Problem 99 - Elementary Algebra

Question

The pressure due to surface tension in a spherical bubble is given by \(P=\frac{4 T}{r},\) where \(T\) is the surface tension of the liquid, and \(r\) is the radius of the bubble. (a) Determine the pressure due to surface tension within a soap bubble of radius \(\frac{1}{2}\) inch and surface tension 22 . (b) Determine the radius of a bubble if the pressure due to surface tension is 57.6 and the surface tension is 18 .

Step-by-Step Solution

Answer

Answer: The pressure within a soap bubble of radius 0.5 inches and surface tension 22 is 176. The radius of a bubble with a pressure of 57.6 and surface tension 18 is 1.25.

Step 1: Given the formula for pressure due to surface tension in a spherical bubble: \(P=\frac{4 T}{r}\) For a soap bubble of radius \(\frac{1}{2}\) inch and surface tension 22, we just need to plug these values into the formula: \(P=\frac{4 \times 22}{\frac{1}{2}}\) #Step 2: Calculate the pressure

Now we just need to do the calculations: \(P=\frac{88}{\frac{1}{2}}=176\) So the pressure due to surface tension within a soap bubble of radius \(\frac{1}{2}\) inch and surface tension 22 is 176. #Step 3: Rearrange the formula to find the radius

Step 2: In order to determine the radius of a bubble if the pressure due to surface tension and the surface tension are given, we need to rearrange the formula to solve for \(r\): \(r = \frac{4 T}{P}\) Now, given the pressure due to surface tension is 57.6 and the surface tension is 18, we can plug in these values: \(r = \frac{4 \times 18}{57.6}\) #Step 4: Calculate the radius

Now we just need to do the calculations: \(r = \frac{72}{57.6}=1.25\) So the radius of a bubble with pressure due to surface tension 57.6 and surface tension 18 is 1.25.

Key Concepts

Spherical Bubble Formula

The spherical bubble formula is foundational in understanding pressure dynamics in bubbles. It stems from the equilibrium of forces present along the bubble's surface. In a spherical bubble, the pressure difference across the surface is particularly significant. This pressure, due to surface tension, is mathematically expressed as:

• \[P = \frac{4T}{r}\]

Here, \(P\) stands for pressure, \(T\) represents surface tension, and \(r\) is the radius of the bubble. This formula indicates that the pressure is directly proportional to the surface tension and inversely proportional to the radius of the bubble. In simple terms, a smaller bubble has a higher pressure for the same surface tension. Likewise, greater surface tension leads to increased pressure. Understanding this relationship is crucial for solving problems involving bubbles and helps in calculating the key variables involved in maintaining a bubble's shape.

Surface Tension

Surface tension is the elastic tendency of a fluid surface, which makes it acquire the least surface area possible. Imagine a thin elastic film stretched over the water that pulls on the surface molecules, keeping the bubble intact. In our spherical bubble formula, it is denoted as \(T\).
Surface tension is crucial because it affects the behavior and stability of the bubble. A substance with high surface tension, like water mixed with soap, allows bubbles to form and persist. The measurement of surface tension is usually in dyn/cm (dynes per centimeter) or in relation to the force exerted over a certain length.
In the bubble context, surface tension works against the internal pressure driving outward, forming a thin, stable layer encasing air. It plays a vital role in the bubble's formation and affects how robust and inflated the bubble appears.

Radius Calculation

To find the radius of a bubble where the pressure and surface tension are provided, we rearrange the formula for spherical bubble pressure to solve for radius \(r\):

• \[r = \frac{4T}{P}\]

Here, \(P\) is the pressure due to surface tension, and \(T\) is the surface tension itself. This formula highlights that the radius is dependent on both the force exerted per unit length and the internal pressure within the bubble.
To calculate the radius given the values of pressure and surface tension:

• Identify the values for pressure \(P\) and surface tension \(T\).
• Substitute the known values into the rearranged formula.
• Perform the calculation to solve for \(r\).

This mathematical process allows one to determine how large or small the bubble will be under specific conditions, given that both surface tension and pressure are known factors.