Show that the pressure difference between the inside and the outside of a bubble of radius r is 4y/r, where y is the surface tension of the liquid from which the bubble is blown.

To demonstrate that the pressure difference between the inside and the outside of a bubble of radius r is given by the formula 4y/r, where y represents the surface tension of the liquid, we can delve into the principles of fluid mechanics and surface tension. This concept can be understood more easily by visualizing the bubble and considering the forces acting on it.

The Role of Surface Tension

Surface tension is a property of liquids that causes their surfaces to behave like a stretched elastic membrane. When a bubble forms, the liquid film surrounding it experiences tension, which creates a pressure difference between the inside of the bubble and the surrounding liquid. This difference is crucial for maintaining the bubble's shape and stability.

Understanding Pressure in a Bubble

In a spherical bubble, the pressure inside is greater than the pressure outside due to the surface tension acting along the surface of the bubble. To analyze this situation, we can apply the Young-Laplace equation, which relates the pressure difference across a curved surface to the curvature of that surface and the surface tension.

Applying the Young-Laplace Equation

The Young-Laplace equation states that:

• ΔP = y (1/R1 + 1/R2)

In the case of a bubble, which is a spherical shape, both principal radii of curvature (R1 and R2) are equal to the radius of the bubble (r). Therefore, we can simplify the equation:

• ΔP = y (1/r + 1/r) = 2y/r

This result gives us the pressure difference for a soap bubble considering only the surface tension on one side of the bubble. However, a soap bubble has two surfaces: the inner and outer surfaces. Each surface contributes to the overall pressure difference.

Doubling the Contribution from Two Surfaces

Since there are two interfaces (the inner and outer surfaces of the bubble), we must account for the pressure contribution from both surfaces. Thus, we multiply the pressure difference derived from one surface by 2:

• Total ΔP = 2(2y/r) = 4y/r

Thus, the final expression for the pressure difference between the inside and outside of a bubble of radius r is:

• ΔP = 4y/r

Real-World Applications

This principle has practical applications in various fields, including the design of soap bubbles, foam stability in food science, and even in understanding the behavior of biological cells. Knowing how pressure and surface tension interact helps us predict and manage the stability of bubbles and foams in both natural and engineered systems.

In summary, the pressure difference across a bubble arises from the combined effects of surface tension at both the inner and outer surfaces. This fundamental understanding of fluid mechanics not only explains the stability of bubbles but also emphasizes the importance of surface tension in various scientific and engineering contexts.