Centrifuge question (Stat Mech)

In summary, the conversation discusses the expression for the dependence of the concentration of an ideal gas on the radial distance in a rotating circular cylinder. It involves the total chemical potential, quantum concentration, and external potential. The second part of the conversation talks about the total number of molecules in the atmosphere at constant temperature, which involves an integral and the concentration at different layers. The connection to quantum corrections is unclear.
  • #1
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A circular cylinder of radius R rotates about hte long axis with angular velocity omega. The cylinder contains an ideal gas of atoms of mass M at temperature tau. Find the expression for the dependence of the concentration n(r) on the radial distance r from the axis in terms of n(0) on the axis. Take the chemical potnetial u to be that of an ideal gas.

[tex] \mu_{tot} = \mu_{int} + \mu_{ext} [/tex]

right?

[tex] \mu_{tot} = \tau\log\frac{n}{n_{Q}} + \mu_{ext} [/tex]
where n is te concentration and nQ is the quantum concentration
[tex] n_{Q} = \frac{M\tau}{2\pi\hbar^2} [/tex]
now I am wondering what external potentisl should be...
should it be the energy of the particle while it is rotation at some radiual distance r with angular velocity omega?
Wold that mean taht
[tex] \mu_{ext} = \frac{1}{2} M\omega^2 r^2 [/tex]

so then
[tex] \mu = \tau\log\frac{n}{n_{Q}} + \frac{1}{2} M\omega^2 r^2[/tex]

[tex] n(r) = n_{Q} \exp\left(\frac{1}{\tau}\left(\mu-\frac{1}{2}M\omega^2 r^2\right)\right) [/tex]
[tex] n(0) = n_{Q} \exp\left(\frac{\mu}{\tau}\right) [/tex]

[tex] n(r) = n(0) \exp\left(-\frac{1}{2\tau} M\omega^2 r^2\right) [/tex]


If n is the concentration of molectules at the surface of the Earth, M the mass of a molecule and g the gravitational acceleration at teh surface, show that at constant temperature the total numbers of molecules in the atmosphere is

[tex] N = 4\pi n(R) \exp\left(-MgR/\tau\right) \int_{R}^{\infty} dr r^2 \exp\left(MgR^2/r\tau\right) [/tex]

i don't understand why the number of molecules at the bottom is multiplied by the concentration at the above layers... shouldn't it be added?

why they change the exponential's argument to what it is in the integrand...? IM a bit


thank you for the input!
 
Last edited:
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  • #2
Just asking: what does the setting of the first problem have to do with any quantum corrections ?

Daniel.
 
  • #3
dextercioby said:
Just asking: what does the setting of the first problem have to do with any quantum corrections ?

Daniel.

well it's a question that appears in the Textbook...
the textbook is Thermal Physics by Kittel and Kroemer

i tihnk they wnat us to solve it in that way
 

1. What is a centrifuge and how does it work?

A centrifuge is a laboratory instrument used to separate particles or substances of different densities by spinning them at high speeds. It works based on the principle of centrifugal force, where denser particles are pushed to the bottom of the tube while less dense particles remain on top.

2. What are the different types of centrifuges available?

There are three main types of centrifuges: fixed-angle, swinging bucket, and ultracentrifuges. Fixed-angle centrifuges hold the tubes at a fixed angle, while swinging bucket centrifuges allow the tubes to swing outwards during centrifugation. Ultracentrifuges spin at higher speeds and are used for separating smaller particles.

3. How do you choose the right centrifuge for your experiment?

The type of centrifuge you need depends on the size and type of particles you are trying to separate, as well as the speed and volume of your samples. Consider the maximum speed, capacity, and rotor options when selecting a centrifuge for your experiment.

4. What is the maximum speed of a centrifuge and how is it measured?

The maximum speed of a centrifuge is measured in revolutions per minute (RPM) or relative centrifugal force (RCF). RCF is the force applied to the particles due to the spinning motion and is calculated using the equation RCF = (1.12 x 10^-5) x (rpm)^2 x radius (in centimeters).

5. What are some safety precautions to take when using a centrifuge?

When using a centrifuge, it is important to carefully balance the samples to avoid damage to the rotor or uneven separation. Wear appropriate personal protective equipment, such as gloves and eye protection, when handling samples. Follow the manufacturer's instructions for proper use and maintenance of the centrifuge.

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