Gas Pressure Calculation In A Rigid Container Applying Physics Principles

In the realm of thermodynamics, understanding the behavior of gases is paramount. Gases, unlike solids and liquids, are highly compressible and their properties are significantly influenced by factors such as temperature, pressure, and volume. One fundamental principle governing gas behavior is the relationship between pressure and temperature, particularly when the volume is held constant. This principle is encapsulated in Gay-Lussac's Law, which forms the bedrock of our exploration.

This article delves into a specific scenario involving a gas confined within a rigid, non-deformable 10-liter container. Initially, the gas is at a temperature of 30°C and exerts a pressure of 4 atm. Our primary objective is to determine the pressure exerted by the gas when its temperature is increased to 60°C. This exploration will not only reinforce our understanding of Gay-Lussac's Law but also provide practical insights into the behavior of gases in real-world applications.

To embark on this journey, we will first lay the theoretical groundwork by elucidating Gay-Lussac's Law and its underlying principles. Subsequently, we will meticulously outline the step-by-step solution to the problem at hand, ensuring clarity and comprehension. Finally, we will extend our discussion to explore the broader implications of this principle and its relevance in various scientific and engineering contexts.

At the heart of our investigation lies Gay-Lussac's Law, a cornerstone of gas laws. This law articulates a direct proportionality between the pressure and absolute temperature of a gas when the volume and number of moles are kept constant. In simpler terms, as the temperature of a gas increases, its pressure increases proportionally, and vice versa, provided the volume remains unchanged. This relationship can be mathematically expressed as:

P₁/T₁ = P₂/T₂

Where:

• P₁ represents the initial pressure of the gas.
• T₁ denotes the initial absolute temperature of the gas.
• P₂ signifies the final pressure of the gas.
• T₂ corresponds to the final absolute temperature of the gas.

It is imperative to note that the temperature in Gay-Lussac's Law must be expressed in an absolute scale, such as Kelvin (K). The conversion from Celsius (°C) to Kelvin (K) is achieved by adding 273.15 to the Celsius temperature:

K = °C + 273.15

The underlying principle behind Gay-Lussac's Law stems from the kinetic molecular theory of gases. This theory posits that gas particles are in constant, random motion, and their kinetic energy is directly proportional to the absolute temperature. As the temperature rises, the gas particles move more vigorously, colliding more frequently and forcefully with the container walls. This heightened collision rate translates into an increased pressure exerted by the gas.

Gay-Lussac's Law finds widespread applications in various domains. For instance, it plays a crucial role in understanding the operation of pressure cookers, where increasing the temperature inside the cooker elevates the pressure, thereby accelerating the cooking process. Similarly, it is instrumental in comprehending the behavior of tires on vehicles, where temperature fluctuations can significantly impact tire pressure. Furthermore, this law is indispensable in numerous industrial processes involving gases, such as chemical reactions and gas storage.

Now, let us apply Gay-Lussac's Law to solve the problem at hand. We have a gas confined within a rigid 10-liter container, with the following initial conditions:

• Initial temperature (T₁) = 30°C
• Initial pressure (P₁) = 4 atm

The gas is then heated to a final temperature of 60°C, and our objective is to determine the final pressure (P₂).

Step 1: Convert Celsius Temperatures to Kelvin

Before we can employ Gay-Lussac's Law, we must convert the temperatures from Celsius to Kelvin:

• T₁ = 30°C + 273.15 = 303.15 K
• T₂ = 60°C + 273.15 = 333.15 K

Step 2: Apply Gay-Lussac's Law

Now, we can substitute the known values into Gay-Lussac's Law equation:

P₁/T₁ = P₂/T₂

4 atm / 303.15 K = P₂ / 333.15 K

Step 3: Solve for the Final Pressure (P₂)

To isolate P₂, we can multiply both sides of the equation by 333.15 K:

P₂ = (4 atm / 303.15 K) * 333.15 K

P₂ ≈ 4.4 atm

Therefore, the final pressure of the gas at 60°C is approximately 4.4 atm.

The solution to this problem underscores the direct relationship between pressure and temperature as described by Gay-Lussac's Law. As the temperature of the gas increased, its pressure also increased proportionally, while the volume remained constant due to the rigid container.

This principle has far-reaching implications in various practical scenarios. Consider the inflation of tires on a vehicle. As the tires roll along the road, friction between the tire and the road surface generates heat, causing the air inside the tires to warm up. This temperature increase leads to a corresponding increase in tire pressure. It is crucial to be mindful of this phenomenon, as excessive tire pressure can compromise vehicle handling and even lead to tire blowouts.

In the realm of cooking, pressure cookers leverage Gay-Lussac's Law to expedite the cooking process. By sealing the cooker, the volume remains constant. As the cooker is heated, the temperature inside rises, leading to a significant increase in pressure. This elevated pressure allows water to boil at a temperature higher than 100°C, thereby reducing cooking times.

Furthermore, Gay-Lussac's Law finds applications in diverse industrial processes. In chemical reactions involving gases, controlling the temperature is essential to maintain the desired pressure and reaction rate. Similarly, in gas storage, understanding the pressure-temperature relationship is crucial for safe and efficient storage practices.

In conclusion, we have explored the relationship between pressure and temperature of a gas confined within a rigid container, guided by Gay-Lussac's Law. We successfully determined the final pressure of the gas when its temperature was increased, reinforcing the direct proportionality between these two variables.

This exploration highlights the significance of understanding gas laws in various scientific, engineering, and everyday applications. From tire inflation to pressure cooking and industrial processes, the principles governing gas behavior play a pivotal role. By grasping these principles, we can gain a deeper appreciation for the world around us and make informed decisions in various practical scenarios. The study of gases and their properties continues to be an active area of research, with ongoing efforts to develop new technologies and applications that harness the unique characteristics of these fascinating substances.