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Kinetic theory of gases

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Kinetic theory of gases

  1. 1. Kinetic Theory of Gases
  2. 2. Ideal GasThe number of molecules is largeThe average separation between molecules is largeMolecules moves randomlyMolecules obeys Newton’s LawMolecules collide elastically with each other and with the wallConsists of identical molecules
  3. 3. The Ideal Gas Law PV = nRT in Kn: the number of moles in the ideal gas N total number n= NA of molecules Avogadro’s number: the number of atoms, molecules, etc, in a mole of a substance: NA=6.02 x 1023/mol.R: the Gas Constant: R = 8.31 J/mol · K
  4. 4. Pressure and Temperature Pressure: Results from collisions of molecules on the surface Force F Pressure: P= A Area dp Force: F= Rate of momentum dt given to the surfaceMomentum: momentum given by each collision times the number of collisions in time dt
  5. 5. Only molecules moving toward the surface hitthe surface. Assuming the surface is normal tothe x axis, half the molecules of speed vx movetoward the surface.Only those close enough to the surface hit itin time dt, those within the distance vxdtThe number of collisions hitting an area A intime dt is 1  N ⋅ A ⋅ vx ⋅ dt 2V  Average densityThe momentum given by each collision tothe surface 2mvx
  6. 6. Momentum in time dt: 1  N dp = (2mv x )⋅ ⋅ ⋅ A ⋅v x dt 2  VForce: dp 1  N F= = (2mvx )⋅ ⋅ ⋅ A⋅ v x dt 2 V Pressure: F N 2 P = = mv x A VNot all molecules have the same v x ⇒ average v2 x N 2 P = mv x V
  7. 7. 2 vx 1 2 1 2 2 3 3 ( 2 = v = v x + v y + vz ) 2 1 2 1 2 vx = v = vrms 3 3 vrms is the root-mean-square speed 2 2 2 vx + vy + vz vrms = v 2 = 3 1N 2 2  N1 2Pressure: P = mv = mv 3V 3  V 2 Average Translational Kinetic Energy: 1 2 1 2 K = mv = mvrms 2 2
  8. 8. 2 NPressure: P = ⋅ ⋅K 3 V 2From PV = ⋅ N ⋅ K and PV = nRT 3 3 nRT 3Temperature: K= ⋅ = ⋅ k BT 2 N 2 R −23Boltzmann constant: k B = = 1.38 × 10 J/K NA
  9. 9. 1 2From PV = ⋅ N ⋅ mvrms 3 Nand PV = nRT = RT NA Avogadro’s number N = nN A 3RT vrms = M Molar mass M = mN A
  10. 10. Internal EnergyFor monatomic gas: the internal energy = sumof the kinetic energy of all molecules: 3 3 Eint = N ⋅ K = nN A ⋅ k BT = nRT 2 2 3 Eint = nRT ∝ T 2
  11. 11. Mean Free Path Molecules collide elastically with other molecules Mean Free Path λ: average distance between two consecutive collisions 1 λ= 2 2πd N / Vthe bigger the molecules the more molecules the more collisions the more collisions
  12. 12. Q = cm ⋅ ∆T Molar Specific Heat ∆Eint = Q − W 3 Eint = nRT 2Definition: For constant volume: Q = nCV ∆T For constant pressure: Q = nC p ∆TThe 1st Law of Thermodynamics: 3 ∆Eint = nR∆T = Q − W (Monatomic) 2
  13. 13. 3 nR∆T = Q − WConstant Volume 2 (Monatomic) Q = nCV ∆TW = ∫ PdV = 0 3 Eint = nRT3 2 nR∆T = nCV ∆T2 3CV = R 2Eint = nCV T
  14. 14. 3 nR∆T = Q − W 2 Constant Pressure (Monatomic) Q = nC p ∆TW = P∆V = nR∆T3 nR∆T = nC p ∆T − nR∆T2 5CV = Cp − R Cp = R 2 Cp 5γ = γ = CV 3
  15. 15. 1st Law dEint = dQ − dW Adiabatic Process Ideal Gas Law pV = nRT (Q=0) Eint = nCV TdEint = −dW = − pdV C p = CV + R = nCV dT Cp γ =  pdV  CVpdV + Vdp = nRdT = nR −   nCV Divide by pV:dV dp  C p − CV  dV dV + = −  = (1 − γ )V p  CV  V V
  16. 16. dV dp dV Ideal Gas Law + = (1 − γ ) pV = nRTV p Vdp dV +γ =0 p V γ γln p + ln V = ln( pV ) = const. γ pV = const. nRT γ( )V = const. V γ −1 TV = const.
  17. 17. Equipartition of EnergyThe internal energy of non-monatomicmolecules includes also vibrational androtational energies besides thetranslational energy.Each degree of freedom has associated with 1it an energy of k BT per molecules. 2
  18. 18. Eint = nCV T Monatomic Gases3 translational degrees of freedom: 3 3 Eint = kBT ⋅nN A = nRT 2 2 1 dEint 3 CV = ⋅ = R n dT 2
  19. 19. Eint = nCV T Diatomic Gases3 translational degrees of freedom2 rotational degrees of freedom2 vibrational degrees of freedomHOWEVER, different DOFs require differenttemperatures to excite. At room temperature,only the first two kinds are excited: 5 5 Eint = nRT CV = R 2 2

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