Question

Two identical spheres are dropped into two different columns: one column contains a liquid of viscosity \(0.5 \mathrm{Pa} \cdot \mathrm{s},\) while the other contains a liquid of the same density but unknown viscosity. The sedimentation velocity in the second tube is \(20 \%\) higher than the sedimentation velocity in the first tube. What is the viscosity of the second liquid?

Short answer

The viscosity of the second liquid is approximately \(0.417 \mathrm{Pa} \cdot \mathrm{s}\).

Step-by-step solution

Understanding the Problem

We have two identical spheres. One falls through a liquid with a known viscosity of \(0.5 \mathrm{Pa} \cdot \mathrm{s}\) (let's call this \(\eta_1\)) and the other through a liquid of unknown viscosity, \(\eta_2\). The sedimentation velocity in the second tube is 20% higher than in the first tube.

Setting up the Relationship

The sedimentation velocity \(v\) of a sphere in a liquid is inversely proportional to the viscosity of that liquid (according to Stokes' Law). This means \(v \propto \frac{1}{\eta}\). If the velocity in the first tube is \(v_1\), then the velocity in the second tube \(v_2 = 1.2v_1\).

Relating Sedimentation Speeds

Since \(v_2 = \frac{1}{\eta_2}\) and \(v_1 = \frac{1}{\eta_1}\), we use the given relationship: \(v_2 = 1.2v_1\). This implies \(\frac{1}{\eta_2} = 1.2 \times \frac{1}{\eta_1}\).

Solving for Unknown Viscosity

By substituting the known viscosity into the relationship, we have \(\frac{1}{\eta_2} = 1.2 \times \frac{1}{0.5}\). This equation can be simplified to find \(\eta_2\).

Calculating the Result

Solve for \(\eta_2\): \(\frac{1}{\eta_2} = \frac{1.2}{0.5} = 2.4\). Therefore \(\eta_2 = \frac{1}{2.4} = 0.4167 \mathrm{Pa} \cdot \mathrm{s}\).

Key concepts

Stokes' Law

Stokes' Law is a fundamental principle in fluid dynamics, especially useful when studying the motion of small particles through viscous fluids. It lays out the relationship between the force acting on a spherical object moving through a fluid and various factors like the radius of the sphere, the viscosity of the fluid, and the sedimentation velocity. The key formula from Stokes' Law is that the drag force, F, on a sphere in a viscous fluid is given by:
\[ F = 6\pi \eta r v \]where:

• \( \eta \) is the viscosity of the fluid,
• \( r \) is the radius of the sphere,
• and \( v \) is the velocity of the sphere.

Stokes' Law applies under conditions of steady, laminar flow, typically for small Reynolds numbers, which means it works well for small particles or at low velocity where fluid inertia is negligible. Understanding this principle helps to explain how particles settle in a liquid, bringing the concept of sedimentation into focus.

Sedimentation Velocity

Sedimentation velocity is the rate at which a particle or sphere falls through a fluid under the influence of gravity. It is a crucial concept in understanding how pollutants, sediments, or even biological cells settle in various environments.
According to Stokes' Law, sedimentation velocity is directly connected to the balance of forces acting on a particle. These include gravitational force pulling the particle down and the viscous drag from the fluid opposing this motion.
Key factors influencing sedimentation velocity include:

• The viscosity of the fluid — higher viscosity means more resistance, hence slower sedimentation.
• The density difference between the particle and the fluid, which determines the gravitational force.
• The radius of the particle — larger particles fall faster.

To find sedimentation velocity using the relationship derived from Stokes' Law, we find that velocity \( v \) is inversely proportional to the fluid's viscosity, explaining why changes in fluid properties directly affect how fast particles settle.

Inverse Proportionality

Inverse proportionality is a mathematical relationship where an increase in one quantity leads to a proportional decrease in another. In the context of Stokes' Law and sedimentation, it is crucially applied to show the relationship between sedimentation velocity and a fluid's viscosity.
In simpler terms, when you have two spheres falling through two liquids with different viscosities, the sphere in the less viscous liquid will fall faster. This is due to the inverse relationship, where sedimentation velocity \( v \) is given by:
\[ v \propto \frac{1}{\eta} \]This implies:

• If viscosity increases, velocity decreases.
• If viscosity decreases, velocity increases.

This principle is vital for applications such as determining unknown viscosities of fluids by observing the sedimentation velocities. For example, if in one experiment the sedimentation velocity is 20% higher in one liquid than in another, inverse proportionality allows us to calculate the viscosity of the unknown fluid, as shown in the problem solution.