Lecture 8 the kinetic theory of gases

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CHAPTER 21 THE KINETIC THEORY OF GASES – THE FIRST LAW OF THERMODYNAMICS The kinetic theory of Gases The number of molecules in the gas is large, and the average separation between them is large compared with their dimensions In other words, the molecules occupy a negligible volume in the container We model the molecules as particles The molecules obey Newton’s laws of motion, but as a whole they move randomly By “randomly,” we mean that any molecule can move in any direction with any speed The molecules interact only by short-range forces during elastic collisions That is consistent with the ideal gas model, in which the molecules exert no long-range forces on each other The molecules make elastic collisions with the walls These collisions lead to the macroscopic pressure on the walls of the container The gas under consideration is a pure substance; that is, all molecules are identical KINETIC THEORY The pressure that a gas exerts is caused by the collisions of its molecules with the walls of the container We try to derive the equation that relates the macroscopic quantity P and the microscopic quantity of the gas – the average Translational Kinetic Energy  P  no Pressure (N/m2) particle density (m^-3) Average Translational Kinetic Energy of particle KINETIC THEORY Consider a collection of N molecules of an ideal gas in a container of volume V=L3 Consider molecule i of mass m, velocity vi, It collides the wall The impulse – momentum theorem gives    mv'i  mvi  Fwall particle   2mv xi  Fwall particle   ex  L v’ v x 2L  v xi  mv2xi  Fwall particle   ex L P  no Pressure (N/m2)    mv'i  mvi  Fwall particle   2mv xi  Fwall particle   ex  2L  v xi  mv 2xi  Fwall particle   ex L   mv 2xi  Fparticle wall   Fwall particle  ex L  mv 2xi m   Fparticle wall   L  N L v 2x ex i v2     Nm v   Fparticle wall  L ex  Fparticle wall N mv P  L3 L P  no N mv no  ;  L v 2x particle density (m^-3) Average Translational Kinetic Energy of particle v 2y v 2z Temperature is a direct measure of average molecular translational kinetic energy P  no nRT nN A k BT P   n o k BT V V   k BT Theorem of equipartition of energy The average translational kinetic energy per molecule is 2 2   m v  m v x  m v y  m v z  k BT 2 2 Translational motion has degree of freedom Each degree of freedom contributes ½ kBT to the energy of a system It is also right for degrees freedom associated translatioal, rotation, and vibration of molecules Degree of Freedom i Monoatomic Gas :i transl  Diatomic Gas : i transl  3; i rotation  2; i oscillation  Manyatomic Gas :i transl  3; i rotation  3; i oscillation  3N  Internal Energy Internal energy  number of particle  energy of particle i U  N k BT N  nN A n : number of mol R  N A k B :Universal Gas Const i i U  nN A k BT  nRT 2 i U  nRT Equation of state for an ideal gas PV=nRT R is called the universal gas constant R=8.31 J/(mol.K) P pressure is expressed in pascals (1 Pa = N/m2) V volume in cubic meters PV has units of joules 20.5 The First Law of Thermodynamics • The first law of thermodynamics states that when a system undergoes a change from one state to another, the change in its internal energy is U  Q  W • where Q is the energy transferred into the system by heat and W is the work done on the system • Although Q and W both depend on the path taken from the initial state to the final state, the quantity U does not depend on the path Work done on the system   dW  F.d s  P.S.dy   PdV V2 W    PdV V1 Appications of the 1rst Law of TD Isothermal process of an Ideal Gas T  const : PV  const U  Q  W U  V2 nRT V2 W    PdV    dV   nRT ln V V1 V1 V2 Q   W  nRT ln V1 isobaric process P=const P  const : V / T  const U  Q  W W    PdV   PV   nRT U  Q  PV i i2 Q  U  PV  nRT  nRT  n RT 2 Q  nC p T i2 Cp  R isovolumetric process V=const V  const : U  Q  W W    PdV  i U  Q  n RT  nC V T i CV  R Adiabatic process Q=0 U  Q  W Q0 i U  W  nRT  PV  const PV  const T  1 TV  const  1  T P  const The molar specific heat at constant volume : CV  The molar specific heat at constant pressure Cp  The ratio of specific heats R  1 R CP   1 CP  CV  R CV  CP i    CV i i R i2 R ADIABATIC PROCESS dU  Q  PdV Adiabatic process : Q  dU   PdV i dU  nRdT PV  nRT PdV  VdP  nRdT i i dU  nRdT  (PdV  VdP) 2 i (PdV  VdP)   PdV i2 i PdV  VdP  2 i  dV dP  0 i V P  ln V  ln P  ln C PV   Const PV   const  TV  1  const PV    1   const  T P  const T  SUMMARY .. .The kinetic theory of Gases The number of molecules in the gas is large, and the average separation between them is large compared with their dimensions In other words, the molecules... with the walls These collisions lead to the macroscopic pressure on the walls of the container The gas under consideration is a pure substance; that is, all molecules are identical KINETIC THEORY. .. of the gas – the average Translational Kinetic Energy  P  no Pressure (N/m2) particle density (m^-3) Average Translational Kinetic Energy of particle KINETIC THEORY Consider a collection of