Problem 30 - College Physics

Question

\- A scuba diver takes a tank of air on a deep dive. The tank's volume is 10 Land it is completely filled with air at an absolute pressure of \(232 \mathrm{~atm}\) at the start of the dive. The air temperature at the surface is \(94^{\circ} \mathrm{F}\) and the diver ends up in deep water at \(60^{\circ} \mathrm{F}\). Assuming thermal equilibrium and neglecting air loss, determine the absolute internal pressure of the air when it is cold.

Step-by-Step Solution

Answer

The final absolute internal pressure is approximately 217.64 atm.

Step 1: Convert Temperatures to Kelvin

First, we need to convert the temperature from Fahrenheit to Kelvin. The formula for converting Fahrenheit to Kelvin is: \[ K = \left( F - 32 \right) \times \frac{5}{9} + 273.15 \]For the surface temperature: \[ K_{surface} = \left( 94 - 32 \right) \times \frac{5}{9} + 273.15 = 307.59 \mathrm{~K} \]For the deep-water temperature:\[ K_{deep} = \left( 60 - 32 \right) \times \frac{5}{9} + 273.15 = 288.71 \mathrm{~K} \]

Step 2: Use Gay-Lussac's Law

To find the final pressure, we use Gay-Lussac’s Law, which relates the pressure and temperature of a gas at constant volume. The law is expressed as:\[ \frac{P_1}{T_1} = \frac{P_2}{T_2} \]where \( P_1 = 232 \mathrm{~atm} \) is the initial pressure, \( T_1 = 307.59 \mathrm{~K} \) is the initial temperature, \( P_2 \) is the final pressure, and \( T_2 = 288.71 \mathrm{~K} \) is the final temperature.

Step 3: Solve for Final Pressure

Rearrange the equation to solve for the final pressure \( P_2 \):\[ P_2 = P_1 \times \frac{T_2}{T_1} \]Plug in the values:\[ P_2 = 232 \mathrm{~atm} \times \frac{288.71 \mathrm{~K}}{307.59 \mathrm{~K}} \approx 217.64 \mathrm{~atm} \]

Step 4: Interpret the Result

The final pressure inside the tank, when the air is cold after reaching thermal equilibrium in the deep water, is approximately \(217.64 \mathrm{~atm}\). This pressure decrease reflects the lower temperature at depth.

Key Concepts

Thermodynamics

Thermodynamics is the study of heat transfer and how it relates to energy and work. It involves the principles governing the conversion of energy from one form to another. In the context of scuba diving, thermodynamics helps us understand how temperature changes can affect the gas in a scuba tank.

The behavior of gases under different conditions is a fundamental aspect of thermodynamics. When a gas is cooled, its molecules move less energetically and occupy less pressure in a fixed volume. This is why a diver's air tank, when subjected to lower temperatures at greater depths, experiences a decrease in pressure.

Key aspects of thermodynamics in this scenario include:

• The idea that no energy is lost, just transformed.
• How temperature and pressure are related due to the energy contained within the gas molecules.
• Understanding the state of equilibrium, where the diver’s tank reaches a point where the temperatures stabilize, resulting in a new pressure level.

Pressure-Temperature Relationship

The pressure-temperature relationship for gases is described by Gay-Lussac's Law. This law outlines that the pressure of a given mass of gas is directly proportional to its temperature (in Kelvin) when the volume remains constant.

In mathematical terms, Gay-Lussac’s Law is represented as \( \frac{P_1}{T_1} = \frac{P_2}{T_2} \), where:

• \( P_1 \) and \( T_1 \) are the initial pressure and temperature of the gas.
• \( P_2 \) and \( T_2 \) are the final pressure and temperature.

In practical terms, this means that as a scuba diver goes deeper and the temperature decreases, the pressure inside the tank decreases if no gas is lost. This explains the empirical relationship where a drop in temperature results in reduced internal pressure in the tank.

Understanding this relationship helps in anticipating how the equipment will respond in different thermal conditions underwater, ensuring safe and efficient diving.

Scuba Diving Physics

Scuba diving physics explores how physical laws affect divers and their equipment. The principles of pressure, temperature, and gas laws are vital for a safe diving experience.

As divers descend, the temperature typically drops, especially in deeper waters where sunlight cannot penetrate. This cooler environment influences both the physiology of the diver and the physics of the equipment, such as the air tank.

The basics of diving physics include understanding how:

• Pressure increases with depth, but the internal tank pressure decreases with lower temperatures.
• Managing buoyancy is crucial, as changes in air pressure affect how buoyant equipment behaves.
• Thermal protection is necessary to maintain the diver’s body temperature and ensure comfort underwater.
• Planning for air supply is critical; understanding how much air is available and how it expands/contracts with temperature changes can dictate how long a dive can safely last.

By applying these physics concepts, divers can manage their dives more effectively, reducing risks and enhancing their overall underwater experience.