# How relative density depends on R in a Centrifuge

• dwintz02
In summary, the problem at hand involves finding the relative density of molecules in a dilute solution at a given radial distance, given the temperature and angular velocity of a centrifuge. By using the Boltzmann factor and assuming non-interacting molecules, we can calculate the probability of finding a molecule at a specific distance and use it to find the relative density compared to the total number of molecules in the solution.
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

, 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!

## 1. How does the size of the centrifuge affect relative density?

The size of the centrifuge, specifically the radius (R), has a direct impact on the relative density. As the radius increases, the centrifugal force also increases, causing denser particles to move towards the outer edge and less dense particles to move towards the center. Therefore, the higher the R value, the higher the relative density in a centrifuge.

## 2. How does the speed of rotation affect relative density in a centrifuge?

The speed of rotation, or angular velocity, also plays a crucial role in determining relative density in a centrifuge. The faster the centrifuge rotates, the higher the centrifugal force and the greater the separation of particles by density. Therefore, a higher speed of rotation will result in a higher relative density.

## 3. Can the relative density of a substance be accurately determined using a centrifuge?

Yes, a centrifuge is a commonly used tool for accurately determining the relative density of a substance. By measuring the position of different particles in the centrifuge and knowing the speed of rotation and radius, scientists can calculate the relative density of a substance with precision.

## 4. How does the viscosity of a substance affect its relative density in a centrifuge?

The viscosity, or thickness, of a substance can impact its relative density in a centrifuge. More viscous substances will experience greater resistance to movement and therefore may not separate as easily by density. This can result in a less accurate measurement of relative density in a centrifuge.

## 5. Are there any limitations to using a centrifuge to determine relative density?

While centrifuges are a powerful tool for determining relative density, there are some limitations to consider. Certain substances, such as gases, may not be suitable for centrifugation. Additionally, the shape and size of particles can also affect their separation in a centrifuge, potentially impacting the accuracy of the results.

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