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Structural Determination Approach for Linear Molecules

Detailed methodology for determining atom position from the Center of Mass in linear molecules using moments of inertia and coordinate calculations.

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Structural Determination Approach for Linear Molecules

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  1. General Method of Structure Determination for Linear Molecules

  2. General Method of Structure Determination for Linear Molecules We wish to determine r2 the position of a particular atom (mass m2) from the Center of Mass (C of M)

  3. General Method of Structure Determination for Linear Molecules We wish to determine r2 the position of a particular atom (mass m2) from the Center of Mass (C of M) a m1 m2 r2

  4. General Method of Structure Determination for Linear Molecules We wish to determine r2 the position of a particular atom (mass m2) from the Center of Mass (C of M) a m1 m2 I = Moment of Inertia of the normal species about a the C of M

  5. General Method of Structure Determination for Linear Molecules We wish to determine r2 the position of a particular atom (mass m2) from the Center of Mass (C of M) a b d m1 m2 I = Moment of Inertia of the normal species about a the C of M I* = Moment of Inertia of the substituted species about b its C of M

  6. General Method of Structure Determination for Linear Molecules We wish to determine r2 the position of a particular atom (mass m2) from the Center of Mass (C of M) a b d m1 m2 → m2+∆m I = Moment of Inertia of the normal species about a the C of M I* = Moment of Inertia of the substituted species about b its C of M

  7. General Method of Structure Determination for Linear Molecules We wish to determine r2 the position of a particular atom (mass m2) from the Center of Mass (C of M) a b m1 m2 I = Moment of Inertia of the normal species about a the C of M I* = Moment of Inertia of the substituted species about b its C of M I’ = Moment of Inertia of the substituted species about a

  8. a m1 m2 r1 r2

  9. a is the axis of the normal molecule a m1 m2 r1 r2

  10. a is the axis of the normal molecule b is the axis of the substituted molecule mass m2+ ∆m a b m1 m2 r1 r2

  11. a is the axis of the normal molecule b is the axis of the substituted molecule mass m2+ ∆m a b m1 m2 → m2+∆m r1 r2

  12. a is the axis of the normal molecule b is the axis of the substituted molecule mass m2+ ∆m a b d m1 m2 → m2+∆m r1 r2

  13. a is the axis of the normal molecule b is the axis of the substituted molecule a b d m1 m2 r1 r2 For the substituted molecule the parallel axis theorem yields

  14. a is the axis of the normal molecule b is the axis of the substituted molecule a b d m1 m2 r1 r2 • For the substituted molecule the parallel axis theorem yields • I’ = I* + (M + ∆m)d2

  15. a is the axis of the normal molecule b is the axis of the substituted molecule a b d m1 m2 r1 r2 • For the substituted molecule the parallel axis theorem yields • I’ = I* + (M + ∆m)d2 • I’ = I +∆mr22

  16. a is the axis of the normal molecule b is the axis of the substituted molecule a b d m1 m2 r1 r2 • For the substituted molecule the parallel axis theorem yields • I’ = I* + (M + ∆m)d2 • I’ = I +∆mr22 • I* + (M + ∆m)d2 = I +∆mr22

  17. a is the axis of the normal molecule b is the axis of the substituted molecule a b d m1 m2 r1 r2 • For the substituted molecule the parallel axis theorem yields • I’ = I* + (M + ∆m)d2 • I’ = I +∆mr22 • I* + (M + ∆m)d2 = I +∆mr22 • I* - I = ∆mr22 – (M + ∆m) d2

  18. a is the axis of the normal molecule b is the axis of the substituted molecule a b d m1 m2 r1 r2 • For the substituted molecule the parallel axis theorem yields • I’ = I* + (M + ∆m)d2 • I’ = I +∆mr22 • I* + (M + ∆m)d2 = I +∆mr22 • I* - I = ∆mr22 – (M + ∆m) d2 = ∆I

  19. ∆I = ∆mr22 – (M + ∆m) d2 m2+∆m a b m1 m2 r1 r2 d

  20. ∆I = ∆mr22 – (M + ∆m) d2 • m1(r1 + d) = (m2 + ∆m)(r2 – d) m2+∆m a b m1 m2 r1 r2 d

  21. ∆I = ∆mr22 – (M + ∆m) d2 • m1(r1 + d) = (m2 + ∆m)(r2 – d) • m1r1 + m1d = m2r2 – m2d + ∆mr2 – ∆md m2+∆m a b m1 m2 r1 r2 d

