3. Centrifuges are often used to separate mixtures of substances, or to enrich i

Question

3. Centrifuges are often used to separate mixtures of substances, or to enrich isotopes. Due to non- inertial rotational effects, lighter particles or molecules are expected to be found at higher concentrations near the center of the centrifuge in equilibrium. Consider a centrifuge consisting of a right circular cylinder of radius R and length L which rotates about its symmetry axis at constant angular frequency 2, the whole apparatus being maintained at a constant temperature T. Imagine that before spinning up, such a centrifuge is filled with a mixture of ideal gases at temperature T and total pressure P. A fraction a1 (by number) of the molecules have mass m1 and adiabatic index 1, and the remainder have mass m2 and adiabatic index 2 You may assume that the viscosity of the gas mixture is sufficiently small so as to be neglected in the corresponding free energy, but large enough so that, in equilibrium, the gas will on average rotate with the cylinder: (a) Find the equilibrium concentrations (number densities) of both species of gas as a function of position within the cylinder, by using the total chemical potential for each gas in the rotating frame (and therefore including the effects of the fictitious forces that arise due to the rotation (b) [PRACTICE] Using the maximum-entropy formalism, verify explicitly that the Gibbs rotational ensemble" yields the same predictions. Recall that in this ensemble, one constrains the total particle number, the average energy, and also the average angular momentum along the symmetry axis

Explanation / Answer

1. Euilibirium concentration is zero in my opinion for species of a gas when calculated with respect to chemical potential.

2. More information required.

3..0025 fraction is collected.

4..05

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