Question

The amount of work done in blowing a soap bubble such that its diameter increases from \(d\) to \(D\) is \((T=\) Surface tension of solution) (A) \(4 \pi\left(\mathrm{D}^{2}-\mathrm{d}^{2}\right) \mathrm{T}\) (B) \(8 \pi\left(\mathrm{D}^{2}-\mathrm{d}^{2}\right) \mathrm{T}\) (C) \(\pi\left(\mathrm{D}^{2}-\mathrm{d}^{2}\right) \mathrm{T}\) (D) \(2 \pi\left(\mathrm{D}^{2}-\mathrm{d}^{2}\right) \mathrm{T}\)

Short answer

The short answer is: The amount of work done in blowing a soap bubble such that its diameter increases from \(d\) to \(D\) is \(4 \pi\left(\mathrm{D}^{2}-\mathrm{d}^{2}\right) \mathrm{T}\).

Step-by-step solution

Understanding the problem

The work done in blowing a soap bubble can be calculated using the formula: Work = Surface tension × Change in surface area. We are given the initial and final diameters, d and D, respectively, and we need to find the change in surface area.

Calculate the surface area of the initial and final bubbles

The surface area of a sphere can be calculated using the formula: A = 4πr². To find the surface areas of the soap bubble initially and finally, we need to convert diameters into radii. The radii for initial and final bubbles can be represented as r₁ = d/2 and r₂ = D/2. Now, calculate the initial and final surface areas: - Initial surface area: A₁ = 4πr₁² = 4π(d/2)² - Final surface area: A₂ = 4πr₂² = 4π(D/2)²

Calculate the change in surface area

We can find the change in surface area by subtracting the initial surface area from the final surface area: ΔA = A₂ - A₁ = 4π(D/2)² - 4π(d/2)² To factor out 4π, we get: ΔA = 4π[(D/2)² - (d/2)²]

Calculate the work done

Using the formula, Work = Surface tension × Change in surface area, we can now find the work done: Work = T × ΔA = T × 4π[(D/2)² - (d/2)²] Comparing this expression to the given options, we find that it matches with option (A): \(4 \pi\left(\mathrm{D}^{2}-\mathrm{d}^{2}\right) \mathrm{T}\) Therefore, the correct answer is (A).

Key concepts

Surface Tension

Surface tension is a fascinating phenomenon that occurs at the surface of a liquid. It is the reason why small objects can sometimes float on water without sinking, even if they are denser than water. The basic idea is that the molecules on the surface of a liquid are pulled inward by other molecules, creating a sort of 'film' that resists external force.
Surface tension is quantified as force per unit length. You can think of it as a membrane stretched over the surface, exerting a force that tries to minimize the surface area. This is why droplets form spheres—nature's way of using the least surface possible for a given volume.
When you blow a soap bubble, you do work against the surface tension to increase its size. This is exactly what happens as you increase the diameter of the bubble from a smaller diameter to a larger one. Then, the work done is equal to the surface tension multiplied by the change in the bubble's surface area.

Change in Surface Area

To understand how the surface area changes when you blow a bubble larger, consider the initial and final states of the bubble. Initially, the bubble has a certain surface area based on its initial diameter, and as you blow into it, the diameter increases and so does the surface area.
The change in surface area is key to calculating the work done on the bubble. First, you compute the surface area of the bubble at the initial and final diameters using the formula for the surface area of a sphere. For this, you'll need the radii derived from these diameters. Then, subtract the initial surface area from the final surface area. This net change represents the added surface that requires energy due to the surface tension of the bubble.
This difference in surface area, when combined with the surface tension, gives the total work done on the bubble. It's important to understand this because it directly ties the geometry of the sphere with the physics of the surface tension.

Sphere Surface Area Calculation

Calculating the surface area of a sphere is an essential skill in geometry, especially when understanding problems involving bubbles and other spherical objects.
The formula to find the surface area of a sphere is \[ A = 4\pi r^2 \]where \( r \) is the radius of the sphere. Since the diameter of a sphere is twice the radius, you can easily find \( r \) by dividing the diameter by two.
For example, when you have a diameter \( d \) for an initial bubble and \( D \) for a final bubble, their respective radii are \( r_1 = d/2 \) and \( r_2 = D/2 \). Plug these into the surface area formula to find the initial surface area \( A_1 = 4\pi (d/2)^2 \) and the final surface area \( A_2 = 4\pi (D/2)^2 \).
This method is straightforward and provides a clear understanding of how a small increase in the radius affects the surface area significantly, a vital factor when performing calculations involving bubbles and surface tension.