This is my problem:
A particle with a density of 0.9g/ml is suspended in a solution that has a density of 1.1g/ml. It cannot be sedimented. Using the equation that describes the movement of a particle in a centrifugal field, describe why this is true.
The issue I'm having is that none of the equations provided in the exercise reference density of the particle vs. density of the solution. Logical deduction tells me that a less dense particle will float in a more dense solution, but without knowing the equation, I can't figure out what effect the application of a centrifugal field will have on it. Based on the wording of the question, I would assume no effect. But why?
To understand why the particle cannot be sedimented in the solution, let's consider the equation that describes the movement of a particle in a centrifugal field. The equation is known as the sedimentation equation:
s = ω^2 * r^2 * (rho_p - rho_f) / (18 * μ)
where:
- s is the sedimentation coefficient,
- ω is the angular velocity of the centrifuge,
- r is the distance from the axis of rotation,
- rho_p is the density of the particle, and
- rho_f is the density of the fluid.
In this case, we are given that the density of the particle is 0.9 g/ml and the density of the solution is 1.1 g/ml.
Let's consider the effect of the centrifugal field on the sedimentation of the particle by analyzing the terms in the equation:
1. Angular Velocity (ω): The angular velocity determines the strength of the centrifugal force. However, the angular velocity does not depend on the density of the particle or the solution. Therefore, it does not have a direct effect on the sedimentation behavior in this context.
2. Distance from the Axis of Rotation (r): The distance from the axis of rotation represents the position of the particle in the centrifuge. However, in this problem, the distance from the axis of rotation is not specified. Thus, it does not provide any information regarding the sedimentation behavior.
3. Density Difference (rho_p - rho_f): This term represents the difference in density between the particle and the fluid. If the density difference is positive, it means that the particle is denser than the fluid and will sediment. If the density difference is zero or negative, the particle will not sediment. In this case, the given densities indicate that the density of the solution (1.1g/ml) is higher than the density of the particle (0.9g/ml). As a result, the density difference is negative, implying that the particle will not sediment.
Therefore, based on the equation, we can conclude that the particle will not sediment in the solution because its density is lower than the density of the solution.