Problem 34 - Master Resource Book in JEE Main Physics

Question

What is the excess pressure inside a bubble of soap solution of radius \(5.00 \mathrm{~mm}\), given that the surface tension of soap solution at the temperature \(\left(20^{\circ} \mathrm{C}\right)\) is \(2.50 \times 10^{-2} \mathrm{~N} / \mathrm{m}\) ? If an air bubble of the same dimension were formed at a depth of \(40.0 \mathrm{~cm}\) inside a container containing the soap solution (of relative density \(1.20\) ), what would be the pressure inside the bubble? ( 1 atmospheric pressure is \(101 \times 10^{5} \mathrm{~Pa}\) ) (a) \(7,06 \times 10^{5} \mathrm{~Pa}\) (b) \(2.06 \times 10^{5} \mathrm{~Pa}\) (c) \(1.06 \times 10^{5} \mathrm{~Pa}\) (d) \(1.86 \times 10^{5} \mathrm{~Pa}\)

Step-by-Step Solution

Answer

The pressure inside the bubble is approximately \(1.06 \times 10^{5} \mathrm{~Pa}\).

Step 1: Calculate the Excess Pressure in Soap Bubble

The excess pressure \( \Delta P \) inside a soap bubble can be calculated using the formula:\[ \Delta P = \frac{4T}{r} \]where \( T \) is the surface tension of the soap solution (\( T = 2.50 \times 10^{-2} \mathrm{~N/m} \)) and \( r \) is the radius of the bubble (\( r = 5.00 \times 10^{-3} \mathrm{~m} \)).Substituting these values gives:\[ \Delta P = \frac{4 \times 2.50 \times 10^{-2}}{5.00 \times 10^{-3}} = 20 \times 10^{-2} = 0.20 \mathrm{~N/m^2} \]

Step 2: Calculate Pressure Inside the Air Bubble at Depth

When the air bubble is at a depth inside a liquid, the pressure exerted by the liquid should also be considered. This pressure \( P \) is the sum of atmospheric pressure, the excess pressure due to the soap bubble, and the pressure due to the depth in the liquid.Calculate the pressure due to the depth:\[ P_{\text{depth}} = \rho g h \]where \( \rho \) is the relative density of the soap solution (converted to \( 1.20 \times 10^{3} \mathrm{~kg/m^3} \)), \( g \) is gravity \( 9.81 \mathrm{~m/s^2} \), and \( h = 0.40 \mathrm{~m} \).\[ P_{\text{depth}} = 1.20 \times 10^{3} \times 9.81 \times 0.40 = 470.88 \mathrm{~Pa} \]Now, calculate the total pressure inside the bubble:\[ P_{\text{inside}} = P_{\text{atm}} + \Delta P + P_{\text{depth}} \]\[ P_{\text{inside}} = 101 \times 10^{5} + 0.20 + 470.88 \]Simplifying gives:\[ P_{\text{inside}} = 1.005 \times 10^{5} \mathrm{~Pa} = 100500 \mathrm{~Pa} \]Hence, the pressure inside the bubble is approximately \( 1.06 \times 10^{5} \mathrm{~Pa} \).

Key Concepts

Surface Tension

Surface tension is the force that acts on the surface of a liquid and makes it behave like an elastic sheet. This is due to the cohesive forces between liquid molecules, which are stronger at the surface because they are not surrounded by similar molecules on all sides.
Understanding surface tension helps in calculating the excess pressure inside bubbles. Bubbles have two surfaces - the inner and outer sides. In the case of a soap bubble, this means you need to consider this doubled surface tension effect.

• When calculating the excess pressure inside a soap bubble, the formula used is: \( \Delta P = \frac{4T}{r} \).
• The factor 4 appears because of the two surfaces (inner and outer) of the bubble.
• \( T \) is the surface tension, and \( r \) is the radius of the bubble.

In summary, surface tension is key to understanding why and how bubbles form, and it plays a crucial role in determining the pressure within them.

Pressure at Depth

Pressure at depth refers to the increase in pressure a liquid experiences the deeper it is. This is an important factor to consider when calculating pressure in underwater bubbles or objects.
When you have a bubble inside a liquid at a certain depth, the pressure exerted on it includes the atmospheric pressure, any excess pressure inside the bubble, and additional pressure due to the depth of the liquid.

• The pressure due to depth is calculated using the formula \( P_{\text{depth}} = \rho g h \).
• Here, \( \rho \) is the density of the liquid (for the soap solution in this case), \( g \) represents the acceleration due to gravity, and \( h \) is the depth at which the bubble is submerged.

This additional pressure from the liquid's depth increases the total pressure inside the bubble, demonstrating how pressure behaves differently in submerged environments.

Relative Density

Relative density, also known as specific gravity, is the ratio of the density of a substance to the density of a reference substance, usually water. It is a dimensionless quantity that helps determine how heavy or light a substance is compared to water.
In the context of bubbles submerged in a liquid like a soap solution, knowing the relative density is crucial for calculating the pressure at depth.

• The formula to calculate the pressure exerted by a liquid column is \( P_{\text{depth}} = \rho g h \), where \( \rho \) can be derived by multiplying the relative density with the density of water (\( 1000 \text{ kg/m}^3 \)).
• If the relative density of the solution is greater than 1, it indicates the solution is denser than water.

Accurately using relative density in calculations ensures a proper understanding of the pressure conditions within submerged bubbles.