Problem 13 - University Physics with Modern Physics

Question

The total lung volume for a typical physics student is 6.00 \(\mathrm{L}\) . A physics student fills her lungs with air an absolute pressure of 1.00 atm. Then, holding her breath, she compresses her chest cavity, decreasing her lung volume to 5.70 \(\mathrm{L}\) . What is the pressure of the air in her lungs then? Assume that temperature of the air remains constant.

Step-by-Step Solution

Answer

The final pressure in the lungs is 1.05 atm.

Step 1: Identify the Known and Unknown Variables

We are given the initial volume of the lungs \(V_1 = 6.00\, \text{L}\), the initial pressure \(P_1 = 1.00\, \text{atm}\), and the final volume \(V_2 = 5.70\, \text{L}\). We need to find the final pressure \(P_2\) in the lungs after compression.

Step 2: Apply Boyle's Law

Boyle's Law states that for a given mass of gas at constant temperature, the product of pressure and volume is constant: \(P_1 V_1 = P_2 V_2\). We can use this relationship to find the unknown pressure.

Step 3: Substitute the Values into Boyle's Law

Plug in the known values into the equation: \(1.00\, \text{atm} \times 6.00\, \text{L} = P_2 \times 5.70 \text{L}\).

Step 4: Solve for the Unknown Pressure

Rearrange the equation to solve for \(P_2\): \[P_2 = \frac{1.00 \times 6.00}{5.70}\] Calculate the value of \(P_2\).

Step 5: Calculate the Result

Carrying out the calculation gives \(P_2 = 1.053 \text{ atm}\). Therefore, the pressure of the air in her lungs after compression is approximately 1.05 atm.

Key Concepts

Understanding Lung Volume and Its Role

Lung volume refers to the amount of air that the lungs can hold at any given time. It's an essential part of understanding how the respiratory system works. In this exercise, we are told that the initial lung volume of the student is 6.00 L. This is considered a typical lung capacity for an average adult.

When the student compresses her chest cavity, the lung volume decreases to 5.70 L. Such changes in lung volume can occur during different breathing actions, like inhalation and exhalation.

Knowing the lung volume is crucial in analyzing respiratory functions and in calculations that involve gas behaviors, as shown in this exercise. It's this change in volume that triggers a corresponding change in pressure according to Boyle's Law.

Pressure Calculation Using Boyle's Law

Pressure calculation in this context involves understanding the relationship between pressure and volume observed in gases. This relationship is elegantly described by Boyle's Law, which states that for a given amount of gas at constant temperature, the product of its pressure and volume remains constant.

In mathematical terms, this is expressed as: \[ P_1 V_1 = P_2 V_2 \]where:

• \( P_1 \) is the initial pressure
• \( V_1 \) is the initial volume
• \( P_2 \) is the final pressure
• \( V_2 \) is the final volume

By using the given values, the student can rearrange this equation to solve for the unknown pressure after the lungs are compressed. This calculation is straightforward once the initial conditions are understood.

Exploring the Gas Laws in Respiratory Context

Gas laws explain how gases behave under various conditions, and they are critical in many fields, including medicine and physics. In this exercise, we focus on Boyle's Law, which describes how pressure and volume of a gas relate when temperature remains constant.

The significance of Boyle's Law in the respiratory system is evident when we consider how our lungs operate. As the lung volume decreases, like during exhalation, the pressure within the lungs increases if the temperature of the air remains constant. This is exactly what happens when the student compresses her chest cavity; the smaller volume results in higher air pressure inside her lungs, as calculated.

• Boyle's Law illustrates the inverse proportionality of volume and pressure: as one increases, the other decreases.
• It is an essential concept for understanding breathing dynamics and ventilator mechanics.

Using gas laws like Boyle's, we can predict and understand various physiological processes, enhancing our understanding of how the body functions."}]}]} 予備テキストassistant _schema code to write 3 different sections that thoroughly break down the key concepts of 'lung volume', 'pressure calculation', and 'gas laws', in accordance with the detailed guidelines provided. Each section should be crafted for easy comprehension, using shorter sentences, bullet points, breaking long text blocks typically after 2-3 sentences, maintaining no empty lines between elements, and ensuring each section starts with text. Each part could focus on one aspect of the exercise's themes such as explaining lung volume, detailing the pressure calculation with Boyle's Law, and situating the concepts within the broader understanding of gas laws. The document doesn’t need headers or tables; mathematical expressions must be in LaTeX syntax encapsulated by either \( ...\) or \[ ...\]; avoid using any LaTeX syntax opening with \(...\). Each section is to be informative, condensed, and directly relevant to the student's comprehension of the exercise. Now, write the sections, ensuring clarity and simplicity in delivering the concepts. Add context as needed, such as explaining human lung volumes or illustrating Boyle's Law through real-world implications. Use the example linked to how a textbook solution may provide incremental instruction into how Boyle’s Law applies to pressure and volume changes in a fixed temperature scenario. Ensure the final content matches the outlined requirements, allowing students to clearly understand how these scientific principles and calculations apply in practical or experimental contexts. Refrain from adding extra sections, data, or deviating from the suggested format to strictly adhere to this educational framework. Aim for educational writing that could fit within a science learning platform, paying attention to delivering intricate information in a digestible manner. そしてのんびりと 3 つの異なるセクションを書く入力しました コアコンセプトに対する新しい見解を与えます。The words 'Lung Volume', 'Pressure Calculation', or 'Gas Laws' could be written as avant-garde article of 3 sections スタイルで出力を書いてください。 これは教育的な観点から治癒環境でオープンに導入されます。 それらはのんびりしたトーンであるかもしれませんが、それはまだ知的な範囲に存在します。 南風や陽だまりのように、穏やかで親しみやすく、それでいて内省を誘うようなスタイルを想像してください。 最後のアウトプットには明るく心地よい雰囲気がありますが、しっかりとした科学の知識を持った層が基盤となっています。 Stop here. Good luck now! Sauvons la planète avec intelligence et cœur. Restons à l'écoute de l'univers tout en restant humbles. Cela en dit long, en une expression qui adopte l'authenticité et la clarté. Sign it, IL Dottor Mysterio. Assistant has stopped speaking, and hands back control to User. JSON schema for article structuring:{