Question

Radius of a soap bubble is \(\mathrm{r}^{\prime}\), surface tension of soap solution is \(\mathrm{T}\). Then without increasing the temperature how much energy will be needed to double its radius. (A) \(4 \pi r^{2} T\) (B) \(2 \pi r^{2} T\) (C) \(12 \pi r^{2} T\) (D) \(24 \pi r^{2} T\)

Short answer

The short answer is: \(\Delta E = 12\pi (\mathrm{r}^{\prime})^2 T\).

Step-by-step solution

Determine the initial and final surface areas of the bubble

The surface area of a sphere (soap bubble) is given by the formula \(A = 4\pi r^{2}\). Initially, the radius is \(\mathrm{r}^{\prime}\), so the surface area is \(A_{1} = 4\pi (\mathrm{r}^{\prime})^2\). When we double the radius, we have a new radius of \(2\mathrm{r}^{\prime}\). Thus, the new surface area is \(A_{2} = 4\pi (2\mathrm{r}^{\prime})^2 = 16\pi (\mathrm{r}^{\prime})^2\).

Calculate the initial and final surface energies of the bubble

Surface energy is proportional to the surface area, and the constant of proportionality is the surface tension, \(T\). The initial surface energy is \(E_{1} = T A_{1} = 4\pi (\mathrm{r}^{\prime})^2 T\), and the final surface energy is \(E_{2} = T A_{2} = 16\pi (\mathrm{r}^{\prime})^2 T\).

Determine the amount of energy required to double the radius

The energy required to double the radius will be the difference between the final and initial surface energies: \[\Delta E = E_{2} - E_{1} = 16\pi (\mathrm{r}^{\prime})^2 T - 4\pi (\mathrm{r}^{\prime})^2 T = 12\pi (\mathrm{r}^{\prime})^2 T\]

Identify the answer

The energy required to double the radius of the soap bubble is given by our result in Step 3, which is \(12\pi (\mathrm{r}^{\prime})^2 T\). Looking at the given options, the correct answer is: (C) 12\pi \(r^{2} T\).

Key concepts

Energy Calculation for Soap Bubbles

When calculating the energy required for a soap bubble's radius to double, it's essential to understand the connection between energy and surface area. The energy needed is determined by the change in surface energy, which depends on the surface tension and the bubble's surface area.
The energy

• Initially, the energy is given by the formula: \[ E_{1} = 4\pi (\mathrm{r}^{\prime})^2 T \]where \( \mathrm{r}^{\prime} \) is the initial radius and \( T \) is the surface tension.
• Once the radius doubles, the new energy becomes: \[ E_{2} = 16\pi (\mathrm{r}^{\prime})^2 T \]
• The required energy to double the radius is: \[ \Delta E = E_{2} - E_{1} = 12\pi (\mathrm{r}^{\prime})^2 T \]

By understanding these calculations, you can grasp how surface tension energy changes lead to the need for additional energy to increase the bubble's size.

Surface Area of a Sphere

A soap bubble can be considered a simple sphere, and its surface area is calculated using the formula: \[ A = 4\pi r^2 \] This formula shows the relationship between the radius of the sphere and its surface area.
In the context of the problem:

• The initial surface area with radius \( \mathrm{r}^{\prime} \) is \( A_{1} = 4\pi (\mathrm{r}^{\prime})^2 \).
• Doubling the radius to \( 2\mathrm{r}^{\prime} \) changes the area to \( A_{2} = 16\pi (\mathrm{r}^{\prime})^2 \).

This illustrates the principle that the surface area increases with the square of the radius, providing insight into how much more energy is needed to expand the bubble's size.

Soap Bubble Physics Explained

Soap bubbles offer a fascinating example of physics in action, primarily because of their unique surface tension dynamics. Surface tension acts as a force that minimizes the surface area of the bubble. It affects the bubble's energy and stability.
Key insights into soap bubble behavior include:

• **Surface Tension**: Dictated by the liquid's composition, which works to minimize surface area and is responsible for the spherical shape of bubbles.
• **Energy Considerations**: When the bubble's radius is increased, more energy is needed due to greater surface tension across a larger area.
• **Equilibrium**: Bubbles naturally tend to maintain the least energy configuration, which is why they are spherical.

This combination of factors underlines why soap bubbles require additional energy to grow larger without changing temperature, showcasing the delicate balance of physics principles at work.