Animated Solution for Physics - Properties of Solids and Liquids: Comprehension Passage

Imagine a tiny world trapped inside a bubble at the bottom of a deep, quiet lake. This bubble is filled with a monoatomic ideal gas, and as it begins its journey upward, it undergoes a fascinating transformation governed by the laws of fluid mechanics and thermodynamics.

Let's dive deep into the physics of this journey, analyzing the forces, the temperature changes, and the dynamic buoyancy force that acts on the bubble as it ascends.

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When the bubble is in motion, it is not just floating passively; it is interacting dynamically with the surrounding liquid.

First, there is the buoyancy force ($F_B$). This force is the net result of the hydrostatic pressure difference between the top and bottom of the bubble. Because pressure increases with depth, the bottom of the bubble experiences a slightly greater upward force than the downward force at the top. This pressure gradient is what pushes the bubble upward.

But what other forces are at play?

- Gravity ($F_g$) acts downwards on the mass of the gas inside the bubble. - Viscous Drag ($F_v$) acts downwards, opposing the bubble's upward velocity.

It is a common misconception to list the "pressure of the liquid" as a separate force. Remember, the pressure gradient is already fully accounted for in the buoyancy force! Therefore, the only other forces acting on the bubble are gravity and viscosity.

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As the bubble rises, the surrounding hydrostatic pressure decreases. This allows the bubble to expand. Because the problem states that the bubble does not exchange any heat with the liquid, this expansion is strictly adiabatic.

For an adiabatic process involving an ideal gas, the relationship between temperature $T$ and pressure $p$ is given by:

Since we are dealing with a monoatomic gas, the adiabatic index is $\gamma =5/3$. Let's calculate the exponent:

Thus, the temperature at any height $y$ (State 2) is related to the temperature at the bottom (State 1) by:

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To find the exact temperature, we must write down the pressures at both states:

1. At the bottom ($y=0$): The depth of the liquid is $H$. The total pressure is the sum of the atmospheric pressure $p_0$ and the hydrostatic pressure:

The initial temperature is $T_1=T_0$.

2. At a height $y$ from the bottom: The depth of the liquid above the bubble is now $H-y$. The pressure is:

Substituting these pressures into our adiabatic temperature equation yields:

This elegant result shows that as the bubble rises (as $y$ increases), the pressure $p_2$ decreases, causing the temperature $T_2$ of the gas inside to drop. The expanding gas does work against the surrounding liquid, and since no heat enters the system, this work comes at the expense of its internal energy, cooling the gas down!

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Now, let's calculate the buoyancy force $F$ acting on the bubble at height $y$. By Archimedes' principle:

To find the volume $V_2$ at height $y$, we apply the ideal gas law:

Substituting our adiabatic temperature expression $T_2=T_0{\left(\frac{p_2}{p_1}\right)}^{2/5}$ into this equation:

Finally, substituting this volume back into the buoyancy force equation:

By substituting the explicit expressions for $p_1$ and $p_2$, we get:

This beautiful formula shows that as the bubble rises, the buoyancy force actually increases because the decrease in pressure ($p_2$) in the denominator causes the volume of the bubble to grow, displacing more liquid and generating a stronger upward lift!