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## Summary:

- The MB Distribution shows that the highest probable energy level is different from the highest probable speed. I have a hard time understanding this physically because energy is tied to speed. Is there a way to show that this is possible physically?

## Main Question or Discussion Point

It appears that the most probable energy level according to the Maxwell-Boltzmann distribution is

$$E_{max} = \frac{k_BT}{2}$$

$$v_{max} = \sqrt{\frac{2k_BT}{m}}$$

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

Is this reasoning correct?

**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?

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