Question

Time Required for Sedimentation by Gravity A certain reagent is added to a suspension of cells \(4 \mu \mathrm{m}\) in diameter. These cells have a density of \(1.08 \mathrm{~g} / \mathrm{cm}^{3}\), and they are suspended in liquid with a density of \(1.00 \mathrm{~g} / \mathrm{cm}^{3}\) and viscosity of \(1.0 \mathrm{cp}\). This reagent causes about half of the cells to form fairly solid aggregates, all of which are \(90 \mu \mathrm{m}\) in diameter and have density midway between that of the liquid and the cells. How much time is required for all the aggregates to sediment to within \(1 \mathrm{~cm}\) of the bottom of a vessel filled with suspension that is \(0.5 \mathrm{~m}\) high? Approximately what fraction of the single cells would have sedimented to this depth in the same amount of time? How much time is required for all the single cells to sediment to within \(1 \mathrm{~cm}\) of the bottom of the vessel?

Short answer

To solve this problem, it's crucial to use Stoke's law to calculate the sedimentation velocities and times for both single cells and aggregates. Then, by comparing the sedimentation times, we can determine which fraction of the single cells would have sedimented within the same amount of time as the aggregates.

Step-by-step solution

Calculate the sedimentation velocity for the aggregates

The velocity of sedimentation can be calculated using Stoke's law: \(v = \frac{2}{9} \frac{(ρ_p-ρ_f)d^2g}{η}\) where \(v\) is the velocity, \(ρ_p\) is the particle density (\(1.04 g/cm^3\)), \(ρ_f\) is the fluid density (\(1.00 g/cm^3\)), \(d\) is the diameter of the particle (\(90 \mu m\)), \(g\) is the acceleration due to gravity (\(981 cm/s^2\)), and \(η\) is the fluid's viscosity (\(1.0 cp\)).

Calculate the sedimentation time for the aggregates

The time it would take for a particle to sediment can be calculated by dividing the total distance that the particle will sediment (\(50 cm - 1 cm\)) by the sedimentation velocity.

Calculate the sedimentation velocity for the single cells

The sedimentation velocity for a single cell can also be calculated by using Stoke's law. The density of the cell (\(1.08 g/cm^3\)) and the diameter of the cell (\(4 \mu m\)) will be used in this calculation.

Calculate the sedimentation time for the single cells

Using the sedimentation velocity for a single cell, the sedimentation time for a single cell can be calculated by dividing the total sedimentation distance (\(50 cm - 1 cm\)) by this velocity.

Determine what fraction of the single cells would have sedimented within the same amount of time as the aggregates

This can be found by dividing the time it would take for an aggregate to sediment by the time it would take for a single cell to sediment. This would provide the ratio of sedimentation times, which is essentially the fraction of single cells that would have sedimented within the same time required for all aggregates to sediment.

Key concepts

Stoke's law

Stoke's Law is a fundamental principle that helps us understand how particles settle in fluids. This law provides a way to calculate the sedimentation velocity of spherical particles through a viscous medium. The formula for Stoke's law is:\[v = \frac{2}{9} \frac{(ρ_p-ρ_f)d^2g}{η}\]This equation tells us several key points:

• **Particles' speed**: The sedimentation velocity, \(v\), is dependent on the size, density, and the difference in density between the particle and the fluid.
• **Gravity's role**: The acceleration due to gravity, \(g\), impacts the velocity.
• **Medium's resistance**: The fluid's viscosity, \(η\), also affects the sedimentation process by resisting the particle's movement.

Understanding these variables helps clarify why some particles settle quickly, while others take longer.

sedimentation velocity

Sedimentation velocity is how fast a particle moves down when dropped in a fluid. It can be influenced by various factors as captured in Stoke's Law.Some things that matter include:

• Particle size: Larger particles tend to settle faster, which is why aggregates with a diameter of \(90 \mu m\) move quicker than smaller single cells.
• Density difference: Particles that have a significant density difference from the fluid in which they are suspended will usually settle faster.
• Viscosity: A thicker fluid slows down sedimentation. For our problem, with a viscosity of \(1.0 cp\), both single cells and aggregates experience the same fluid resistance.

By calculating the sedimentation velocity through Stoke's Law, we can predict how long it will take for particles to settle in a container.

particle density

Particle density is a vital parameter when determining how particles will behave in a fluid. It is a measure of the mass of particles relative to their volume.In our context:

• The **density of single cells** is \(1.08 \text{ g/cm}^3\), which is slightly denser than the liquid they are in.
• For the **aggregates**, their density is determined to be between that of the cells and the liquid, approximated to \(1.04 \text{ g/cm}^3\).

The difference in density between the particles and the fluid drives the sedimentation process. Particles with a higher density than the fluid tend to settle down due to gravity pulling them more strongly compared to the resistive force of the fluid's viscosity. Making sense of particle density is crucial for predicting sedimentation outcomes, especially when different particles are involved.

sedimentation time

Sedimentation time is how long it takes for particles to settle to a specified depth in a fluid. This is calculated by dividing the distance the particles need to travel by their sedimentation velocity.In the exercise given:

• For aggregates, the total distance to sediment was \(49 \text{ cm}\) which is **\(50 \text{ cm} - 1 \text{ cm}\)** since they must fall to within \(1 \text{ cm}\) of the bottom.
• Knowing their sedimentation velocity lets us compute the time it takes to reach this point.
• Similarly, sedimentation time for single cells is also computed using their respective velocity.

By understanding sedimentation time, we can determine how efficiently particles settle in various conditions, and how adjustments in parameters like particle size or fluid viscosity can affect the process.