Density distribution of gas in a centrifugal field

[Fin_de_Siecle]

The problem asks me to find the density of gas in a cylinder of radius R and length l rotating about its axis with angular velocity ω, there being a total of N molecules in the cylinder.

What I have done is shown as following
I choose to look this scenario in a static frame with the cylinder, which is rotating at an unchanging angular speed ω. Then we can consider the field is a centrifugal field.

Thus the energy of a particle can be written as
$$E=E_0-\frac{1}{2}m\omega ^2 r^2$$ (1)

I haven't determined E_0 yet as I'm not sure about the thermal energy of the particle and the energy caused by the centrifugal field differs with the choosing of zero potential point.

After that, I wrote the probability as
$$dP=\frac{\rho dV}{N}=\frac{\rho \cdot 2\pi rdr\cdot l}{N}$$ (2)
where ρ is what I want to find.

My goal is to convert all r and dr into E and dE and finally compare it with Boltzmann distribution
$$P(E)=C\cdot e^{-E/kT}$$ (3)
where C is a constant.

However, I have met with some problems doing this.
From (1), it can be derived
$$E=-m\omega ^2 r$$ (4)

After I plug this into (2), I find P(E) in dP=P(E)dE is always negative. Thus is cannot be compared with the Boltzmann distribution.

How can I fix this?
Thank you.

 

[Chestermiller]

Hi Fin_de_siecle. Welcome to Physics Forums!

Irrespective of whether the cylinder is rotating, what is the average density of the gas in the cylinder? With the cylinder rotating, the gas near the axis is going to have less than the average density (because the pressure near the axis will be lower), and the gas near the surface will have greater than the average density (because the pressure near the outside will be higher). You can use the ideal gas law in combination with a differential radial force balance to solve for how the pressure and density are varying with radius, under the constraint that the average density remains unchanged.

Chet

 

[mfb]

You can use LaTeX if you format them like [tex]formula[/tex] or $$formula$$.

I'm not sure why you try to solve this via a particle energy.
You can consider a force equilibrium at distance r ("between r and r+dr"). This gives a differential equation for the pressure, depending only on pressure and constants. That method is very similar to a (simplified) model of the atmosphere.

 

[Fin_de_Siecle]

You can use LaTeX if you format them like [tex]formula[/tex] or $$formula$$.

I'm not sure why you try to solve this via a particle energy.
You can consider a force equilibrium at distance r ("between r and r+dr"). This gives a differential equation for the pressure, depending only on pressure and constants. That method is very similar to a (simplified) model of the atmosphere.

Thank you for your information on typing formulas.

I think maybe the thermal fluctuation should be taken into consideration and this is a problem assigned in my statistical mechanics class. I have read the solution in Landau's Statistical Physics (P115). He just simply plugged the centrifugal potential into the Boltzmann distribution, not considering that the farther r is, the more volume it takes. I don't quite understand his solution, so I tried this.

Notwithstanding, I think your strategy may derive the result of Landau's as well. I've read a solution use the mechanic equilibrium to derive the Boltzmann distribution of atmosphere pressure rather than using the ideas in statistical mechanic.

 

[mfb]

In equilibrium, temperature will be the same everywhere. And if the system is not in equilibrium and you don't know its temperature distribution, you are lost.