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## Homework Statement

I have to find the boltzmann ditribution of a 1 dimensional ideal gas.

The answer is given as:

[itex]\frac{dn}{n}=\sqrt{\frac{m}{2piKT}}e^{(\frac{-mc^2}{2KT})}[/itex]

For the second part I have to find the mean kinetic energy.

**2. Homework Equations / Attempt**

For part 1:

I know how to work out the Boltzmann distribution for a 3D and 2D gas. However, for a 1D gas, I can't figure out what the constant has to be. I know the form to solve it is:

[itex]\int_0^\infty C e^{(\frac{-mc^2}{2KT})} dv = 1 [/itex] (1)

Where [itex]C[/itex] is a constant.

However, when I do this and solve for [itex]C[/itex], I get a factor of 2 in front of my equation. Is there something wrong in my logic here? Am I meant to use a factor infront of my [itex]C[/itex]

For the second part, I know that I need to get a [itex]v^2[/itex] infront of the exponential, but I cannot figure out how to do this for the 1D case and even for the 3D case.

Any help would be much appreciated and please tell me if I need to clarify anything.

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