  22. ∆I = ∆mr22 – (M + ∆m) d2 • m1(r1 + d) = (m2 + ∆m)(r2 – d) • m1r1 + m1d = m2r2 – m2d + ∆mr2 – ∆md • m1r1 = m2r2 m2+∆m a b m1 m2 r1 r2 d

  23. ∆I = ∆mr22 – (M + ∆m) d2 • m1(r1 + d) = (m2 + ∆m)(r2 – d) • m1r1 + m1d = m2r2 – m2d + ∆mr2 – ∆md • m1r1 = m2r2 • m1r1+ m1d = m2r2 – m2d + ∆mr2 – ∆md m2+∆m a b m1 m2 r1 r2 d

  24. ∆I = ∆mr22 – (M + ∆m) d2 • m1(r1 + d) = (m2 + ∆m)(r2 – d) • m1r1 + m1d = m2r2 – m2d + ∆mr2 – ∆md • m1r1 = m2r2 • m1r1+ m1d = m2r2 – m2d + ∆mr2 – ∆md • m1d = – m2d + ∆mr2 – ∆md m2+∆m a b m1 m2 r1 r2 d

  25. ∆I = ∆mr22 – (M + ∆m) d2 • m1(r1 + d) = (m2 + ∆m)(r2 – d) • m1r1 + m1d = m2r2 – m2d + ∆mr2 – ∆md • m1r1 = m2r2 • m1r1+ m1d = m2r2 – m2d + ∆mr2 – ∆md • m1d = – m2d + ∆mr2 – ∆md • d(m1 + m2 + ∆m) = ∆mr2 m2+∆m a b m1 m2 r1 r2 d

  26. ∆I = ∆mr22 – (M + ∆m) d2 • m1(r1 + d) = (m2 + ∆m)(r2 – d) • m1r1 + m1d = m2r2 – m2d + ∆mr2 – ∆md • m1r1 = m2r2 • m1r1+ m1d = m2r2 – m2d + ∆mr2 – ∆md • m1d = – m2d + ∆mr2 – ∆md • d(m1 + m2 + ∆m) = ∆mr2 • d = [∆m/(M + ∆m)]r2 m2+∆m a b m1 m2 r1 r2 d

  27. ∆I = ∆mr22 – (M + ∆m) d2 • m1(r1 + d) = (m2 + ∆m)(r2 – d) • m1r1 + m1d = m2r2 – m2d + ∆mr2 – ∆md • m1r1 = m2r2 • m1r1+ m1d = m2r2 – m2d + ∆mr2 – ∆md • m1d = – m2d + ∆mr2 – ∆md • d(m1 + m2 + ∆m) = ∆mr2 • d = [∆m/(M + ∆m)]r2 = ∆mr2/(M + ∆m) m2+∆m a b m1 m2 r1 r2 d

  28. ∆I = ∆mr22 – (M + ∆m) d2 • d = ∆m r2 /(M + ∆m)

  29. ∆I = ∆mr22 – (M + ∆m) d2 • d = ∆m r2 /(M + ∆m) • ∆I = ∆mr22 – (M + ∆m) ∆m2 r22/(M + ∆m)2

  30. ∆I = ∆mr22 – (M + ∆m) d2 • d = ∆m r2 /(M + ∆m) • ∆I = ∆mr22 – (M + ∆m) ∆m2 r22/(M + ∆m)2 • ∆I = ∆mr22 – ∆m2 r22/(M + ∆m)

  31. ∆I = ∆mr22 – (M + ∆m) d2 • d = ∆m r2 /(M + ∆m) • ∆I = ∆mr22 – (M + ∆m) ∆m2 r22/(M + ∆m)2 • ∆I = ∆mr22 – ∆m2 r22/(M + ∆m) • ∆I = r22 [∆m – ∆m2 /(M + ∆m)]

  32. ∆I = ∆mr22 – (M + ∆m) d2 • d = ∆m r2 /(M + ∆m) • ∆I = ∆mr22 – (M + ∆m) ∆m2 r22/(M + ∆m)2 • ∆I = ∆mr22 – ∆m2 r22/(M + ∆m) • ∆I = r22 [∆m – ∆m2 /(M + ∆m)] • ∆I = r22[∆m(M + ∆m) – ∆m2 ]/(M + ∆m)

  33. ∆I = ∆mr22 – (M + ∆m) d2 • d = ∆m r2 /(M + ∆m) • ∆I = ∆mr22 – (M + ∆m) ∆m2 r22/(M + ∆m)2 • ∆I = ∆mr22 – ∆m2 r22/(M + ∆m) • ∆I = r22 [∆m – ∆m2 /(M + ∆m)] • ∆I = r22[∆m(M + ∆m) – ∆m2 ]/(M + ∆m) • ∆I = r22[∆mM + ∆m2 – ∆m2 ]/(M + ∆m)]

