Question

Two soap bubbles \(A\) and \(B\) are kept in a closed chamber where the air is maintained at pressure \(8 \mathrm{~N} / \mathrm{m}^{2}\). The radii of bubbles \(\mathrm{A}\) and \(\mathrm{B}\) are \(2 \mathrm{~cm}\) and \(4 \mathrm{~cm}\), respectively. Surface tension of the soap-water used to make bubbles is \(0.04 \mathrm{~N} / \mathrm{m}\). Find the ratio \(n_{\mathrm{B}} / n_{A}\), where \(n_{\Lambda}\) and \(n_{\mathrm{B}}\) are the number of moles of air in bubbles \(\mathrm{A}\) and \(\mathrm{B}\), respectively. [Neglect the effect of gravity.]

Short answer

The ratio of the number of moles of air in Bubble B to that in Bubble A is 1.2.

Step-by-step solution

Determine the Pressure Inside Each Bubble

First, calculate the pressure inside each bubble due to surface tension using the formula: Pressure inside bubble = Pressure outside + (2 * Surface Tension) / Radius. For Bubble A, this is: P_A = 8 N/m^2 + (2 * 0.04 N/m) / (0.02 m). For Bubble B, the same formula is applied: P_B = 8 N/m^2 + (2 * 0.04 N/m) / (0.04 m).

Calculate Pressures P_A and P_B

Substitute the given values to calculate pressures for both bubbles: P_A = 8 + (2 * 0.04) / 0.02 = 8 + 4 = 12 N/m^2. P_B = 8 + (2 * 0.04) / 0.04 = 8 + 2 = 10 N/m^2.

Use the Ideal Gas Law

Apply the ideal gas law to relate pressure and number of moles inside the bubble, which states PV = nRT. Here P is pressure, V is volume, n is number of moles, R is the gas constant, and T is the temperature. Since R, T, and V for a bubble of a given radius are constants (and equal for bubbles in the same chamber), we can relate the pressures and mole numbers in two bubbles by the formula (P_A / n_A) = (P_B / n_B).

Relate Pressures and Number of Moles

From the ideal gas law, we have P_A / n_A = P_B / n_B, which yields n_B / n_A = P_A / P_B. Substitute P_A and P_B with the calculated values to find the ratio n_B / n_A.

Calculate the Ratio n_B / n_A

Substitute the calculated pressures into the ratio: n_B / n_A = P_A / P_B = 12 N/m^2 / 10 N/m^2 = 1.2.

Key concepts

Understanding the Ideal Gas Law

When studying soap bubbles, like in our textbook problem, we often rely on the ideal gas law, which is a fundamental principle in chemistry and physics. This law is crucial for understanding how gases behave under different conditions of pressure, volume, and temperature.

The ideal gas law is generally represented by the equation \(PV = nRT\), where \(P\) denotes the pressure of the gas, \(V\) its volume, \(n\) the number of moles of the gas, \(R\) the ideal gas constant, and \(T\) the absolute temperature.

In the context of soap bubbles, this law helps us deduce the number of moles of air inside each bubble, assuming that the air inside behaves like an ideal gas. By comparing the ratio of pressure to number of moles between two bubbles, as in our exercise, we can determine the relative amount of air in each one. This comparison is made possible because the temperature and the ideal gas constant are the same for both bubbles. Hence, differences in the moles of gases are reflected in changes in pressure when the bubble's volume is held constant.

The Role of Surface Tension in Soap Bubbles

Surface tension is a central concept not only in our exercise but also in understanding fluid dynamics and phenomena involving liquids. In simple terms, surface tension is the elastic tendency of a fluid surface to acquire the least surface area possible.

Surface tension explains various daily life occurrences such as the shape of water droplets and, pertinent to our discussion, the formation of soap bubbles. When creating a soap bubble, the surface tension acts to minimize the bubble's surface area, creating a spherical shape.

In the given exercise, the surface tension of the soap-water solution adds extra pressure to the air trapped inside the bubble. The formula for the pressure inside a soap bubble is given by \(P_{inside} = P_{outside} + \frac{2 \times \text{Surface Tension}}{\text{Radius}}\). It's this additional pressure from surface tension that directly affects the number of moles inside the bubble according to the ideal gas law. Therefore, understanding surface tension allows us to accurately analyze the pressure conditions not only within soap bubbles but in any situation where liquids and gases interact.

Pressure Calculations Inside Soap Bubbles

Calculating the pressure inside a soap bubble is not as straightforward as one might think due to the involvement of surface tension. As shown in the step by step solution of our exercise, the pressure inside the bubble is not just the ambient pressure but also includes a contribution from surface tension.

The pressure inside a soap bubble can be calculated by adding the external air pressure to twice the ratio of the surface tension to the radius of the bubble, expressed as \(P_{inside} = P_{outside} + \frac{2 \times \text{Surface Tension}}{\text{Radius}}\).

Understanding Pressure Differences

With two soap bubbles in different sizes, the pressure inside each bubble will differ due to their varying radii. As the pressure inside a smaller bubble is higher, owing to a lower radius, and assuming that the temperature and the amount of substance are consistent, the smaller bubble will contain fewer moles of gas compared to the larger one. This is why in our exercise, despite bubble B being larger and thus having a larger volume, it has a lower internal pressure compared to bubble A.

By understanding how to conduct pressure calculations and how they relate to the number of moles within the ideal gas law, students can solve complex problems that involve gas laws and fluid mechanics with more confidence and clarity.