Problem 38 - An Introduction to Thermal Physics

Question

Two identical bubbles of gas form at the bottom of a lake, then rise to the surface. Because the pressure is much lower at the surface than at the bottom, both bubbles expand as they rise. However, bubble \(A\) rises very quickly, so that no heat is exchanged between it and the water. Meanwhile, bubble \(B\) rises slowly (impeded by a tangle of seaweed), so that it always remains in thermal equilibrium with the water (which has the same temperature everywhere). Which of the two bubbles is larger by the time they reach the surface? Explain your reasoning fully.

Step-by-Step Solution

Answer

Bubble B is larger at the surface due to isothermal expansion.

Step 1: Understanding the Initial Conditions

At the bottom of the lake, both bubbles A and B have the same initial conditions including pressure, volume, and temperature. Because they are identical, we assume they start with the same number of moles of gas.

Step 2: Analyze Bubble A's Rise

Bubble A rises rapidly with no heat exchange to its surroundings, indicating it undergoes an adiabatic process. In an adiabatic process, pressure-volume relationship is given by: \( PV^\gamma = \text{constant} \) where \( \gamma \) is the heat capacity ratio \( \frac{C_p}{C_v} \). As the bubble rises and external pressure decreases, volume increases but not as much as in an isothermal process due to the lack of heat exchange.

Step 3: Analyze Bubble B's Rise

Bubble B rises slowly, maintaining thermal equilibrium with the surrounding water. This indicates an isothermal process, meaning temperature remains constant throughout. The relationship here is described by Boyle's Law: \( PV = \text{constant} \). As external pressure decreases, volume increases significantly to maintain constant temperature.

Step 4: Comparisons Using Thermodynamics

Compare the final states of bubbles A and B. The adiabatic expansion of bubble A results in less volume increase than the isothermal expansion of bubble B because the temperature of bubble B remains constant, requiring pressure times volume to remain constant, and allowing greater volume increase with decreasing pressure.

Step 5: Conclusion Based on Analysis

By the time they reach the surface, Bubble B, which expands isothermally, has a larger volume compared to Bubble A, which expands adiabatically. This is because heat exchange in Bubble B allows for greater expansion with decreasing pressure.

Key Concepts

Adiabatic Process

An adiabatic process is a type of thermodynamic process where no heat is exchanged between the system and its surroundings. This means

• the system does not gain or lose energy through heat transfer.
• All energy changes are due to work done by or on the system.

For a gas undergoing an adiabatic process, the relationship between pressure and volume can be described by the equation:\[PV^\gamma = \text{constant}\]
where \(P\) is the pressure, \(V\) is the volume, and \(\gamma\) is the heat capacity ratio (\(\frac{C_p}{C_v}\)).
In the case of bubble A, as it rises quickly from the bottom of the lake, it undergoes an adiabatic expansion. Due to the decrease in external pressure, the volume of the bubble increases but not as much as it would in a process that allowed heat exchange. This results in a smaller expansion compared to bubble B, which undergoes an isothermal process.

Isothermal Process

An isothermal process maintains a constant temperature throughout the transition.

• In isothermal processes, any heat absorbed or released by the system is used to do work, keeping the internal temperature steady.
• The pressure-volume relationship is described by Boyle's Law: \(PV = \text{constant}\).

For bubble B, which rises slowly and maintains thermal equilibrium with the surrounding water, the slow rise ensures that the temperature remains constant.
This means that, as the pressure decreases with ascent, the volume of the bubble can increase significantly more than in an adiabatic process, allowing for greater expansion.

Boyle's Law

Boyle's Law is a fundamental principle in thermodynamics that describes the inverse relationship between the pressure and volume of a gas under constant temperature. The law can be stated as:

• When temperature is constant, pressure multiplied by volume remains constant.

In mathematical terms:\[ PV = \text{constant} \]
This law is particularly relevant to the isothermal process that bubble B experiences. As bubble B ascends, the external pressure decreases while maintaining a constant temperature due to thermal equilibrium with the water. To satisfy Boyle's Law, the volume of bubble B increases, allowing it to expand more than bubble A, ultimately making it larger when it reaches the surface.

Heat Capacity Ratio

The heat capacity ratio, also known as the adiabatic index or \(\gamma\), is a crucial factor in determining the behavior of gases in thermodynamic processes.

• It is defined as the ratio of the heat capacity at constant pressure \((C_p)\) to the heat capacity at constant volume \((C_v)\).
• It plays a vital role in characterizing adiabatic processes.

In the context of bubble A, the heat capacity ratio \(\gamma\) helps describe how the pressure and volume relate to each other during its adiabatic expansion. A higher heat capacity ratio means that for a given change in pressure, there will be a smaller change in volume, as work done on or by the bubble occurs without heat exchange. The value of \(\gamma\) thus directly impacts how much the volume can expand under an adiabatic process.