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Maxwell Velocity Distribution & Molecular Energy Levels

Explore Maxwell-Boltzmann Distribution, Velocity Distribution of Gases, and Molecular Internal Energy concepts in physics. Learn about probability distributions, normalized functions, and energy levels. Understand the significance of rotational, translational, and vibrational motions in complex molecules.

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Maxwell Velocity Distribution & Molecular Energy Levels

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  1. PROBABILITY DISTRIBUTIONS FINITECONTINUOUS ∑ Ng = N NvΔv = N

  2. PROBABILITY DISTRIBUTIONS FINITECONTINUOUS ∑ Ng = N NvΔv = N Pg = Ng /N ∫Nv dv = N Pv = Nv /N

  3. PROBABILITY DISTRIBUTIONS FINITECONTINUOUS ∑ Ng = N NvΔv = N Pg = Ng /N ∫Nv dv = N Normalized Pv = Nv /N ∑ Pg = 1 ∫Pv dv = 1

  4. PROBABILITY DISTRIBUTIONS FINITECONTINUOUS ∑ Ng = N NvΔv = N Pg = Ng /N ∫Nv dv = N Normalized Pv = Nv /N ∑ Pg = 1 ∫Pv dv = 1 < g> = ∑ g Pg < v > = ∫vPv dv

  5. PROBABILITY DISTRIBUTIONS FINITECONTINUOUS ∑ Ng = N NvΔv = N Pg = Ng /N ∫Nv dv = N Normalized Pv = Nv /N ∑ Pg = 1 ∫Pv dv = 1 < g> = ∑ g Pg < v > = ∫vPv dv <g2> = ∑ g2 Pg < v2> = ∫v2 Pv dv

  6. Velocity Distribution of Gases • Maxwell Velocity Distribution for gases is • N(v) dv = N4πv2 (m/2πkT)3/2 e –mv^2/2kTdv • where N is the number of molecules of mass m and temperature T.

  7. Velocity Distribution of Gases • Maxwell Velocity Distribution for gases is • N(v) dv = N4πv2 (m/2πkT)3/2 e –mv^2/2kTdv • where N is the number of molecules of mass m and temperature T. If one divides by N and changes the differential element dv to d3v = dvx dvy dvz ,

  8. Velocity Distribution of Gases • Maxwell Velocity Distribution for gases is • N(v) dv = N4πv2 (m/2πkT)3/2 e –mv^2/2kTdv • where N is the number of molecules of mass m and temperature T. If one divides by N and changes the differential element dv to d3v = dvx dvy dvz , then the normalized probability function F(v) is: • F(v) = (m/2πkT)3/2 e –mv^2/2kT

  9. Velocity Distribution of Gases • This velocity probability distribution has all the properties given before: ∫ F(v) d3v = 1

  10. Velocity Distribution of Gases • This velocity probability distribution has all the properties given before: ∫ F(v) d3v = 1 and the mean velocity and the mean of the square velocity are: <v> = ∫ v F(v) d3v <v2 > = ∫ v2F(v) d3v

  11. Velocity Distribution of Gases • This velocity probability distribution has all the properties given before: ∫ F(v) d3v = 1 and the mean velocity and the mean of the square velocity are: <v> = ∫ v F(v) d3v <v2 > = ∫ v2F(v) d3v (remember d3v means one must do a triple integration over dvx dvy dvz )

  12. Velocity Distribution of Gases • The results of this are: • <v> = √(8kT/(πm)) = 1.59 √kT/m

  13. Velocity Distribution of Gases • The results of this are: • <v> = √(8kT/(πm)) = 1.59 √kT/m • <v2> = √(3kT/m) = 1.73 √kT/m

  14. Velocity Distribution of Gases • The results of this are: • <v> = √(8kT/(πm)) = 1.59 √kT/m • <v2> = √(3kT/m) = 1.73 √kT/m • If one sets the derivative of the probability function to zero (as was done for the Planck Distribution) one obtains the most probable value of v

