Problem 14 - University Physics with Modern Physics

Question

A diver observes a bubble of air rising from the bottom of a lake (where the absolute pressure is 3.50 atm \()\) to the surface (where the pressure is 1.00 atm). The temperature at the bottom is \(4.0^{\circ} \mathrm{C},\) and the temperature at the surface is \(23.0^{\circ} \mathrm{C}\) . (a) What is the ratio of the volume of the bubble as it reaches the surface to its volume at the bottom? (b) Would it be safe for the diver to hold his breath while ascending from the bottom of the lake to the surface? Why or why not?

Step-by-Step Solution

Answer

(a) The volume ratio is approximately 3.74. (b) It would not be safe for the diver to hold their breath while ascending due to the risk of lung expansion.

Step 1: Understand the problem and given data

We need to find the ratio of volumes as a bubble rises from the bottom to the surface. We are given: \( P_1 = 3.50 \) atm (pressure at bottom), \( P_2 = 1.00 \) atm (pressure at surface), \( T_1 = 4.0 \degree C = 277 \) K (temperature at bottom), and \( T_2 = 23.0 \degree C = 296 \) K (temperature at surface). We will use the Ideal Gas Law to solve this problem.

Step 2: Set up the Ideal Gas Law equation

The Ideal Gas Law in relation to pressure, volume, and temperature is \( PV = nRT \). For a constant amount of gas (the bubble), if we are looking at the change, we can express it in a ratio form as \( \frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2} \). We are interested in the ratio \( \frac{V_2}{V_1} \).

Step 3: Solve for the volume ratio

Using \( \frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2} \), rearrange to find the volume ratio: \( \frac{V_2}{V_1} = \frac{P_1 T_2}{P_2 T_1} \). Substituting the known values gives: \( \frac{V_2}{V_1} = \frac{3.50 \times 296}{1.00 \times 277} \). Calculating this gives \( \frac{V_2}{V_1} = \frac{1036}{277} \approx 3.74 \).

Step 4: Conclusion on holding breath

As the bubble volume increases by approximately 3.74 times from the bottom to the surface, the diver's lungs would expand by a similar factor if the air is trapped. This significant expansion could potentially burst a lung if the diver holds his breath while ascending, making it unsafe.

Key Concepts

Gas Laws

The Ideal Gas Law is a fundamental principle in understanding the behavior of gases. It combines several simpler gas laws into one equation: \( PV = nRT \). This equation relates pressure \( P \), volume \( V \), and temperature \( T \) of a gas, with \( n \) being the number of moles and \( R \) the universal gas constant.
The Ideal Gas Law helps predict how a gas will behave under varying conditions of pressure and temperature, which is crucial for scenarios like the ascent of a bubble in water where pressure and temperature change significantly.

• Boyle's Law: At constant temperature, increases in pressure will decrease volume and vice versa - \( PV = \text{constant} \).
• Charles's Law: At constant pressure, gas volume increases with temperature - \( \frac{V}{T} = \text{constant} \).
• Combined into the Ideal Gas Law, these laws provide a comprehensive framework to analyze gas behavior in dynamic conditions.

The problem uses this law to examine the volume change of an air bubble rising from a lake's bottom, indicating the law's applicability in real-world situations.

Volume and Pressure Relationships

The relationship between volume and pressure is a key theme in the analysis of gas behavior and is exemplified by Boyle's Law. Boyle's Law states that if the temperature of a gas is held constant, the volume is inversely proportional to its pressure
. This means that as pressure increases, volume decreases, and vice versa:
\[ PV = \text{constant} \]
In the exercise, as the bubble rises, the surrounding water pressure decreases from 3.50 atm at the bottom to 1.00 atm at the surface. Consequently, the volume of the bubble increases. This is calculated using the rearranged Ideal Gas Law equation:
\[ \frac{V_2}{V_1} = \frac{P_1 T_2}{P_2 T_1} \]
The decrease in pressure as the bubble moves upwards leads to a significant increase in its volume, expanding approximately 3.74 times its original size. This demonstrates the inverse relationship accurately in a practical context, highlighting the importance of carefully managing pressure changes in situations involving gases.

Thermodynamics

Thermodynamics encompasses the principles that govern energy and heat transformations, providing a broader context for understanding technologies and natural phenomena.
In the context of gases, thermodynamic principles aid in predicting how variables like temperature affect gas volumes and pressures. The concept of thermodynamics tells us:

• Internal energy: It involves the energy contained within the gas due to molecular motion.
• Heat transfer: Changes in volume and pressure through heat exchange lead to energy transformations.
• The First Law of Thermodynamics: Energy within a closed system is conserved, relating work done and heat absorbed by the system.

In the given exercise, the difference in temperatures from 4.0 °C at the bottom to 23.0 °C at the surface exemplifies how gases expand with heat. As per Charles's Law, the volume increases linearly with an increase in temperature if the pressure is stable. By incorporating temperature changes into the analysis, we apply thermodynamic concepts to assess the behavior of gases, guiding practical activities such as safe diving practices.