Problem 32 - Elementary Algebra

Question

The pressure due to surface tension in a spherical drop of liquid is given by \(P=\frac{2 T}{r},\) where \(T\) is the surface tension of the liquid and \(r\) is the radius of the drop. If the liquid is a bubble, it has two surfaces and the surface tension is given by $$ P=\frac{2 T}{r}+\frac{2 T}{r}=\frac{4 T}{r} $$ (a) Determine the pressure due to surface tension within a soap bubble of radius 2 inches and surface tension 28 . (b) Determine the radius of a bubble if the pressure due to surface tension is 52 and the surface tension is 39 .

Step-by-Step Solution

Answer

Question: Calculate the pressure due to surface tension within a soap bubble of radius 2 inches and surface tension 28. Also, find the radius of a bubble if the pressure due to surface tension is 52 and the surface tension is 39. Answer: The pressure due to surface tension within the soap bubble is 56. The radius of the bubble is 3.

Step 1: Part (a): Pressure due to surface tension in a soap bubble of radius 2 inches and surface tension 28

For this part, we are given the radius r = 2 inches and surface tension T = 28. We can use the formula for the pressure in a bubble: \(P = \frac{4T}{r}\) Now, plug in the given values for T and r: \(P = \frac{4 \times 28}{2}\) Calculate the pressure: \(P = \frac{112}{2}\) \(P = 56\) The pressure due to surface tension within the soap bubble is 56.

Step 2: Part (b): Radius of a bubble given pressure due to surface tension is 52 and surface tension is 39

For this part, we know the pressure P = 52 and the surface tension T = 39. We will use the bubble pressure formula and rearrange it to solve for the radius r: \(P = \frac{4T}{r}\) First, let's solve for the expression \(\frac{4T}{r}\): \(\frac{4T}{r} = \frac{4 \times 39}{r}\) \(\frac{4T}{r} = \frac{156}{r}\) Now, we are given that the pressure P = 52. Set the expression equal to P: \(52 = \frac{156}{r}\) To solve for r, multiply both sides of the equation by r: \(52r = 156\) Now, divide both sides of the equation by 52 to isolate r: \(r = \frac{156}{52}\) Calculate the radius: \(r = 3\) The radius of the bubble is 3.

Key Concepts

spherical drop

A spherical drop is a small, round body of liquid that maintains its shape due to surface tension. Surface tension is a physical property of the liquid which acts as a "skin" at the surface, holding the liquid molecules together. This phenomenon occurs because liquid molecules are attracted to each other, causing the surface to contract and minimize the area.
For a spherical drop, pressure due to surface tension is an important concept. It is the force exerted on the interior of the drop due to the molecules straining against each other. This pressure is inversely proportional to the radius, as given by the formula:

• Pressure, \(P = \frac{2 T}{r}\),

where \(T\) is the surface tension and \(r\) is the radius of the drop. In essence, smaller drops will have higher pressure because the molecules are packed more closely together.

soap bubble

Soap bubbles are fascinating due to their thin, spherical film of soap water enclosing air. Unlike simple spherical drops, soap bubbles involve two liquid-gas interfaces: one inner and one outer. This means the surface tension equation for bubbles needs to be adjusted.
For soap bubbles, the pressure difference is experienced across both interfaces, and is expressed as:

• Pressure, \(P = \frac{4 T}{r}\),

indicating that the bubbles have twice the surface tension force compared to singular drops, given the two layers of liquid. This equation helps us determine crucial information about the soap bubble's behavior, such as how different radii and surface tension values affect the pressure experienced inside the bubble.

radius calculation

Calculating the radius of a spherical object involves algebraic rearrangement of the formula for pressure. In the context of bubbles, this means using the formula \(P = \frac{4 T}{r}\) to solve for the radius \(r\) when the pressure \(P\) and surface tension \(T\) are known.
If given the pressure and surface tension values, you can find the radius using the following steps:

• Rearrange the formula to isolate \(r\): \(r = \frac{4 T}{P}\).
• Substitute the known values of \(P\) and \(T\) into this equation.
• Calculate the result to determine the radius.

These steps effectively allow you to understand how various properties of a soap bubble are connected, such as how a larger radius may correspond to a lower internal pressure given the same surface tension.

algebraic manipulation

Algebraic manipulation is a valuable skill in solving physics problems, particularly when dealing with equations involving pressure and surface tension like the ones used in sphere and bubble models. The goal of this exercise is to practice rearranging the given formulae to solve for unknown variables.
For example:

• Start with the known formula \(P = \frac{4 T}{r}\).
• To find \(r\), multiply both sides by \(r\): \(P \times r = 4 T\).
• Then divide by \(P\) to isolate \(r\): \(r = \frac{4 T}{P}\).

Algebraic manipulation allows us to solve problems by simplifying and reordering expressions, making it a crucial tactic in understanding relationships between different physical properties such as pressure, surface tension, and radius.