  34. ∆I = ∆mr22 – (M + ∆m) d2 • d = ∆m r2 /(M + ∆m) • ∆I = ∆mr22 – (M + ∆m) ∆m2 r22/(M + ∆m)2 • ∆I = ∆mr22 – ∆m2 r22/(M + ∆m) • ∆I = r22 [∆m – ∆m2 /(M + ∆m)] • ∆I = r22[∆m(M + ∆m) – ∆m2 ]/(M + ∆m) • ∆I = r22[∆mM + ∆m2 – ∆m2 ]/(M + ∆m)] • ∆I = r22[∆mM]/(M + ∆m)]

  35. ∆I = r22[∆mM]/(M + ∆m)]

  36. ∆I = r22[∆mM]/(M + ∆m)] ∆I = μr22

  37. ∆I = r22[∆mM]/(M + ∆m)] ∆I = μr22 where μ = M∆m/(M + ∆m)

  38. ∆I = r22[∆mM]/(M + ∆m)] ∆I = μr22 where μ = M∆m/(M + ∆m) The reduced mass on substitution

  39. ∆I = r22[∆mM]/(M + ∆m)] ∆I = μr22 where μ = M∆m/(M + ∆m) The reduced mass on substitution ~ Analogue of m1m2/(m1+m2)

  40. ∆I = r22[∆mM]/(M + ∆m)] ∆I = μr22 where μ = M∆m/(M + ∆m) The reduced mass on substitution ~ Analogue of m1m2/(m1+m2) or Mm/(M+m)

  41. Problem Determine the bond lengths for the molecule H-C≡C-H

  42. Problem Determine the bond lengths for the molecule H-C≡C-H H-C≡C-H B = 1.17692 cm-1

  43. Problem Determine the bond lengths for the molecule H-C≡C-H H-C≡C-H B = 1.17692 cm-1 H-C≡C-D B = 0.99141 cm-1

  44. ∆I = ∆mr22 – (M + ∆m) d2 m1(r1 + d) = (m2 + ∆m)(r2 – d) m1r1 + m1d = m2r2 – m2d + ∆mr2 – ∆md m1r1 = m2r2 m1r1+ m1d = m2r2 – m2d + ∆mr2 – ∆md m1d = – m2d + ∆mr2 – ∆md d(m1 + m2 + ∆m) = ∆mr2 d = [∆m/(M + ∆m)]r2 m2+∆m a b m1 m2 r1 r2 d

  45. I’ = I* + (M + ∆m)d2 • I’ = I +∆mr22 • I* + (M + ∆m)d2 = I +∆mr22 • I’ = I* + (M + ∆m)d2 • I’ = I +∆mr22 • I* - I = ∆mr22 – (M + ∆m) d2 I • m1r1 = m2r2 • M1(r1 + d) = (m2 + ∆m)(r2 – d) • m1r1 + m1d = m2r2 – m2d + ∆mr2 – ∆md • d(m1 + m2 + ∆m) = ∆mr2 • d = {∆m/(M + ∆m)}r2 a b m1 m2 r1 r2 d

  46. a is the axis of the normal molecule b is the axis of the substituted molecule a b d m1 m2 r1 r2 For the substituted molecule the parallel axis theorem yields I’ = I* + (M + ∆m)d2 I’ = I +∆mr22 I* + (M + ∆m)d2 = I +∆mr22 I* - I = ∆mr22 – (M + ∆m) d2

  47. I’ = I* + (M + ∆m)d2 • I’ = I +∆mr22 • I* + (M + ∆m)d2 = I +∆mr22 • I’ = I* + (M + ∆m)d2 • I’ = I +∆mr22 • I* - I = ∆mr22 – (M + ∆m) d2 I • m1r1 = m2r2 • M1(r1 + d) = (m2 + ∆m)(r2 – d) • m1r1 + m1d = m2r2 – m2d + ∆mr2 – ∆md • d(m1 + m2 + ∆m) = ∆mr2 • d = {∆m/(M + ∆m)}r2 a b d m1 m2

  48. General Method of Structure Determination for Linear Molecules We wish to determine r2 the position of a particular atom (mass m2) from the Center of Mass (C of M) a is the axis of the normal molecule b is the axis of the substituted molecule a b d m1 m2 r1 r2 • For the substituted molecule the parallel axis theorem yields • I’ = I* + (M + ∆m)d2 • I’ = I +∆mr22 • I* - I = ∆mr22 – (M + ∆m) d2

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