation of state of a perfect gas, work done in compressing a gas.<\/p>\n

 <\/p>\n

 <\/p>\n

Unit-9 Kinetic Theory of Gases<\/u><\/strong><\/p>\n

    \n
  1. Boyle’s Law:- <\/u><\/strong><\/li>\n<\/ol>\n

    According to Boyle’s law, at constant temperature, the volume of a gas is inversely proportional to its pressure. i.e.<\/p>\n

    Or<\/p>\n

    Or<\/p>\n

    Here \u00a0are constant of proportionality depending upon mass, temperature and units of<\/p>\n

    If \u00a0is initial pressure and volume of the gas and \u00a0are final pressure and volume of the gas then<\/p>\n

    As graph shows that when pressure decreases then volume increases or vice versa.<\/p>\n

      \n
    1. Charle’s Law:<\/u><\/strong><\/li>\n<\/ol>\n

      According to Charle’s law at constant pressure, the volume of gas is directly proportional to the temperature of the gas. \u00a0\u00a0i.e.<\/p>\n

      Or\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0 Or<\/p>\n

      If \u00a0is the initial volume and temperature of the gas and \u00a0is the final volume and temperature of the gas then<\/p>\n

       <\/p>\n

      The graph between T\/V<\/em> is as shown in fig. The graph is a straight line graph.<\/p>\n

      The other form of Charle\u2019s law states that if pressure remains constant then volume of gas increases or decreases by \u00a0of its volume at \u00a0for each \u00a0rise or fall of temperature. I.e. volume at one degree is \u00a0which can be given as<\/p>\n

       <\/p>\n

      Example: 1.The volume of a given mass of a gas at 27\u02daC, 1 atm is 100 cc. What will be its volume at 327\u02daC ?<\/strong><\/p>\n

      Ans. Given that<\/p>\n

      If P is constant then \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 or<\/p>\n

        \n
      1. Gay Lussac’s Law:<\/u><\/strong><\/li>\n<\/ol>\n

        According to this law at constant volume, the pressure of a gas is directly proportional to the temperature of the gas.\u00a0\u00a0 i.e.<\/p>\n

        Or<\/p>\n

        If \u00a0are the initial pressure and temperature of the gas and \u00a0are the final pressure and temperature of the gas then\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0 at constant volume<\/p>\n

        The Gay Lussac’s law may also be given as<\/p>\n

        Where \u00a0is pressure at \u00a0and \u00a0is pressure at \u00a0of the gas<\/p>\n

          \n
        1. Perfect Gas and Perfect gas equation:<\/u><\/strong><\/li>\n<\/ol>\n

          Perfect Gas or Ideal Gas:<\/u><\/strong><\/p>\n

          A gas which strictly obeys the Boyle’s law Charle’s law and Gay Lussac’s law is called perfect gas. In reality there is no any gas is perfect gas.<\/p>\n

          Derivation:<\/u><\/strong><\/p>\n

          According to Boyle’s Law, at constant temperature<\/p>\n

          And according to Charle’s law, at constant pressure<\/p>\n

          Combing the above eq. we get<\/p>\n

          Or<\/p>\n

          Or<\/p>\n

          Now we may write for one mole of a gas \u00a0\u00a0\u00a0 as<\/p>\n

          For \u00a0mole of gas\u00a0\u00a0\u00a0 \u00a0 \u00a0\u00a0\u00a0 this eq. is called perfect gas equation.<\/p>\n

          \u00a0<\/u><\/strong><\/p>\n

            \n
          1. Universal Gas Constant:-<\/u><\/strong><\/li>\n<\/ol>\n

            As we know for ideal gas<\/p>\n

            Hence universal gas constant may be defined as the work done by a gas per unit mole per unit Kelvin.<\/em><\/p>\n

            Example: 2 calculate the number of molecules in each cubic metre of a gas at 1 atm and 27 \u00b0C<\/strong><\/p>\n

            Solution: We have<\/p>\n

            Example:3. Molar volume is the volume occupied by 1 mol of any (ideal) gas at standard temperature and Pressure (STP: 1 atm pressure, 0\u00b0C). Show that it is 22.4 liters.\u00a0 <\/strong><\/p>\n

            Ans.\u00a0 Here n =1 mol, T = 273 K,<\/p>\n

             <\/p>\n

            Using the relation<\/p>\n

            Or,<\/p>\n

              \n
            1. Kinetic Theory of An Ideal Gas:<\/u><\/strong><\/li>\n<\/ol>\n

              The Kinetic theories given by Claussius and Maxwell have the following assumptions.<\/p>\n

