Macromolecules (Thermodynamics)

[Slepton]

Homework Statement

A dilute solution of macromolecules at temperature T is placed in an untracentrifuge rotating with angular velocity w. The centripetal acceleration w²r acting on a particle of mass m may then be replaced by an equivalent centrifuge force mw²r in the rotating frame of reference.
a) Find how the relative density $$\rho$$(r) of molecules varies with their radial distance r from the axis of rotation.
b) Show quantitatively how the molecular weight of the macromolecules can be determined if the density ratio [tex]\rho₁[/tex] / [tex]\rho₂[/tex]at the radii r1 and r2 is measured by optical means.

Homework Equations

F = mv²/r

density = mass /volume

The Attempt at a Solution

I do not know how to even start the problem. Please help.

 

[anathema]

Thank you for your question. I can provide you with some guidance on how to approach this problem.

For part a), we can use the equation for centrifugal force, F = mv2/r, to determine how the relative density of molecules varies with their radial distance. Since we know that the centrifugal force is equal to the centripetal acceleration (mw2r), we can set the two equations equal to each other and solve for the density, \rho. This will give us an expression for how \rho varies with r.

For part b), we can use the equation for density, \rho = m/V, where m is the mass of the macromolecules and V is the volume. We can then use the given information about the density ratios at different radii (r1 and r2) to set up an equation and solve for the molecular weight, m.

I hope this helps to get you started on solving the problem. Let me know if you have any further questions or need clarification on any of the steps. Good luck!

 

[nlightner]

As a scientist, it is important to first understand the key concepts and equations involved in the problem. In this case, we are dealing with macromolecules in a rotating frame of reference, which means that we need to consider both centrifugal and centripetal forces. The key equation to use here is F = mv2/r, where F is the force, m is the mass, v is the velocity, and r is the distance from the axis of rotation. We also need to keep in mind that density is equal to mass divided by volume.

a) To find the variation of relative density \rho(r) with radial distance r, we can use the equation for centrifugal force, F = mw2r, and equate it to the centripetal force, F = mv2/r. This will give us:

mw2r = mv2/r

Solving for \rho(r), we get:

\rho(r) = m / (w2r2)

Since we are dealing with a dilute solution of macromolecules, we can assume that the mass of each molecule is the same. Therefore, the relative density will depend on the distance from the axis of rotation, r, and the angular velocity, w.

b) To determine the molecular weight of the macromolecules, we can use the fact that the density ratio \rho1 / \rho2 is equal to the mass ratio m1 / m2. This means that:

\rho1 / \rho2 = m1 / m2

Substituting the equation for \rho(r) from part a, we get:

(m1 / w2r12) / (m2 / w2r22) = m1 / m2

Simplifying, we get:

m1 / m2 = r12 / r22

This means that the mass of the molecule at a distance r1 from the axis of rotation is equal to the mass at a distance r2 squared. Therefore, by measuring the density ratio at two different radii, we can determine the molecular weight of the macromolecule using the above equation.

In summary, by understanding the key concepts and equations involved, we can solve for the variation of relative density with radial distance and determine the molecular weight of the macromolecules using optical measurements of the density ratio at different radii.