# How relative density depends on R in a Centrifuge

1. Feb 21, 2008

### dwintz02

1. The problem statement, all variables and given/known data

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.

2. Relevant 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.

3. 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