                \n
              1. All the gases are consisting of molecules. The molecules of a gas are identical but are different of different gases.<\/li>\n
              2. The size of the molecules is negligible compared with the distance between the molecules<\/li>\n
              3. The molecules of the gas are in a stating continuous random motion, moving in all directions with all possible velocities.<\/li>\n
              4. During the random motion, the molecules collide with one another and with the wall of container.<\/li>\n
              5. The collision of gas molecules are perfect elastic.<\/li>\n
              6. There is no any force of attraction or repulsion between the molecules.<\/li>\n
              7. Between two collisions, the gas molecule moves in a straight line.<\/li>\n
              8. The average distance travelled by particle between two successive collision is called mean free path.<\/li>\n
              9. The collision are almost instantaneous, the time during collision is very negligible small.<\/li>\n
              10. The density remains uniform in the container containing gas.<\/li>\n<\/ol>\n

                 <\/p>\n

                 <\/p>\n

                  \n
                1. How does a gas exert pressure:<\/u><\/strong><\/li>\n<\/ol>\n

                  As gases collides the container containing gas and impart momentum on the wall of the container, now according to Newton’s second law of motion, the rate of momentum exerted on the wall is equal to the force.<\/p>\n

                  This force per unit on the wall of the container gives the pressure.<\/p>\n

                  Express for pressure exerted by a gas:<\/u><\/strong><\/p>\n

                  Suppose \u00a0numbers of gas molecules each of mass \u00a0are moving with velocities \u00a0in a cubical vessel of each side<\/p>\n

                  Now suppose one molecule move toward \u00a0axis and collide with the container and impart momentum = \u00a0and regain its original path with back momentum \u00a0.<\/p>\n

                  So total change in momentum<\/p>\n

                  By conversion of total change in momentum<\/p>\n

                  Total time taken by the gas molecules to move \u00a0distance with \u00a0velocity is<\/p>\n

                  So force exerted by gas molecules on the container<\/p>\n

                  If there are \u00a0number of molecules per unit volume then total number of molecules<\/p>\n

                  So total force<\/p>\n

                  Or<\/p>\n

                  Or<\/p>\n

                  Similarly\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0 &<\/p>\n

                  As we may let<\/p>\n

                  Where \u00a0is mean square speed of the gas molecules.<\/p>\n

                  Hence<\/p>\n

                  (\u00a0\u00a0 For unit volume\u00a0\u00a0\u00a0\u00a0\u00a0 )<\/p>\n

                  This is the required expression for pressure exerted by gas molecules.<\/p>\n

                    \n
                  1. Relation between Pressure and K.E. of Gas Molecules:<\/u><\/strong><\/li>\n<\/ol>\n

                    As we know<\/p>\n

                    Mass of the unit volume of gas<\/p>\n

                    so<\/p>\n

                    Dividing eq. (i) by (ii) we get<\/p>\n

                    or<\/p>\n

                    Hence pressure exerted by a gas is equal to two-third of kinetic energy of the gas.<\/p>\n

                      \n
                    1. Kinetic Interpretation of temperature:<\/u><\/strong><\/li>\n<\/ol>\n

                      As we know<\/p>\n

                      And from kinetic theory of gases<\/p>\n

                      For unit volume of gas molecules<\/p>\n

                      or<\/p>\n

                      Or\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 average K.E. of gas molecules<\/p>\n

                      Hence K.E. of gas molecules is directly proportional to the temperature of gas.<\/p>\n

                        \n
                      1. Maxwell’s Speed Distribution:<\/u><\/strong><\/li>\n<\/ol>\n

                        James Clerk Maxwell was the first to derive a mathematically relation for the most probable distribution of speed among the molecules of gas.<\/p>\n

                        The graph between number of molecules \u00a0and speed of the gas molecules is as shown in fig. the important feature of speed distribution curve are.<\/p>\n

                          \n
                        • At any temperature the speed of molecules varies from zero or infinity.<\/li>\n
                        • The speed passed by the larger number of molecules is called most probable speed.<\/li>\n<\/ul>\n
                            \n
                          1. Average, Root means square and most probable speeds:<\/u><\/strong><\/li>\n<\/ol>\n

                            Average Speed:-<\/strong><\/p>\n

                            It is defined as the arithmetic mean of the speed of the molecules of a gas.<\/em><\/p>\n

                            If \u00a0are the speed of \u00a0molecules, then average speed \u00a0can be given as<\/p>\n

                            From Maxwell’s speed distribution law, the average speed may be given as<\/p>\n

                            Root Mean Square Speed:- <\/u><\/strong><\/p>\n

                            It is defined as the root of mean of square of speed of gas molecules.<\/p>\n

                            And from Maxwell\u2019s speed distribution law, the root mean square speed of gas is<\/p>\n

