- #1
JohnnyGui
- 796
- 51
It appears that the most probable energy level according to the Maxwell-Boltzmann distribution is not equal to the most probable speed squared multiplied by ##\frac{1}{2}m##. The most probable speed has a different value.
$$E_{max} = \frac{k_BT}{2}$$
$$v_{max} = \sqrt{\frac{2k_BT}{m}}$$
I am trying to understand this physically rather than mathematically, but I have a hard time comprehending this.
An energy level with the highest probability means it has the most particles out of all other energy levels, therefore that same largest number of particles has the speed corresponding to that energy level.
If I’m imagining energy levels as containers and I draw a specific number of particles in each container, there is no way I can distribute the particles in a way to have the most probable speed in an energy container different from the most probable energy. Energy is tied to speed.
The only way I can think of how this can be possible is if in the continuous approach a certain energy level covers a range of different speed values. If that’s the case then I’d deduce further that the reason is because an infinitesimally small ##dE## covers a larger range of speed values dan an infinitesimally small speed ##dv## does.
Is this reasoning correct?
$$E_{max} = \frac{k_BT}{2}$$
$$v_{max} = \sqrt{\frac{2k_BT}{m}}$$
I am trying to understand this physically rather than mathematically, but I have a hard time comprehending this.
An energy level with the highest probability means it has the most particles out of all other energy levels, therefore that same largest number of particles has the speed corresponding to that energy level.
If I’m imagining energy levels as containers and I draw a specific number of particles in each container, there is no way I can distribute the particles in a way to have the most probable speed in an energy container different from the most probable energy. Energy is tied to speed.
The only way I can think of how this can be possible is if in the continuous approach a certain energy level covers a range of different speed values. If that’s the case then I’d deduce further that the reason is because an infinitesimally small ##dE## covers a larger range of speed values dan an infinitesimally small speed ##dv## does.
Is this reasoning correct?
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