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Velocity Distributions of Molecules

Velocity Distributions of Molecules. Velocity Distributions. Let dN vx be the number of molecules moving with velocity v x , etc.; Each is represented by a slice in Fig. 2. The fraction of molecules moving with a given velocity in that direction is then; dN vx /N = f(v x )dv x (1)

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Velocity Distributions of Molecules

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  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

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