  15. Velocity Distribution of Gases • The results of this are: • <v> = √(8kT/(πm)) = 1.59 √kT/m • <v2> = √(3kT/m) = 1.73 √kT/m • If one sets the derivative of the probability function to zero (as was done for the Planck Distribution) one obtains the most probable value of v • vmost prob = √(2kT/m) = 1.41√kT/m

  16. Maxwell-Boltzmann Distribution • Molecules with more complex shape have internal molecular energy. Boltzmann realized this and changed Maxwell’s Distribution to include all the internal energy. FM (v)  FMB (v) FMB (v) = (1/Z) e –E/kT where Z = the normalization factor

  17. Maxwell-Boltzmann Distribution • Molecules with more complex shape have internal molecular energy.

  18. Maxwell-Boltzmann Distribution • Molecules with more complex shape have internal molecular energy. Boltzmann realized this and changed Maxwell’s Distribution to include all the internal energy. FM (v)  FMB (v)

  19. Maxwell-Boltzmann Distribution • Molecules with more complex shape have internal molecular energy. Boltzmann realized this and changed Maxwell’s Distribution to include all the internal energy. FM (v)  FMB (v) FMB (v) = (1/Z) e –E/kT where Z = the normalization factor

  20. MOLECULAR INTERNAL ENERGY • Diatomic, Triatomic and more complex molecules can rotate and vibrate about their symmetrical axes.

  21. MOLECULAR INTERNAL ENERGY • Diatomic, Triatomic and more complex molecules can rotate and vibrate about their symmetrical axes. • EINT = < E > = ETRANS + EROT + EVIBR

  22. MOLECULAR INTERNAL ENERGY • Diatomic, Triatomic and more complex molecules can rotate and vibrate about their symmetrical axes. • EINT = < E > = ETRANS + EROT + EVIBR • ETRANS = < ETRANS > = ½ m <v2>

  23. MOLECULAR INTERNAL ENERGY • Diatomic, Triatomic and more complex molecules can rotate and vibrate about their symmetrical axes. • EINT = < E > = ETRANS + EROT + EVIBR • ETRANS = < ETRANS > = ½ m <v2> • EROT = ½ Ixωx2 + ½ Iyωy2 + ½ Izωz2

  24. MOLECULAR INTERNAL ENERGY • Diatomic, Triatomic and more complex molecules can rotate and vibrate about their symmetrical axes. • EINT = < E > = ETRANS + EROT + EVIBR • ETRANS = < ETRANS > = ½ m <v2> • EROT = ½ Ixωx2 + ½ Iyωy2 + ½ Izωz2 • Diatomic (2 axes) Triatomic (3 axes)

  25. MOLECULAR INTERNAL ENERGY • Diatomic, Triatomic and more complex molecules can rotate and vibrate about their symmetrical axes. • EINT = < E > = ETRANS + EROT + EVIBR • ETRANS = < ETRANS > = ½ m <v2> • EROT = ½ Ixωx2 + ½ Iyωy2 + ½ Izωz2 • Diatomic (2 axes) Triatomic (3 axes) • EVIBR = - ½ k x2VIBR (for each axis)

  26. INTERNAL MOLECULAR ENERGY • For a diatomic molecule then <E> = 5/2 kT

  27. INTERNAL MOLECULAR ENERGY • For a diatomic molecule then <E> = 5/2 kT • One of the basic principles used in the Kinetic Theory of Gases is that each degree of freedom has an average energy • of ½ kT.

  28. INTERNAL MOLECULAR ENERGY • For a diatomic molecule then <E> = 5/2 kT • One of the basic principles used in the Kinetic Theory of Gases is that each degree of freedom has an average energy • of ½ kT. Or <E> = (s/2) kT

  29. INTERNAL MOLECULAR ENERGY • For a diatomic molecule then <E> = 5/2 kT • One of the basic principles used in the Kinetic Theory of Gases is that each degree of freedom has an average energy • of ½ kT. Or <E> = (s/2) kT where s = the number of degrees of freedom

  30. INTERNAL MOLECULAR ENERGY • For a diatomic molecule then <E> = 5/2 kT • One of the basic principles used in the Kinetic Theory of Gases is that each degree of freedom has an average energy • of ½ kT. Or <E> = (s/2) kT where s = the number of degrees of freedom • This is called the • EQUIPARTION THEOREM

  31. INTERNAL MOLECULAR ENERGY • For dilute gases which still obey the ideal gas law, the internal energy is:

  32. INTERNAL MOLECULAR ENERGY • For dilute gases which still obey the ideal gas law, the internal energy is: • U = N<E> = (s/2) NkT

  33. INTERNAL MOLECULAR ENERGY • For dilute gases which still obey the ideal gas law, the internal energy is: • U = N<E> = (s/2) NkT • Real gases undergo collisions and hence can transport matter called diffusion.

