Energy Distribution in Molecular Systems

Energy Distribution in Molecular Systems
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Energy distribution in molecular systems is crucial for determining thermodynamic properties. The most probable configuration of energy at equilibrium leads to the Boltzmann distribution. Configurations represent the total energy available to the system, while microstates describe individual oscillator energies. Energy levels for oscillators are determined by the quantum number, and a modified harmonic oscillator model is used for analysis.

  • Energy Distribution
  • Molecular Systems
  • Thermodynamics
  • Boltzmann Distribution
  • Quantum Number

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  1. CHM 331

  2. PHYSICAL CHEMISTRY III

  3. PART B

  4. MAXWELL BOLTZMANS STATISTICS

  5. most probable configuration of energy for a molecular system at equilibrium. It is used to derive important thermodynamics properties of the system.

  6. The distribution of energy associated with the dominant configuration is known as Boltzmann distribution.

  7. Introduction

  8. Configuration is a general arrangement of total energy available to the system available to the system.

  9. of energy that describes the energy contained by each individual oscillator. Microstates are equivalent to permutations.

  10. Energy levels for oscillators are given by

  11. E = h (n + ), n = 0, 1, 2 ---- (i)

  12. = oscillator frequency

  13. n = quantum number associated with a given energy level of the oscillator.

  14. A modified version of the harmonic oscillator is given as

  15. E n = h n, n = 0, 1, 2 ---- (ii)

  16. For a ground state energy n = 0

  17. Microstates and Configurations

  18. probability theory to chemical systems, the configurations with the largest number of corresponding permutations is the most probable configuration.

  19. PE= E/ N,

  20. Where, PE= probability of the configuration trial outcome

  21. E = number of permutation associated with the event of interest.

  22. N = total number of possible permutations

  23. The most likely configurationally outcome for a trial is the configuration the greatest number of associated permutations.

  24. Number of microstates = = (iii)

  25. the number of units occupying a given energy level e.g. in the configuration 3, 0, 0, a0 = 2, a3 =1, and all other an = 0 which is 0!, = 1.

  26. The probability of observing a configuration is given as

  27. Pi = = (iv)

  28. Configuration with the largest weight is called the predominant configuration.

  29. Dominant configuration = dln W/d X = 0 (v)

  30. The sum of all probabilities is unity

  31. P1 + P2 + ---+Pm = (vi)

  32. Pi = 1/n , n = total number of variables

  33. PE= 1/N = 1/N = E/N (vii)

  34. Mj having nj ways to perform the entire series manipulations, the total number of ways to perform the entire series of manipulations (total M) is

  35. Total M = (n1)(n2)...(nj) (viii)

  36. Total number of permutation = n! of n objects.

  37. P (n, j) represents the number of permutations possible using a subset of j objects from the total group of n,

  38. P (n, j) = n (n 1).... (n j + 1) = (ix)

  39. Configuration is an unordered arrangement of objects manipulated = n

  40. Subset of objects = j

  41. Configuration is j)/ j! = C (n, j) = P (n, (x)

  42. Stirlings Approximation

  43. Provides a simple method of calculating the natural log of N! It is written as

  44. All microstates are equally probable; with a microstate associated with the dominant configuration.

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