Question

Using a soap solution for which the surface tension is \(0.025 \mathrm{~N} / \mathrm{m}\), a child blows a soap bubble of radius \(1.40 \mathrm{~cm} .\) How much energy is expended in stretching the soap surface?

Short answer

The energy expended in stretching the soap surface is \(7.75e-5 Joules\).

Step-by-step solution

List the Given Information

The surface tension (T) is \(0.025 N/m\) and the radius (r) of the bubble is \(1.40 cm\), which is converted to meters it's \(0.014 m\)

Calculate the Change in Surface Area

A bubble has two layers, so the surface area has doubled. The total change in surface area (\(\Delta A\)) of a sphere of radius r is given by the equation \(\Delta A = 2 \times 4\pi r^2 \). Plugging in the values gives \(\Delta A = 2 \times 4 \times 3.1416 \times (0.014 m)^2 = 0.0031 m^2\).

Calculate the Work Done

The work done (or energy expended) in stretching the bubble is given by the equation \(Work = T \times \Delta A\). Plugging the values in gives; \(Work = 0.025 N/m \times 0.0031 m^2 = 7.75e-5 Joules\).

Key concepts

Soap Bubble

A soap bubble is a thin film of liquid enclosing air. When we blow soap bubbles, they form spherical shapes mainly because a sphere has the smallest surface area for a given volume. This is nature’s way of minimizing energy expenditure and achieving equilibrium.
Soap bubbles are fascinating because they consist of two layers with air trapped in the middle. This dual-layer contributes to certain properties of soap bubbles, such as their lustrous, reflective surface and vibrant colors due to light interference. Upon forming a bubble, the air pressure inside it creates an outward force on the liquid film. Meanwhile, surface tension, the force that occurs at the surface of a liquid due to cohesive forces, tries to minimize the bubble’s surface area by pulling inwards.
Naturally, this balance of forces maintains a bubble's distinct shape and size during its short lifespan in the air. Understanding the nature and behavior of soap bubbles can help in various scientific explorations and in practical applications like designing lightweight structures.

Energy Expenditure

Energy expenditure is the amount of energy used to stretch the soap bubble when blowing it. The surface tension of the soap solution, which is measured in Newtons per meter (N/m), plays a crucial role in this process. Surface tension is the force per unit length exerted by a liquid in contact with a gas or another immiscible liquid.
In the context of a soap bubble, energy expenditure refers to the work done to expand the bubble and increase its surface area. For any bubble, work is calculated by multiplying the surface tension by the change in the surface area of the bubble.

• In our exercise, the surface tension is given as 0.025 N/m.
• The change in surface area is computed using the formula for a sphere.
• Finally, applying the work formula, we find the energy expenditure to be approximately 7.75e-5 Joules.

This calculation helps understand how energy is transformed and used in blowing a bubble—a classic illustration of physics in daily life.

Spherical Surface Area

When dealing with soap bubbles, the concept of spherical surface area is vital. A bubble tends to become spherical because this shape requires the least energy to maintain.
The formula for the surface area of a sphere is given by:\[A = 4\pi r^2\]
For a soap bubble, the surface area essentially doubles because of its two film layers. So, the total surface area is actually twice the area given by the equation above.Calculating the surface area involves squaring the radius and then multiplying by \(4\pi\). This result is then doubled, hence the change in surface area (\(\Delta A\)) for a fully inflated bubble.Understanding the spherical shape and its surface area importance aids in comprehending how things like minimal energy and surface tension coalesce to give bubbles their physical properties. It’s an engaging way to learn about geometry and the principles of physics that govern everyday phenomena.