                             <\/p>\n

                            Most Probable Speed:-<\/u><\/strong><\/p>\n

                            It is defined as the speed of the maximums number of molecules in gas and Maxwell\u2019s distribution law<\/p>\n

                            Relation between <\/u><\/strong> :-<\/u><\/strong><\/p>\n

                            :<\/p>\n

                            Or<\/p>\n

                            Or<\/p>\n

                            Clearly<\/p>\n

                            Example: 4 Find the rms speed of oxygen molecules in a gas at300 K.<\/strong><\/p>\n

                            Solution:<\/p>\n

                            Example: 5. Two molecules of a gas have speed of <\/strong> \u00a0and \u00a0<\/strong> \u00a0respectively. What is the root mean source speed of these molecules?<\/strong><\/p>\n

                            Ans.\u00a0 Root mean square speed<\/p>\n

                            For two molecules<\/p>\n

                             <\/p>\n

                              \n
                            1. Degree of Freedom:-<\/u><\/strong><\/li>\n<\/ol>\n

                              The total number of independent co-ordinate required to describe completely the position and configuration of the system is called degree of freedom. e.g:-<\/p>\n

                                \n
                              • A particle moving along a straight line can change only one axis so have only one degree freedom?<\/li>\n
                              • A particle moving in plane needs two co-ordinates to change so have two degree of freedom<\/li>\n
                              • A particle moving in space can change three co-ordinates so have three degree of freedom.<\/li>\n
                              • There are N<\/em> number of particle, each having three degree of freedom and K<\/em> number of constrains or independent relations, then total number of degree of freedom<\/strong><\/li>\n
                              • A rigid body has six degree of freedom, 3 Translatory and three rotatory.<\/li>\n
                              • Degree of freedom of monatomic Gas:<\/u><\/strong><\/li>\n<\/ul>\n

                                As monatomic gas like \u00a0are consist of single molecules having only translatory motion. So have three degree of freedom.<\/p>\n

                                  \n
                                • Degree of freedom of diatomic Gas: <\/u><\/strong><\/li>\n<\/ul>\n

                                  The molecules like \u00a0are consist of two atoms, connected by one bond so degree of freedom<\/p>\n

                                    \n
                                  • Degree of freedom of Tri-atomic Gas:<\/u><\/strong><\/li>\n
                                  • Non linear gas like \u00a0 In this case the degree of freedom<\/li>\n
                                  • Linear gas In this case there are only two constrains so<\/li>\n<\/ul>\n
                                      \n
                                    1. Law of Equipartion of Energy:<\/u><\/strong><\/li>\n<\/ol>\n

                                      According to law of equipartition of energy, the energy is equally distributed among the various degrees of freedom and the average energy associated with each degree of freedom is<\/p>\n

                                      Proof: –<\/em><\/strong>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 as we know from kinetic theory of gases.<\/p>\n

                                      Here<\/p>\n

                                      or<\/p>\n

                                      or<\/p>\n

                                      Thus average kinetic energy per degree of freedom \u00a0this is obtained by Boltzmann, so also known as Boltzman’s equip partition of energy.<\/p>\n

                                        \n
                                      1. Specific heat of Monatomic, Diatomic and Poly atomic Gases:<\/u><\/strong><\/li>\n<\/ol>\n

                                        \u00a0(a)\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Specific heat of monatomic gas:-<\/u><\/strong><\/p>\n

                                        As monatomic gas have three degree of freedom. So energy associated with monatomic gas is<\/p>\n

                                        U= \u00a0 \u00a0\u00a0\u00a0 Where<\/p>\n

                                        So total energy associated with one mole of gas<\/p>\n

                                        Now Molar specific heat of gas at constant volume \u00a0\u00a0\u00a0 or<\/p>\n

                                        And Molar specific heat at constant pressure<\/p>\n

                                        So specific heat ratio<\/p>\n

                                        (b)<\/em><\/strong>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Specific heat of Diatomic Gas<\/u>:<\/strong><\/p>\n

                                        The total degree of freedom of diatomic gas is<\/p>\n

                                        So energy associated with one mole of diatomic gas<\/p>\n

                                        Now \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0 And<\/p>\n

                                        So<\/p>\n

                                        (c)\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Specific Heat of Tri-atomic Gas:<\/u><\/strong><\/p>\n

                                        \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 (i)\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 For non linear tri-atomic gas:<\/strong><\/p>\n

                                         <\/p>\n

                                        And\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 And<\/p>\n

                                        So<\/p>\n