Question

A soap bubble of radius \(r\) is blown up to form a bubble of radius \(2 \mathrm{r}\) under isothermal conditions if the \(\mathrm{T}\) is the surface tension of soap solution the energy spent in the slowing is. (A) \(3 \pi \mathrm{Tr}^{2}\) (B) \(6 \pi \mathrm{Tr}^{2}\) (C) \(12 \pi \mathrm{Tr}^{2}\) (D) \(24 \pi \mathrm{Tr}^{2}\)

Short answer

The energy spent in expanding the soap bubble of radius r to form a bubble of radius 2r under isothermal conditions is \(12 \pi Tr^2\), which matches option (C).

Step-by-step solution

Understand the problem and its relevant formula

First, let's understand the problem. We have a soap bubble of radius r, and we want to expand it to form a bubble with radius 2r under isothermal conditions, meaning the temperature remains constant during the entire process. Now, we want to find the energy spent in doing this. The key concept to understand here is surface tension (T) of the soap solution. The work done against the surface tension force will be equal to the energy spent in expanding the bubble. The surface tension work is given by the formula: Work done = T * (change in surface area) Now, let's figure out the change in surface area for the given bubble expansion.

Find the initial and final surface area of the soap bubble

The bubble is a sphere, and the surface area of a sphere is given by: Surface area = 4 * π * r^2 For our bubble, the initial surface area (A₁) is: A₁ = 4 * π * r^2 And, after expansion, the final surface area (A₂) is: A₂ = 4 * π * (2r)^2 = 4 * π * 4 * r^2 = 16 * π * r^2 Now that we have both the initial and final surface areas, we can find the change in surface area (∆A).

Calculate the change in surface area of the soap bubble

To find the change in surface area, ∆A, we simply need to subtract the initial surface area (A₁) from the final surface area (A₂). ∆A = A₂ - A₁ ∆A = (16 * π * r^2) - (4 * π * r^2) ∆A = 12 * π * r^2

Calculate the energy spent

Now that we have the change in surface area, we can find the energy spent in expanding the bubble. Recall our formula for work done against surface tension: Work done = T * (change in surface area) Energy spent = T * ∆A Energy spent = T * (12 * π * r^2) Now let's match this result with the given options: (A) 3 * π * Tr^2 (B) 6 * π * Tr^2 (C) 12 * π * Tr^2 (D) 24 * π * Tr^2 Our result matches option (C). Therefore, the energy spent in expanding the soap bubble is 12 * π * Tr^2.

Key concepts

Soap Bubble Expansion

When a soap bubble is blown, it forms a spherical shape with its surface defined by a thin film of soapy water. The expansion of a soap bubble involves changing its radius from a smaller size to a larger size. In this exercise, we are examining a bubble expanding from radius \( r \) to \( 2r \). This is an intriguing phenomenon because during the expansion, the bubble's surface area dramatically increases, which also involves spending energy.
The force that resists the bubble's expansion is known as surface tension, which acts to minimize the bubble's surface area. As the bubble expands, more energy is needed to overcome this surface tension and increase the surface area. This process is very common in daily life but involves complex physics to fully understand.

Isothermal Process

An isothermal process is a type of thermodynamic process where the temperature remains constant, even though other properties such as pressure and volume may change. In the context of our soap bubble exercise, this means that the temperature of the air inside the bubble and surrounding it stays the same as the bubble expands.
This consistency in temperature is important because it implies that any energy added to the system does not go toward heating the air or changing its state, but solely into expanding the bubble. Isothermal conditions are essential for calculations because they simplify the mathematical treatment of the problem, focusing only on changes in pressure and volume without temperature fluctuations.

Sphere Surface Area

The surface area of a sphere is determined by the formula \(4 \pi r^2\), where \(r\) is the radius of the sphere. This formula gives us the total area covered by the surface of the sphere.
In our problem, we're dealing with two radii: an initial radius \(r\) and a final radius \(2r\). Initially, the surface area is \(4 \pi r^2\), and after expansion, the larger sphere's surface area becomes \(4 \pi (2r)^2 = 16 \pi r^2\).
The change in the sphere's surface area is found by subtracting the initial surface area from the final surface area. This difference provides the measure of increased area that needs additional energy to be maintained against the surface tension, which is critical when calculating energy requirements in our exercise.

Energy Calculation

To calculate the energy spent in expanding the soap bubble, we use the concept of work done against surface tension. The work done, or energy required, is calculated using the change in the surface area and the surface tension \( T \).
Given the change in surface area \( \Delta A = 12 \pi r^2 \), the energy (or work done) is given by the formula:

• Energy spent = \( T \times \Delta A \)
• Energy spent = \( T \times 12 \pi r^2 \)

This calculation corresponds to the option \( C \) from the problem choices, which is \( 12 \pi Tr^2 \). Energy calculations like this are vital in physics to understand the interplay between forces such as surface tension and changes in physical properties like area and volume.