- #1
strauser
- 37
- 4
I'm having trouble understanding the role of the volume of velocity space when deriving the Maxwell-Boltzmann speed distribution.
1. If we wish to compute the speed distribution from the velocity distribution we work with, say, [tex]p(v) dv \propto v^2e^{-av^2} dv[/tex] where the ##v^2 dv## comes from considering the volume of velocity space available to the states with speeds in ##[v, v+dv]##. This makes sense, at least mathematically. However ..
2. This distribution seems to say that, if we consider the states close to ##v=0##, we will find no particles with speed 0 (and not many close by), as there is no volume space available to them there (and this appears to be independent of the underlying Boltzmann distribution itself, since ##v^2 f(v) \to 0## as ##v \to 0## for any physically realistic ##f(v)##)
3. But this seems to clash with what the Boltzmann distribution predicts: the probability of finding a particle in a microstate of 0 energy (hence 0 velocity) is greater than finding it with any other energy since the probability of it being in a microstate ##r##, energy ##E_r## is ##p(r) \propto e^{E_r/kT}##.
So I'm confused. The Boltzmann distribution seems to predict that we will often find gas particles with 0 velocity. But the Maxwell-Boltzmann speed distribution seems to say that we will find *no* particles with 0 speed (##\Leftrightarrow## 0 velocity), since there is no velocity space available to them.
Can anyone clear up my confusion?
1. If we wish to compute the speed distribution from the velocity distribution we work with, say, [tex]p(v) dv \propto v^2e^{-av^2} dv[/tex] where the ##v^2 dv## comes from considering the volume of velocity space available to the states with speeds in ##[v, v+dv]##. This makes sense, at least mathematically. However ..
2. This distribution seems to say that, if we consider the states close to ##v=0##, we will find no particles with speed 0 (and not many close by), as there is no volume space available to them there (and this appears to be independent of the underlying Boltzmann distribution itself, since ##v^2 f(v) \to 0## as ##v \to 0## for any physically realistic ##f(v)##)
3. But this seems to clash with what the Boltzmann distribution predicts: the probability of finding a particle in a microstate of 0 energy (hence 0 velocity) is greater than finding it with any other energy since the probability of it being in a microstate ##r##, energy ##E_r## is ##p(r) \propto e^{E_r/kT}##.
So I'm confused. The Boltzmann distribution seems to predict that we will often find gas particles with 0 velocity. But the Maxwell-Boltzmann speed distribution seems to say that we will find *no* particles with 0 speed (##\Leftrightarrow## 0 velocity), since there is no velocity space available to them.
Can anyone clear up my confusion?