Question

A bubble rises from the bottom of a lake \(90\) m deep. On reaching the surface, its volume becomes \((\text{Atmospheric pressure is }10\text{ m of water})\)

• A. \(4\) times
• B. \(8\) times
• C. \(10\) times
• D. \(3\) times

Hint:

Pressure in water increases with depth; total pressure at depth \(h\) equals atmospheric pressure plus liquid pressure.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
As an air bubble rises from the bottom of a lake to the surface, the external pressure on it decreases. Assuming the temperature of the lake water is constant, we can use Boyle's Law to relate the pressure and volume of the bubble at the bottom and at the surface.
Step 2: Key Formula or Approach:
1. Boyle's Law: $P_1V_1 = P_2V_2$. 2. Pressure at a depth: The pressure at a depth 'h' in a liquid is given by $P = P_{atm} + h\rho g$. It's often convenient to express pressures in terms of the height of a water column. 3. The atmospheric pressure is given as equivalent to 10 m of water.
Step 3: Detailed Explanation:
Let's define the initial and final states of the bubble: State 1 (Bottom of the lake): - Depth, $h = 90$ m. - Pressure at the bottom, $P_1$: This is the sum of the atmospheric pressure and the pressure due to the water column. - Atmospheric pressure = 10 m of water. - Water column pressure = 90 m of water. - $P_1 = (10 + 90)$ m of water = 100 m of water. - Volume at the bottom = $V_1$. State 2 (Surface of the lake): - At the surface, the pressure is just the atmospheric pressure. - Pressure at the surface, $P_2 = 10$ m of water. - Volume at the surface = $V_2$. Now, apply Boyle's Law ($P_1V_1 = P_2V_2$): \[ (100 \text{ m of water}) \times V_1 = (10 \text{ m of water}) \times V_2 \] We need to find how many times the volume becomes, which is the ratio $V_2/V_1$. \[ \frac{V_2}{V_1} = \frac{100}{10} = 10 \] So, $V_2 = 10 V_1$. The volume becomes 10 times the original volume. Step 4: Final Answer:
The volume of the bubble becomes 10 times its initial volume upon reaching the surface. Therefore, option (C) is correct.