Deriving equations for pressure & number density for centrif

In summary, the conversation discusses the use of a gas centrifuge to separate gases of different molar mass. The task is to derive equations for pressure (P) and particle number density (nv) as functions of M, r, w, and T. This is achieved by applying Newton's 2nd law to the circular motion of a segment of gas and using the formula for centripetal acceleration. The resulting equations are: nv = n0 * e^(m * r^2 * w^2 / (2 * k * T)) for number density and Fr = (m * v^2)/r for pressure. However, there is uncertainty about the specific form of the number density equation and further research is needed for clarification
  • #1
issacweirdo
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Homework Statement


Consider particles in a gas centrifuge. This device is used to separate gases of different molar mass by rotating a cylinder at high rates. Derive two equations: one for the pressure (P) and one for the particle number density (nv) as functions of M, r, w and T, where r is the radial distance from the center point and w is the angular frequency of the rotation. M and T are molar mass and temperature. Do this by applying Newton's 2^nd law to the circular motion of a segment of gas of mass delta(m) and width delta(r). Recall that centripetal acceleration is given by w^2 r and that the positive direction for r is radially outward from the center of the circle.

Homework Equations


nv = n0 * e\^(m * r^2 * w^2 / (2 * k * T))
Net Fr = (m * v^2)/r

The Attempt at a Solution


I have no clue what I'm supposed to do. I don't even know how I'm going to draw a FBD for this. I don't know what the pressure equation is supposed to look like, but I know the number density equation is supposed to look like this: nv = n0 * e\^(m * r^2 * w^2 / (2 * k * T)), where k is the Boltzmann constant. I only know this because I was able to look online for this answer (though it did not explain how they derived this).
 
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  • #2
Er, I mean nv = n0 * e^(m * r^2 * w^2 / (2 * k * T)). This part may also be wrong, since I think it should be nv = n0 * e^(-m * r^2 * w^2 / (2 * k * T)) since it reminds me of the Boltzmann factor.
 

Related to Deriving equations for pressure & number density for centrif

1. What is the formula for pressure in a centrifuge?

The formula for pressure in a centrifuge is P = ρω^2r, where P is the pressure, ρ is the number density, ω is the angular velocity, and r is the distance from the center of rotation to the point of interest.

2. How do you calculate the number density in a centrifuge?

The number density in a centrifuge can be calculated using the formula ρ = N/V, where N is the number of particles and V is the volume of the centrifuge chamber.

3. What is the relationship between pressure and number density in a centrifuge?

The relationship between pressure and number density in a centrifuge is direct, meaning that as the number density increases, so does the pressure.

4. How does the angular velocity affect pressure in a centrifuge?

The pressure in a centrifuge is directly proportional to the square of the angular velocity. This means that as the angular velocity increases, the pressure also increases.

5. What units are typically used to measure pressure and number density in a centrifuge?

Pressure is typically measured in units of Pascals (Pa), while number density is measured in units of particles per cubic meter (m^-3).

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