Boyle's Law Example Calculating Pressure Change With Volume Expansion

Let's delve into the fascinating world of gas behavior with a classic problem that perfectly illustrates Boyle's Law. We're presented with a scenario involving a 4.60 L container equipped with a moveable piston, initially holding a gas at a pressure of 845 mm Hg. The core question we aim to answer is What happens to the pressure when the container's volume is expanded to 10.6 L? To solve this, we'll invoke the fundamental principle of Boyle's Law, which governs the inverse relationship between pressure and volume for a fixed amount of gas at constant temperature.

To truly grasp the significance of Boyle's Law, we need to understand its underpinnings. Imagine a gas confined within a container. The gas molecules are in constant, random motion, colliding with each other and the walls of the container. These collisions exert a force on the container walls, which we perceive as pressure. Now, if we increase the volume of the container while keeping the temperature constant, the gas molecules have more space to move around. This means they will collide with the walls less frequently, resulting in a decrease in pressure. Conversely, if we decrease the volume, the molecules will collide more frequently, leading to an increase in pressure. Boyle's Law mathematically expresses this inverse relationship.

The formula that embodies Boyle's Law is elegantly simple: P₁V₁ = P₂V₂. Here, P₁ and V₁ represent the initial pressure and volume, respectively, while P₂ and V₂ denote the final pressure and volume. The beauty of this equation lies in its ability to predict pressure changes in gases under isothermal conditions, where the temperature remains constant. In our problem, we are explicitly told that the number of moles and the temperature remain constant, making Boyle's Law the perfect tool for the job. Let's now carefully extract the given information from the problem statement. The initial volume, V₁, is 4.60 L, and the initial pressure, P₁, is 845 mm Hg. The final volume, V₂, is 10.6 L, and we are tasked with finding the final pressure, P₂. Now that we have all the pieces, we can plug them into Boyle's Law equation and solve for the unknown.

Applying Boyle's Law to our specific scenario, we substitute the known values into the equation P₁V₁ = P₂V₂. This gives us (845 mm Hg)(4.60 L) = P₂(10.6 L). Our next step is to isolate P₂ on one side of the equation. To do this, we divide both sides of the equation by 10.6 L. This yields P₂ = (845 mm Hg)(4.60 L) / (10.6 L). Now, a simple calculation will give us the value of P₂. Performing the calculation, we find that P₂ ≈ 366.5 mm Hg. This result tells us that when the container is expanded from 4.60 L to 10.6 L, the pressure decreases from 845 mm Hg to approximately 366.5 mm Hg, which is a direct consequence of Boyle's Law. In the realm of gas behavior, Boyle's Law stands as a cornerstone, providing a clear and concise relationship between pressure and volume. Its applications extend far beyond simple textbook problems, playing a vital role in various fields, including engineering, meteorology, and even medicine. Understanding Boyle's Law is essential for anyone seeking to unravel the mysteries of gases and their behavior.

In the realm of chemistry and physics, Boyle's Law emerges as a fundamental principle governing the behavior of gases. Specifically, it elucidates the inverse relationship between the pressure and volume of a gas, given a constant temperature and number of moles. This means that as the volume of a gas increases, its pressure decreases proportionally, and vice versa. Let's embark on a journey to master Boyle's Law through a systematic, step-by-step approach. This will empower you to confidently tackle a wide array of gas pressure problems.

The cornerstone of Boyle's Law is its mathematical representation: P₁V₁ = P₂V₂. This elegantly simple equation encapsulates the essence of the pressure-volume relationship. Here, P₁ represents the initial pressure, V₁ is the initial volume, P₂ denotes the final pressure, and V₂ signifies the final volume. The subscripts '1' and '2' serve to distinguish between the initial and final states of the gas. Before we dive into problem-solving, it's crucial to understand the units involved. Pressure can be expressed in various units, such as atmospheres (atm), Pascals (Pa), millimeters of mercury (mm Hg), or torr. Volume, on the other hand, is typically measured in liters (L) or milliliters (mL). The key is to ensure that the units are consistent on both sides of the equation. For example, if P₁ is in mm Hg, then P₂ must also be in mm Hg. Similarly, if V₁ is in liters, then V₂ must be in liters. Maintaining consistency in units is paramount to obtaining accurate results.

