An air bubble of radius 1 cm is rising at a steady rate of 0.2cm/s through a liqud of density 1.47 × 10^3. . Calculate coefficient of viscosity of liquid. The density of air may be neglected.

To determine the coefficient of viscosity of the liquid through which the air bubble is rising, we can apply Stokes' law. This law describes the motion of a sphere through a viscous medium and relates the forces acting on the sphere to the viscosity of the fluid. Let's break down the problem step by step.

Understanding the Forces at Play

When the air bubble rises, it experiences two main forces: the buoyant force acting upwards and the viscous drag force acting downwards. At steady state, these forces balance each other out. The buoyant force can be calculated using Archimedes' principle, while the viscous drag force can be calculated using Stokes' law.

Calculating the Buoyant Force

The buoyant force (F_b) can be calculated using the formula:

F_b = V * ρ * g

• V = volume of the bubble = (4/3)πr³
• ρ = density of the liquid = 1.47 × 10³ kg/m³
• g = acceleration due to gravity ≈ 9.81 m/s²

First, we need to convert the radius from centimeters to meters:

r = 1 cm = 0.01 m

Now, calculate the volume:

V = (4/3)π(0.01)³ ≈ 4.19 × 10⁻⁶ m³

Now, substituting the values into the buoyant force equation:

F_b = 4.19 × 10⁻⁶ m³ * 1.47 × 10³ kg/m³ * 9.81 m/s² ≈ 0.0606 N

Calculating the Viscous Drag Force

According to Stokes' law, the viscous drag force (F_d) is given by:

F_d = 6πηrv

• η = coefficient of viscosity (what we want to find)
• r = radius of the bubble = 0.01 m
• v = velocity of the bubble = 0.2 cm/s = 0.002 m/s

At steady state, the buoyant force equals the viscous drag force:

F_b = F_d

Substituting the expressions we have:

0.0606 N = 6πη(0.01 m)(0.002 m/s)

Solving for the Coefficient of Viscosity

Now, we can rearrange the equation to solve for η:

η = F_b / (6 * r * v)

Substituting the known values:

η = 0.0606 N / (6 * 0.01 m * 0.002 m/s)

η = 0.0606 N / (0.00012 m²/s)

η ≈ 505 N·s/m²

Final Result

The coefficient of viscosity of the liquid is approximately 505 N·s/m². This value indicates how resistant the liquid is to flow, which is influenced by its molecular structure and temperature.