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A bubble of a gas released at the bottom of a lake increases to eight times its original volume when it reaches the surface. Assuming that atmospheric pressure is equivalent to the pressure exerted by a column!of water 10 m height, the depth of the lake is
- a)70 m
- b)10 m
- c)40 m
- d)80 m
Correct answer is option 'A'. Can you explain this answer?
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A bubble of a gas released at the bottom of a lake increases to eight ...
**Given:**
- The gas bubble increases to eight times its original volume when it reaches the surface.
- Atmospheric pressure is equivalent to the pressure exerted by a column of water 10 m height.
**To find:**
The depth of the lake.
**Explanation:**
Let's assume the initial volume of the gas bubble at the bottom of the lake is V1, and its final volume at the surface is V2. We are given that V2 is eight times V1.
According to Boyle's Law, the volume of a gas is inversely proportional to the pressure exerted on it, assuming the temperature remains constant.
Mathematically, Boyle's Law can be represented as:
P1 * V1 = P2 * V2
Where P1 and P2 are the initial and final pressures, and V1 and V2 are the initial and final volumes.
In this case, the initial pressure (P1) is the pressure exerted by the column of water at the bottom of the lake, and the final pressure (P2) is the atmospheric pressure at the surface.
We are given that the atmospheric pressure is equivalent to the pressure exerted by a column of water 10 m height. This means P2 is equal to the pressure exerted by a column of water 10 m height.
Let's assume the depth of the lake is D.
Now, let's substitute the given values into Boyle's Law equation:
(Pressure at the bottom of the lake) * V1 = (Pressure at the surface) * V2
(Density of water * g * Depth of the lake) * V1 = (Pressure at the surface) * (8 * V1)
Since the pressure at the surface is equivalent to the pressure exerted by a column of water 10 m height, we can write:
(1000 kg/m³ * 9.8 m/s² * D) * V1 = (1000 kg/m³ * 9.8 m/s² * 10 m) * (8 * V1)
Simplifying the equation:
9800 * D * V1 = 98000 * (8 * V1)
D = 98000 * (8 * V1) / 9800 * V1
D = 8 * 10 = 80 m
Therefore, the depth of the lake is 80 m.
- The gas bubble increases to eight times its original volume when it reaches the surface.
- Atmospheric pressure is equivalent to the pressure exerted by a column of water 10 m height.
**To find:**
The depth of the lake.
**Explanation:**
Let's assume the initial volume of the gas bubble at the bottom of the lake is V1, and its final volume at the surface is V2. We are given that V2 is eight times V1.
According to Boyle's Law, the volume of a gas is inversely proportional to the pressure exerted on it, assuming the temperature remains constant.
Mathematically, Boyle's Law can be represented as:
P1 * V1 = P2 * V2
Where P1 and P2 are the initial and final pressures, and V1 and V2 are the initial and final volumes.
In this case, the initial pressure (P1) is the pressure exerted by the column of water at the bottom of the lake, and the final pressure (P2) is the atmospheric pressure at the surface.
We are given that the atmospheric pressure is equivalent to the pressure exerted by a column of water 10 m height. This means P2 is equal to the pressure exerted by a column of water 10 m height.
Let's assume the depth of the lake is D.
Now, let's substitute the given values into Boyle's Law equation:
(Pressure at the bottom of the lake) * V1 = (Pressure at the surface) * V2
(Density of water * g * Depth of the lake) * V1 = (Pressure at the surface) * (8 * V1)
Since the pressure at the surface is equivalent to the pressure exerted by a column of water 10 m height, we can write:
(1000 kg/m³ * 9.8 m/s² * D) * V1 = (1000 kg/m³ * 9.8 m/s² * 10 m) * (8 * V1)
Simplifying the equation:
9800 * D * V1 = 98000 * (8 * V1)
D = 98000 * (8 * V1) / 9800 * V1
D = 8 * 10 = 80 m
Therefore, the depth of the lake is 80 m.
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A bubble of a gas released at the bottom of a lake increases to eight times its original volume when it reaches the surface. Assuming that atmospheric pressure is equivalent to the pressure exerted by a column!of water 10 m height, the depth of the lake isa)70 mb)10 mc)40 md)80 mCorrect answer is option 'A'. Can you explain this answer?
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