Problem 83 - Master Resource Book in JEE Main Physics

Question

A bubble rises from bottom of a lake \(90 \mathrm{~m}\) deep. On reaching the surface, its volume becomes (take atmospheric pressure correspond upto \(10 \mathrm{~m}\) of water) [BVP Engg, 2006] (a) 18 times (b) 4 times (c) 8 times (d) 10 times

Step-by-Step Solution

Answer

The bubble's volume becomes 10 times its original volume.

Step 1: Understand Boyle's Law

Boyle's Law states that for a given mass of gas at constant temperature, the volume of the gas is inversely proportional to the pressure. Mathematically, it is expressed as \( P_1 V_1 = P_2 V_2 \), where \( P \) is pressure and \( V \) is volume.

Step 2: Determine Initial and Final Pressures

Initially, at the bottom of the lake, the pressure is due to both atmospheric pressure and the water column. So, \( P_1 = 10 \mathrm{~m} + 90 \mathrm{~m} = 100 \mathrm{~m} \) of water. At the surface, the pressure is just the atmospheric pressure which corresponds to \( P_2 = 10 \mathrm{~m} \) of water.

Step 3: Apply Boyle's Law

Using Boyle's Law, \( P_1 V_1 = P_2 V_2 \), we substitute the known pressures: \( 100 V_1 = 10 V_2 \). Simplifying this equation, we find \( V_2 = 10 V_1 \).

Step 4: Calculate the Volume Change

The volume of the bubble when it reaches the surface becomes \( 10 \) times its original volume at the bottom of the lake. Therefore, the volume becomes 10 times.

Key Concepts

Pressure

Pressure is the force exerted by gas particles as they strike the walls of their container. It can be thought of as how hard these particles push against the container's surfaces. In the measurement of pressure, we often use units like atmospheres (atm), pascals (Pa), or meters of water, as in the original example.

In the context of Boyle's Law and our problem, we can identify the key role pressure plays in determining how gases behave under different conditions. Initially, the pressure on the bubble at the lake's bottom is much higher due to the weight of the water above it. This higher pressure compresses the gas, reducing its volume compared to when it is subjected to just atmospheric pressure at the surface.

To calculate the initial pressure, we summed the atmospheric pressure equivalent to the weight of a 10 m column of water and the pressure from the additional 90 m of water above the bubble, totaling to 100 m water pressure. Understanding the changes in pressure as the bubble rises explains why its volume increases drastically at the surface.

Volume

Volume is a measure of the space occupied by a gas. Under constant temperature and using Boyle's Law (which applies to ideal gases), volume and pressure share an inverse relationship. This means if the pressure increases, the volume decreases, and vice versa. This principle helps us comprehend the behavior of the bubble as it ascends to the lake's surface.

At the bottom of the lake, where pressure is high, the volume of the bubble is compressed. As it rises and pressure drops, the volume increases. By the time the bubble reaches the surface, where only atmospheric pressure acts on it, its volume dramatically expands.

With the application of Boyle's Law, we calculated that once the bubble surfaces, the volume becomes ten times larger than its original volume at the bottom. This volume expansion is critical to processes in different fields, such as understanding buoyancy and why similar effects are observed across various applications of gas laws.

Gas Laws

Gas laws describe the relationships between the pressure, volume, and temperature of gases. Boyle's Law, a fundamental gas law utilized in solving our original exercise, states that for a given amount of gas at a constant temperature, its pressure is inversely proportional to its volume. It's expressed by the equation: \[ P_1 V_1 = P_2 V_2 \]This relationship is crucial for understanding how gases respond to changes in their environment.

The other key gas laws include Charles's Law, which looks at the volume and temperature relationship, and Avogadro's Law, which deals with volume and the amount of gas. These laws collectively form the foundation of the ideal gas law, a more comprehensive relationship that combines all these variables.

In real-life applications, such as the behavior of a rising bubble in water, we rely heavily on these principles to predict outcomes, like whether the bubble will burst as it reaches lower pressures and why its volume increases. Mastery of these fundamental concepts enables us to understand and predict gas behavior in various practical and experimental situations.