Maxwell Distribution

# Maxwell Distribution - THE MAXWELL-BOLTZMANN DISTRIBUTION...

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THE MAXWELL-BOLTZMANN DISTRIBUTION CHEMISTRY 213 The molecular description of the bulk properties of a gas depends upon our knowing the mathematical form of the velocity distribution; That is, the probability, F ( v x , v y , v z ) v x v y v z , of finding a molecule with velocity components in the range v x to v x + ∆ v x , v y to v y + ∆ v y , and v z to v z + ∆ v z (see the last handout). This can be found by making two very simple assumptions: 1. The three components of the velocity are independent of each other. This implies that the velocity probability density can be written as: F ( v x , v y , v x ) = f ( v x ) f ( v y ) f ( v z ) (1) 2. All directions are equivalent (space is isotropic). This implies that the probability of find- ing a molecule moving with a certain velocity cannot depend on the direction; it is equally probable to find a molecule with any speed v moving in any direction. Mathematically, this implies that the probability density can only depend on the magnitude of the velocity, or the molecular speed. Hence, F ( v x , v y , v x ) = F ([ v 2 x + v 2 y + v 2 z ] 1/2 ). (2) By comparing Eqs. (1) and (2), we have f ( v x ) f ( v y ) f ( v z ) = F
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