How relative density depends on R in a Centrifuge

[dwintz02]

Homework Statement

A dilute solution of macromolecules at temperature T is placed in a centrifuge rotating with angular velocity w. The mass of each molecule is m. The equivalent centrifugal force on each particle in the rotating frame of reference is mw^2r, where r is the radial distance from the axis of rotation.
Find how the relative density of molecules p(r) varies with r.

Homework Equations

So this problem was designed for us after the chapter in our book on Helmholz Free Energy, the Partition function, the Boltzmann factor, and pressure.

The Attempt at a Solution

I'm having trouble starting the problem. Once I get some equations I know are correct to use I should be fine. I've started by getting

U = -mw$$^{2}$$r$$^{2}$$/2 and this relates to the Partition function, Z, by

U =($$\Sigma \epsilon_{s}exp(\epsilon_{s}/\tau$$))/Z

where $$\epsilon_{s}$$ is the energy of substate s and
$$\tau$$ is the temperature given by the relation $$\tau=k_{B}*T$$

I'm thinking of changing the subscript to epsilon sub r, because the energy of the molecules will only be dependent on R and then changing the summation to an integral but I don't see how that will help me. Maybe give me a relation between the probability and U and then I can divide that by another probability to get the relative probabilities of the particles being at various radii which will in turn be the relative densities?

Any help is greatly appreciated, thanks!

Daniel

 

[Jhero]

, as a fellow scientist, I understand the struggle of starting a problem without a clear direction. Your approach of relating the energy of the molecules to the partition function is a good start. However, I think you might be overcomplicating things a bit.

Since we are dealing with a dilute solution, we can assume that the molecules are non-interacting, and therefore the energy of each molecule will be independent of the others. This means we can simply use the Boltzmann factor to calculate the probability of finding a molecule at a certain radial distance r, given by P(r) = exp(-\epsilon_{r}/\tau)/Z, where \epsilon_{r} is the energy of the molecule at that distance.

To find the relative density, we can simply divide this probability by the total probability of finding a molecule at any distance, which is given by the integral of P(r) over all possible radial distances. This will give us the relative density of molecules at a specific distance r, compared to the total number of molecules in the solution.

I hope this helps you move forward with the problem. Let me know if you need any further clarification or assistance. Good luck!