Now, let's delve into the practical application of Boyle's Law with a step-by-step approach to solving gas pressure problems. The first crucial step is to carefully read the problem statement and identify the knowns and the unknown. What information is provided? What quantity are you being asked to calculate? Typically, you will be given three of the four variables (P₁, V₁, P₂, V₂) and asked to find the fourth. Once you have identified the knowns and the unknown, the next step is to write down Boyle's Law equation: P₁V₁ = P₂V₂. This serves as the foundation for your solution. With the equation in hand, the next step is to substitute the known values into the equation. Replace the symbols (P₁, V₁, P₂, V₂) with their corresponding numerical values, ensuring that you include the units. Remember to maintain consistency in units throughout the calculation. Once you have substituted the values, the next step is to solve the equation for the unknown variable. This typically involves algebraic manipulation to isolate the unknown on one side of the equation. Depending on the problem, you may need to divide, multiply, or take reciprocals to achieve this. Finally, after solving the equation, the last step is to check your answer. Does the answer make sense in the context of the problem? If the volume increased, did the pressure decrease, as predicted by Boyle's Law? Are the units of your answer consistent with what was asked? Checking your answer helps to ensure accuracy and catch any potential errors.

To solidify your understanding, let's consider a few illustrative examples. Imagine a scenario where a gas occupies a volume of 5.0 L at a pressure of 2.0 atm. If the volume is compressed to 2.5 L while keeping the temperature constant, what is the new pressure? Following our step-by-step approach, we first identify the knowns: V₁ = 5.0 L, P₁ = 2.0 atm, V₂ = 2.5 L. The unknown is P₂. We then write down Boyle's Law equation: P₁V₁ = P₂V₂. Substituting the values, we get (2.0 atm)(5.0 L) = P₂(2.5 L). Solving for P₂, we divide both sides by 2.5 L, yielding P₂ = (2.0 atm)(5.0 L) / (2.5 L) = 4.0 atm. Finally, we check our answer. The volume decreased, so the pressure should have increased, which it did. The units are also consistent. As another example, suppose a gas has a pressure of 760 mm Hg and a volume of 10.0 L. If the pressure is increased to 1520 mm Hg while keeping the temperature constant, what is the new volume? Following the same procedure, we identify the knowns: P₁ = 760 mm Hg, V₁ = 10.0 L, P₂ = 1520 mm Hg. The unknown is V₂. We write down Boyle's Law equation: P₁V₁ = P₂V₂. Substituting the values, we get (760 mm Hg)(10.0 L) = (1520 mm Hg)V₂. Solving for V₂, we divide both sides by 1520 mm Hg, yielding V₂ = (760 mm Hg)(10.0 L) / (1520 mm Hg) = 5.0 L. Checking our answer, the pressure increased, so the volume should have decreased, which it did. The units are consistent. By consistently applying this step-by-step approach, you can master Boyle's Law and confidently solve a wide range of gas pressure problems.

Boyle's Law, a fundamental principle in the realm of gas behavior, extends its influence far beyond the confines of textbooks and laboratory experiments. This seemingly simple equation, P₁V₁ = P₂V₂, which describes the inverse relationship between pressure and volume at constant temperature, governs a plethora of phenomena we encounter in our daily lives and in various technological applications. Let's embark on a journey to explore the real-world applications of Boyle's Law, from the very act of breathing to the complexities of scuba diving and beyond.

One of the most fundamental and readily observable applications of Boyle's Law is in the mechanics of human breathing. The lungs, acting as a variable-volume container, expand and contract to facilitate the exchange of gases. When we inhale, the diaphragm muscle contracts and moves downward, increasing the volume of the chest cavity. This increase in volume leads to a decrease in pressure within the lungs, as dictated by Boyle's Law. The pressure inside the lungs becomes lower than the atmospheric pressure, causing air to rush into the lungs until the pressures equalize. Conversely, when we exhale, the diaphragm relaxes and moves upward, decreasing the volume of the chest cavity. This decrease in volume results in an increase in pressure within the lungs, making it higher than the atmospheric pressure. Consequently, air is forced out of the lungs until the pressures equalize. Thus, the rhythmic process of breathing is a direct manifestation of Boyle's Law in action.