  34. INTERNAL MOLECULAR ENERGY • For dilute gases which still obey the ideal gas law, the internal energy is: • U = N<E> = (s/2) NkT • Real gases undergo collisions and hence can transport matter called diffusion. The average distance a molecule moves between collisions is <λ>

  35. COLLISIONS OF MOLECULES • Let D be the diameter of a molecule.

  36. COLLISIONS OF MOLECULES • Let D be the diameter of a molecule. The collision cross section is merely the cross-sectional area σ = π D2 .

  37. COLLISIONS OF MOLECULES • Let D be the diameter of a molecule. The collision cross section is merely the cross-sectional area σ = π D2 . If there is a collision then the molecule traveles a distance λ = vt.

  38. COLLISIONS OF MOLECULES • Let D be the diameter of a molecule. The collision cross section is merely the cross-sectional area σ = π D2 . If there is a collision then the molecule traveles a distance λ = vt. If one averages this • <λ> = vRMSτ where τ = mean collision time.

  39. COLLISIONS OF MOLECULES • Let D be the diameter of a molecule. The collision cross section is merely the cross-sectional area σ = π D2 . If there is a collision then the molecule traveles a distance λ = vt. If one averages this • <λ> = vRMSτ where τ = mean collision time. During this time there are N collisions in a volume V.

  40. MOLECULAR COLLISIONS • The molecule sweeps out a volume which is V = AvRMSτ =

  41. MOLECULAR COLLISIONS • The molecule sweeps out a volume which is V = AvRMSτ = σ vRMSτ

  42. MOLECULAR COLLISIONS • The molecule sweeps out a volume which is V = AvRMSτ = σ vRMSτ Since there are number / volume (ndens ) molecules undergoing a collision then the average number of collisions per unit time is τ = 1/N

  43. MOLECULAR COLLISIONS • The molecule sweeps out a volume which is V = AvRMSτ = σ vRMSτ Since there are number / volume (ndens ) molecules undergoing a collision then the average number of collisions per unit time is τ = 1/N = 1/(nV/τ)

  44. MOLECULAR COLLISIONS • The molecule sweeps out a volume which is V = AvRMSτ = σ vRMSτ Since there are number / volume (ndens ) molecules undergoing a collision then the average number of collisions per unit time is τ = 1/N = 1/(nV/τ) = 1/nσ vRMS .

  45. MOLECULAR COLLISIONS • The molecule sweeps out a volume which is V = AvRMSτ = σ vRMSτ Since there are number / volume (ndens ) molecules undergoing a collision then the average number of collisions per unit time is τ = 1/N = 1/(nV/τ) = 1/nσ vRMS . However, both molecules are moving and this increases the velocity by √ 2.

  46. MOLECULAR COLLISIONS • Thus τ = 1/ (√2 nσvRMS )

  47. MOLECULAR COLLISIONS • Thus τ = 1/ (√2 nσvRMS ) • and <λ> = vRMS τ = 1/(√2 nσ)

  48. MOLECULAR COLLISIONS • Thus τ = 1/ (√2 nσvRMS ) • and <λ> = vRMS τ = 1/(√2 nσ) In 1827 Robert Brown observed small particles moving in a suspended atmosphere. This was later hypothesised to be due to collisions by gas molecules.

  49. MOLECULAR COLLISIONS • The movement of these particles was observed to be random and was similar to the mathematical RANDOM WALK problem.

  50. MOLECULAR COLLISIONS • The movement of these particles was observed to be random and was similar to the mathematical RANDOM WALK problem. See the link below: • http://www.aip.org/history/einstein/brownian.htm

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