Air Bubble Volume Change In A Lake: Physics Explained

Let's dive into a fascinating physics problem involving an air bubble making its journey from the depths of a lake to its surface. This scenario allows us to explore the concepts of pressure, volume, and temperature relationships as described by the ideal gas law. So, grab your thinking caps, and let's break down what happens to this little bubble.

Understanding the Initial Conditions

Our air bubble begins its journey at the bottom of a lake, precisely 45.0 meters deep. At this depth, the water exerts significant pressure on the bubble. The temperature down there is a chilly 5.0°C. The bubble itself has an initial volume of 10.0 cm³. To solve this problem, we need to consider how pressure and temperature change as the bubble ascends, and how these changes affect its volume.

Pressure at Depth

First, let's calculate the pressure at the bottom of the lake. The total pressure is the sum of atmospheric pressure and the pressure due to the water column above. Atmospheric pressure (${P_{atm}}$) is approximately 101325 Pascals (Pa). The pressure due to the water column (${P_{water}}$) can be calculated using the formula:

${P_{water} = \rho \cdot g \cdot h}$

Where:

• ${\rho}$ (rho) is the density of water (approximately 1000 kg/m³)
• ${g}$ is the acceleration due to gravity (approximately 9.81 m/s²)
• ${h}$ is the depth (45.0 m)

So,

${P_{water} = 1000 \cdot 9.81 \cdot 45.0 = 441450 \text{ Pa}}$

Therefore, the total pressure at the bottom of the lake (${P_1}$) is:

${P_1 = P_{atm} + P_{water} = 101325 + 441450 = 542775 \text{ Pa}}$

Initial Temperature

The initial temperature (${T_1}$) is 5.0°C. However, we need to convert this to Kelvin for our calculations:

${T_1 = 5.0 + 273.15 = 278.15 \text{ K}}$

Initial Volume

The initial volume (${V_1}$) is given as 10.0 cm³. It's good to keep it in these units for now, as we'll be comparing it to the final volume in cm³.

Understanding the Final Conditions

As the bubble rises, it reaches the surface where the temperature is a balmy 40.0°C. At the surface, the pressure acting on the bubble is simply the atmospheric pressure.

Pressure at the Surface

The final pressure (${P_2}$) is equal to the atmospheric pressure:

${P_2 = P_{atm} = 101325 \text{ Pa}}$

Final Temperature

The final temperature (${T_2}$) is 40.0°C. Converting this to Kelvin:

${T_2 = 40.0 + 273.15 = 313.15 \text{ K}}$

Applying the Ideal Gas Law

To find the final volume of the bubble, we can use the combined gas law, which is derived from the ideal gas law. The combined gas law states:

${\frac{P_1V_1}{T_1} = \frac{P_2V_2}{T_2}}$

Where:

• ${P_1}$ is the initial pressure
• ${V_1}$ is the initial volume
• ${T_1}$ is the initial temperature
• ${P_2}$ is the final pressure
• ${V_2}$ is the final volume (what we want to find)
• ${T_2}$ is the final temperature

We can rearrange this formula to solve for ${V_2}$:

${V_2 = \frac{P_1V_1T_2}{P_2T_1}}$

Now, let's plug in the values we have:

${V_2 = \frac{542775 \cdot 10.0 \cdot 313.15}{101325 \cdot 278.15}}$

${V_2 = \frac{1699674462.5}{28187486.25}}$

${V_2 \approx 60.3 \text{ cm}^3}$

Conclusion

So, the final volume of the air bubble as it reaches the surface of the lake is approximately 60.3 cm³. This significant increase from its initial volume of 10.0 cm³ is due to the decrease in pressure and the increase in temperature as the bubble rises. Understanding these principles helps us appreciate the physics at play in everyday phenomena!

In summary, the volume of the air bubble increases substantially as it rises to the surface, influenced by both the reduction in pressure and the rise in temperature. This is a classic example of how gases behave under changing conditions, governed by the principles of thermodynamics.

Additional Considerations

Real Gases vs. Ideal Gas Law

It's important to note that the ideal gas law provides a good approximation, but real gases may deviate from this behavior, especially at high pressures or low temperatures. However, for this scenario, the ideal gas law provides a reasonably accurate estimate.

Surface Tension

Surface tension also plays a minor role. As the bubble expands, the surface tension forces might slightly affect the final volume, but this effect is generally negligible compared to the pressure and temperature changes.

Heat Transfer

We assume that the air bubble quickly reaches thermal equilibrium with the surrounding water. In reality, there might be a slight delay in heat transfer, but this is usually insignificant for small bubbles.

Implications and Applications

Understanding how gases behave under different pressures and temperatures has numerous practical applications:

• Diving: Divers need to understand pressure changes to avoid decompression sickness.
• Meteorology: Predicting weather patterns requires knowledge of how air masses behave under varying conditions.
• Engineering: Designing pressure vessels and systems involves understanding gas behavior.

Final Thoughts

This air bubble problem serves as a great example of how physics concepts can be applied to understand everyday phenomena. By considering the initial and final conditions and applying the ideal gas law, we can accurately predict the bubble's volume change. Keep exploring, and you'll find physics all around you!

Remember, the key to solving these problems is to break them down into manageable steps and understand the underlying principles. Happy problem-solving, everyone!

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