Scuba diving, an activity that allows humans to explore the underwater world, provides another compelling example of Boyle's Law in action. As a diver descends deeper into the ocean, the pressure surrounding them increases significantly. For every 10 meters (approximately 33 feet) of descent, the pressure increases by one atmosphere (1 atm). This increased pressure has a direct impact on the volume of air in the diver's lungs and other air-filled spaces in their body. According to Boyle's Law, as pressure increases, volume decreases proportionally. Therefore, the air in a diver's lungs compresses as they descend. This compression can pose a significant risk if the diver ascends too quickly. If the diver ascends rapidly, the pressure surrounding them decreases quickly, causing the air in their lungs to expand rapidly. If this expansion is not properly managed by exhaling continuously, it can lead to lung overexpansion and potentially serious injuries, such as a burst lung. To prevent this, scuba divers are trained to exhale continuously during ascent, allowing the expanding air to escape from their lungs and maintaining a safe pressure gradient. Furthermore, Boyle's Law also plays a role in the buoyancy of a scuba diver. A diver's buoyancy compensator (BCD) is an inflatable vest that allows them to control their buoyancy. By adding air to the BCD, the diver increases their overall volume, which increases their buoyancy, making them float more easily. Conversely, by releasing air from the BCD, the diver decreases their overall volume, which decreases their buoyancy, making them sink more easily. This control over buoyancy is essential for maintaining proper depth and stability underwater.

Beyond breathing and scuba diving, Boyle's Law finds applications in a wide array of other areas. In the field of medicine, ventilators, which are used to assist patients with breathing difficulties, rely on Boyle's Law to deliver air into and out of the lungs. The ventilator controls the pressure and volume of air delivered to the patient, ensuring proper ventilation. In the automotive industry, the internal combustion engine, the heart of most cars, operates on the principles of thermodynamics, which include Boyle's Law. The compression and expansion of gases within the engine cylinders are governed by Boyle's Law, playing a crucial role in the engine's power output and efficiency. In meteorology, Boyle's Law helps explain atmospheric phenomena. For example, the expansion and cooling of air as it rises in the atmosphere, a key process in cloud formation, is influenced by Boyle's Law. As air rises, it encounters lower pressure, causing it to expand. This expansion leads to a decrease in temperature, which can result in the condensation of water vapor and the formation of clouds. From the mundane act of breathing to the complexities of engineering and meteorology, Boyle's Law permeates our world, providing a fundamental understanding of gas behavior and its myriad applications. Its simplicity belies its profound impact on our daily lives and the technologies we rely upon.

In conclusion, we've embarked on a comprehensive exploration of Boyle's Law, a cornerstone principle governing the behavior of gases. We've delved into its theoretical underpinnings, its mathematical representation (P₁V₁ = P₂V₂), and its practical applications in diverse real-world scenarios. From the simple act of breathing to the complexities of scuba diving and beyond, Boyle's Law demonstrates its pervasive influence on our daily lives and the technologies we utilize.

We began by unraveling the essence of Boyle's Law, understanding that it describes the inverse relationship between the pressure and volume of a gas, assuming a constant temperature and number of moles. As the volume of a gas increases, its pressure decreases proportionally, and vice versa. This fundamental concept forms the basis for countless phenomena we observe in the world around us. We then meticulously dissected the application of Boyle's Law through a step-by-step approach to solving gas pressure problems. We learned the importance of identifying the knowns and the unknown, writing down the equation, substituting values, solving for the unknown, and, crucially, checking our answer. This systematic approach empowers us to tackle a wide range of problems with confidence and accuracy.

Furthermore, we ventured beyond the theoretical and explored the real-world applications of Boyle's Law. We discovered its critical role in human breathing, where the expansion and contraction of the lungs, governed by Boyle's Law, facilitate the exchange of gases. We examined its significance in scuba diving, where the pressure changes with depth necessitate a thorough understanding of Boyle's Law to ensure diver safety. We also touched upon its applications in medicine, the automotive industry, and meteorology, highlighting the breadth and depth of its influence.

Mastering Boyle's Law is not merely an academic exercise; it's an essential step towards understanding the world around us. It provides a framework for comprehending the behavior of gases, a fundamental component of our atmosphere, our bodies, and many technological systems. By grasping the principles of Boyle's Law, we gain a deeper appreciation for the intricate interplay of pressure, volume, and temperature in shaping our physical reality. As we continue to explore the vast realm of science and engineering, the knowledge of Boyle's Law will undoubtedly serve as a valuable tool, enabling us to unravel complex phenomena and innovate novel solutions. So, let's embrace the power of Boyle's Law and continue our journey of scientific discovery.