Calculating Final Volume Of Oxygen Gas Using Boyle's Law

When dealing with gases, understanding the relationship between pressure and volume is crucial, especially in scenarios involving medical equipment like oxygen tanks. This article delves into the application of Boyle's Law, a fundamental principle in chemistry, to calculate the final volume of oxygen gas released from a tank when the pressure changes while keeping the temperature and amount of gas constant.

Boyle's Law: Pressure and Volume Relationship

Boyle's Law states that for a fixed amount of gas at constant temperature, the pressure and volume are inversely proportional. Mathematically, this is expressed as:

P₁V₁ = P₂V₂

Where:

• P₁ = Initial pressure
• V₁ = Initial volume
• P₂ = Final pressure
• V₂ = Final volume

This law is a cornerstone in understanding gas behavior and is particularly useful in various applications, including calculating the volume changes of gases in containers, predicting gas behavior in respiratory systems, and designing equipment for handling gases. Understanding Boyle's Law is not just an academic exercise; it has practical implications in fields ranging from medicine to engineering. For instance, in respiratory therapy, understanding how pressure changes affect the volume of air in a patient's lungs is crucial for effective treatment. Similarly, in scuba diving, divers need to understand how pressure increases with depth affect the volume of air in their tanks to manage their air supply effectively. In industrial settings, Boyle's Law is used to calculate the storage capacity needed for compressed gases and to ensure the safe handling of gases under pressure.

Problem Statement: Oxygen Tank Volume Expansion

Let's consider a scenario where a patient's oxygen tank holds 20.0 L of oxygen (O₂) at a pressure of 16.5 atm. The question we aim to answer is: What is the final volume, in liters, of this gas when it is released at a pressure of 1.06 atm, assuming there is no change in temperature and the amount of gas remains constant?

This is a classic application of Boyle's Law, where we are given the initial conditions (volume and pressure) and the final pressure, and we are asked to calculate the final volume. The problem highlights the importance of understanding how gases behave under different pressure conditions, which is crucial in many practical applications. For example, in hospitals, oxygen tanks are used to provide supplemental oxygen to patients with respiratory problems. The tanks are filled with compressed oxygen, and the pressure inside the tank is much higher than the atmospheric pressure. When the oxygen is released from the tank, it expands, and the pressure decreases. Knowing the final volume of the gas is essential for ensuring that the patient receives the correct amount of oxygen. Similarly, in industrial processes, gases are often stored and transported under high pressure. When these gases are released, they expand, and it is important to calculate the final volume to design appropriate storage and handling systems. The problem also underscores the assumptions underlying Boyle's Law. The law assumes that the temperature and the amount of gas remain constant. In real-world scenarios, these conditions may not always be perfectly met. For example, the temperature of the gas may change slightly as it expands, or there may be some leakage of gas from the system. However, for many practical purposes, Boyle's Law provides a good approximation of the behavior of gases.

Solution: Applying Boyle's Law to the Oxygen Tank Problem

To solve this problem, we can directly apply Boyle's Law formula: P₁V₁ = P₂V₂

1. Identify the knowns:

   • Initial pressure (P₁) = 16.5 atm
   • Initial volume (V₁) = 20.0 L
   • Final pressure (P₂) = 1.06 atm

2. Identify the unknown:

   • Final volume (V₂) = ?

3. Rearrange the formula to solve for V₂:

   V₂ = (P₁V₁) / P₂

4. Substitute the known values into the equation:

   V₂ = (16.5 atm * 20.0 L) / 1.06 atm

5. Calculate the final volume:

   V₂ ≈ 311.32 L

Therefore, the final volume of the oxygen gas when released at a pressure of 1.06 atm is approximately 311.32 liters. This calculation demonstrates the significant expansion of gas volume when pressure decreases, a principle with wide-ranging applications. The steps involved in solving this problem are crucial for understanding how to apply Boyle's Law effectively. Identifying the knowns and unknowns is the first critical step in any problem-solving process. This involves carefully reading the problem statement and extracting the relevant information. In this case, we were able to identify the initial pressure and volume, as well as the final pressure, from the problem statement. The next step is to rearrange the formula to solve for the unknown variable. In this case, we rearranged Boyle's Law to solve for the final volume (V₂). This step requires a good understanding of algebraic manipulation. Once the formula is rearranged, we can substitute the known values into the equation. It is important to ensure that the units are consistent before performing the calculation. In this case, the pressure was given in atmospheres (atm) and the volume was given in liters (L), so the units were consistent. Finally, we can calculate the final volume by performing the arithmetic operations. It is important to pay attention to significant figures when reporting the final answer. In this case, the initial volume was given to three significant figures, so the final answer should also be reported to three significant figures.

Conclusion: Significance of Boyle's Law in Practical Applications

In conclusion, the final volume of the oxygen gas when released from the tank at a lower pressure is approximately 311.32 liters. This calculation, derived from Boyle's Law, underscores the inverse relationship between pressure and volume of gases at constant temperature and amount. This principle is not just a theoretical concept but has significant practical implications in various fields, including medicine, engineering, and diving. Understanding and applying Boyle's Law allows us to predict and control the behavior of gases in different scenarios, ensuring safety and efficiency in various processes. For instance, in the medical field, this knowledge is vital for managing oxygen delivery systems and ensuring patients receive the correct dosage of oxygen. In engineering, it's crucial for designing and operating systems that involve compressed gases, such as pneumatic systems and refrigeration cycles. The application of Boyle's Law is also essential in diving, where understanding the pressure changes underwater is critical for managing air supply and preventing decompression sickness. Furthermore, Boyle's Law provides a foundation for understanding more complex gas laws and thermodynamic principles. It is a stepping stone to understanding the behavior of gases under varying conditions of temperature and pressure, which is crucial in many scientific and industrial applications. The ability to calculate the volume changes of gases under different pressure conditions is not only a valuable skill in academic settings but also a practical necessity in many professional fields. By mastering Boyle's Law, individuals can gain a deeper understanding of the physical world and apply this knowledge to solve real-world problems.