SlideServe
Télécharger Obtenir Pro
Menu
Accueil Pro
Préférences
ApparenceLight Aide et commentaires
© SlideServe · v2026

Velocity Distributions of Molecules

monita avatar
monita
17 diapositives · 294 Views · Aug 14, 2014

This document explores the velocity distributions of molecules in three-dimensional space, focusing on the probability functions associated with molecular velocities in the x, y, and z directions. It describes methods for deriving the Maxwell Velocity Distribution Function through various equations that account for the random motion of particles. The analysis includes defining a velocity density and examining isotropic velocity fields, ultimately detailing the mathematical background necessary for understanding molecular movement and distribution in statistical mechanics.

#velocity distributions of molecules #velocity distributions of molecules ppt #powerpoint #presentation
Velocity Distributions of Molecules - slide 1
01 / 17Velocity Distributions of Molecules
1 / 17

Télécharger la présentation

Velocity Distributions of Molecules
Une image / un lien ci-dessous est fourni (tel quel) pour télécharger la présentation

Politique de téléchargement : le contenu du site vous est fourni TEL QUEL pour votre information et votre usage personnel ; il ne peut être vendu, concédé sous licence ni partagé sur d’autres sites sans l’accord de son auteur.Le contenu vous est fourni TEL QUEL, pour votre information et votre usage personnel uniquement. Téléchargez la présentation en cliquant sur ce lien. Lors du téléchargement, si vous ne parvenez pas à télécharger une présentation, l’éditeur a peut-être supprimé le fichier de son serveur.Pendant le téléchargement, si vous ne parvenez pas à obtenir une présentation, le fichier a peut-être été supprimé par l’éditeur.

F I N

Plus de contenu associé

Transcription de la présentation

  1. Velocity Distributions of Molecules

  2. Velocity Distributions • Let dNvx be the number of molecules moving with velocity vx, etc.; • Each is represented by a slice in Fig. 2. • The fraction of molecules moving with a given velocity in that direction is then; dNvx /N = f(vx)dvx (1) dNvy /N = f(vy)dvy (2) dNvz /N = f(vz)dvz (3)

  3. Velocity Distributions Let d2Nvx,vy be the number of molecules moving with velocity vx and vy. Then d2Nvx,vy/dNvx = dNvy/N = f(vy)dvy since motion is random Substituting for dNvx we obtain: d2Nvx,vy = Nf(vx)f(vy)dvxdvy (4)

  4. Velocity Distributions Extending this to three dimensions yields: d3Nvx,vy,vz = Nf(vx)f(vy)f(vz)dvxdvydvz (5) where d3Nvx,vy,vz isthe number of particles with velocities between vx, vx+ dvx, vy, vy, + dvy, vz, vz+dvz and is represented by volume dvx, dvy, dvz located at vx, vy, vz in Fig. 2.

  5. Velocity Distributions Define a Velocity Density: r = d3Nvx,vy,vz/ dvxdvydvz = Nf(vx)f(vy)f(vz) (6) If velocity field is isotropic then r is constant in any shell representing a given velocity. Now lets move a small distance dx, dy, dz and observe the resulting change in r:

  6. Velocity Distributions dr = (dr/dvx)dvx + (dr/dvy)dvy + (dr/dvz)dvz (7) From Equ (6): dr/dvx = Nf’(vx)f(vy)f(vz) dr/dvy = Nf(vx)f’(vy)f(vz) dr/dvz = Nf(vx)f(vy)f’(vz)

  7. Velocity Distributions For the special case where the new point also lies in the same shell: dr = 0 Now substitute dr/dv terms into Equ (7) and divide through by Equ (6) yields: {f’(vx)/ f(vx)}dvx + {f’(vy)/ f(vy)}dvy + {f’(vz)/ f(vz)}dvz =0 (8) Here dvx, dvy and dvz are not independent since v = constant

  8. Velocity Distributions But: vx2 + vy2 + vz2 = v = constant 2 vx d vx + 2 vy d vy + 2 vz d vz = 0 (9) Now use the Lagrangean Method of Undetermined Multipliers; i.e., multiply Equ (9) by l/2 and add to Equ (8): [{f’(vx)/ f(vx)}+lvx]dvx + [{f’(vy)/ f(vy)}+ lvy]dvy + [{f’(vz)/ f(vz)}+ lvz]dvz =0 (10)

  9. Velocity Distributions Now all terms are independent; i.e., each coefficient = 0: {f’(vx)/ f(vx)}+lvx = 0 (11) {f’(vy)/ f(vy)}+lvy = 0 (12) {f’(vz)/ f(vz)}+lvz = 0 (13)

  10. Velocity Distributions Rewriting Equ (11): {df(vx)/dvx} {f(vx)}-1 + lvx = 0 or {df(vx)} {f(vx)}-1= lvx dvx Integrate: ln f(vx) = - lvx2/2 + ln a or f(vx) = a exp(- lvx2/2) = a exp(- bvx2) (15) f(vy) = a exp(- bvy2) (16) f(vz) = a exp(- bvz2) (17)

  11. Velocity Distributions Substituting the f(v)s into Equ (6) results in d3Nvx,vy,vz = Na3exp{-b2(vx2 + vy2 + vz2)} dvxdvydvz or d3Nvx,vy,vz = Na3exp{-b2(v2)}dvxdvydvz (18)

  12. Velocity Distributions The number of vectors per unit velocity interval is: r = d3Nvx,vy,vz/ dvxdvydvz = Na3exp{-b2(v2)} (19) The RHS of Equ (19) is referred to as the Maxwell Velocity Distribution Function

  13. Velocity Distributions The large value of r for low velocities does not mean that most molecules are moving slowly. Instead there are few dvxdvydvz ’cubes’ in the inner shells.

  14. Velocity Distributions Better representation: Plot # of molecules in a given velocity interval (dv); in a shell. Volume of shell = (4pv2)dv Number of molecules in shell = {(4pv2)dv}r (20) (dNv/dv) = 4pNa3v2exp{-b2(v2)} (21)

  15. Speed Distributions Maxwell-Boltzman Speed Distribution Shaded area = # molecules v, v+dv Area to right of dashed line = #molecules v>vo

  16. Velocity Distributions Now switch from speeds to velocities: dNvx = Nf(vx)dvx = Naexp{-b2(vx2)} dvx (22) dNvx/ dvx = Naexp{-b2(vx2)}(23a) dNvy/ dvy = Naexp{-b2(vy2)} (23b) dNvz/ dvz = Naexp{-b2(vz2)} (23c)

  17. Velocity Distributions • Symmetric • Otherwise similar to speed distribution