Question 68: Properties of Matter

Question

An air bubble of volume $$1 \mathrm{~cm}^{3}$$ rises from the bottom of a lake $$40 \mathrm{~m}$$ deep to the surface at a temperature of $$12^{\circ} \mathrm{C}$$. The atmospheric pressure is $$1 	imes 10^{5} \mathrm{~Pa}$$ the density of water is $$1000 \mathrm{~kg} / \mathrm{m}^{3}$$ and $$\mathrm{g}=10 \mathrm{~m} / \mathrm{s}^{2}$$. There is no difference of the temperature of water at the depth of $$40 \mathrm{~m}$$ and on the surface. The volume of air bubble when it reaches the surface will be:

Accepted Answer

The volume of the air bubble changes due to the change in pressure as it rises from the bottom of the lake to the surface. We can use Boyle's Law to calculate the change in volume, which states that the product of pressure and volume is constant for a given mass of confined gas held at a constant temperature:
$P_1V_1 = P_2V_2$
where $P_1$ and $V_1$ are the pressure and volume at the bottom of the lake and $P_2$ and $V_2$ are the pressure and volume at the surface of the lake.
At the bottom of the lake, the pressure is the atmospheric pressure plus the pressure due to the water column above the bubble:
$P_1 = P_{	ext{atm}} + ho gh$
where $ho$ is the density of water, $g$ is the acceleration due to gravity, and $h$ is the height of the water column. Substituting the given values, we get:
$P_1 = 1 	imes 10^{5} 	ext{ Pa} + 1000 	ext{ kg/m}^3 	imes 10 	ext{ m/s}^2 	imes 40 	ext{ m} = 5 	imes 10^{5} 	ext{ Pa}$
At the surface of the lake, the pressure is the atmospheric pressure:
$P_2 = P_{	ext{atm}} = 1 	imes 10^{5} 	ext{ Pa}$
The initial volume of the bubble is:
$V_1 = 1 	ext{ cm}^3 = 1 	imes 10^{-6} 	ext{ m}^3$
Substituting these values into Boyle's Law and solving for $V_2$, we get:
$V_2 = \frac{P_1V_1}{P_2} = \frac{5 	imes 10^{5} 	ext{ Pa} 	imes 1 	imes 10^{-6} 	ext{ m}^3}{1 	imes 10^{5} 	ext{ Pa}} = 5 	imes 10^{-6} 	ext{ m}^3 = 5 	ext{ cm}^3$
So, the volume of the air bubble when it reaches the surface is 5 cmÃÂ³.

Answer

$$4 \mathrm{~cm}^{3}$$

Answer

$$3 \mathrm{~cm}^{3}$$

Answer

$$2 \mathrm{~cm}^{3}$$

Answer

$$5 \mathrm{~cm}^{3}$$

Properties of Matter

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