Gás Em Transformação: Volume E Pressão Alterados, Temperatura Calculada

Hey guys! Let's dive into a classic physics problem: a gas undergoing a transformation. We're gonna figure out how the temperature changes when both the volume and pressure get a makeover. Imagine a scenario where a gas starts with a certain volume, pressure, and temperature. Then, things change – the volume shrinks and the pressure drops. Our mission? To calculate the new temperature after all this has happened. This type of problem is super common in physics, especially when you're preparing for exams like the ENEM. It really tests your understanding of the ideal gas law and how these properties (volume, pressure, and temperature) relate to each other. The coolest part? We can use a simple set of formulas to find the solution.

So, what's the deal? We have a gas, right? And this gas is going through a transformation. During this transformation, the volume of the gas decreases, specifically by 3/4 of its initial volume. This means the gas is getting compressed. At the same time, the pressure of the gas also decreases. The pressure drops by 1/4 of its initial value. Finally, we know the initial temperature of the gas, which is 47°C. The big question is: what's the final temperature of the gas after all these changes? To solve this, we will use the combined gas law, which relates the pressure, volume, and temperature of a gas. To use this law, we need to convert the temperature from Celsius to Kelvin, the absolute temperature scale. Ready to crunch some numbers? Let's get started. Remember, understanding this is key to mastering gas behavior problems. Don't worry if it seems a little tricky at first. With a bit of practice, you'll be solving these problems like a pro! This is a great exercise to learn, because it touches upon important concepts in physics, like the relationships between pressure, volume, and temperature, and how they affect each other. Understanding these relationships is critical in a wide variety of scenarios, from how a car engine works to how weather patterns behave. By working through this problem, you'll be building a solid foundation in thermodynamics. Understanding these concepts also gives you a deeper appreciation for the world around you, allowing you to understand everyday phenomena from a scientific perspective.

Entendendo as Condições Iniciais e Finais do Gás

Alright, let's break down the problem step by step. First things first, we need to clearly identify the initial and final conditions of the gas. This is like setting the stage before the play begins. We have the initial volume (let's call it V1), the initial pressure (P1), and the initial temperature (T1). These are our starting points. The problem tells us that the volume decreases by 3/4. This means the final volume (V2) is only 1/4 of the initial volume. Mathematically, V2 = (1/4) * V1. The pressure decreases by 1/4, which means the final pressure (P2) is 3/4 of the initial pressure. So, P2 = (3/4) * P1. We also know the initial temperature, T1, which is 47°C. However, in gas law calculations, we must use the absolute temperature scale, which is Kelvin. To convert Celsius to Kelvin, you add 273.15 to the Celsius value. Therefore, T1 in Kelvin is 47 + 273.15 = 320.15 K. Now we have all the pieces we need to start. Always, always, start by clearly stating what you know and what you're trying to find. This organized approach helps prevent confusion and makes the problem easier to solve.

Why is it so crucial to convert Celsius to Kelvin? Because the Kelvin scale is an absolute temperature scale. It starts at absolute zero, which is the point where all molecular motion theoretically stops. Using Kelvin ensures that our calculations are accurate and that we are correctly accounting for the behavior of the gas. Remember, the combined gas law directly relates pressure, volume, and absolute temperature. Using the wrong temperature scale can lead to major errors in your results. This conversion is a small but essential step that keeps everything consistent and correct. This initial setup is super important for several reasons. Firstly, it ensures that we fully understand the problem before attempting to solve it. It ensures that we are clear about the relationship between all the given parameters. Secondly, by clearly writing down these conditions, we create a reference that we can go back to at any time during the problem-solving process. Finally, it makes it easier to spot potential mistakes. Because, when we have everything written down clearly, it becomes much easier to identify errors in our approach or calculation. Always remember to break down complex problems into smaller, manageable parts. It makes the whole process less daunting.

Aplicando a Lei Combinada dos Gases

Now, let's get down to the main event: applying the combined gas law. This law is the hero of our story, linking pressure, volume, and temperature. The combined gas law states that (P1 * V1) / T1 = (P2 * V2) / T2, where: P1 and V1 are the initial pressure and volume, T1 is the initial temperature (in Kelvin), P2 and V2 are the final pressure and volume, and T2 is the final temperature (which is what we want to find). We know all the values except for T2. We can substitute the expressions for V2 and P2 (which we found earlier) into the combined gas law. Remember, V2 = (1/4) * V1 and P2 = (3/4) * P1. So, the equation becomes: (P1 * V1) / 320.15 = ((3/4) * P1 * (1/4) * V1) / T2. See how we've replaced the final values with their equivalent expressions based on the initial values? Now, we can simplify this equation. Notice that P1 and V1 appear on both sides of the equation. We can cancel them out. This simplifies the equation to: 1 / 320.15 = (3/16) / T2.

To solve for T2, we can cross-multiply, which gives us: T2 = (3/16) * 320.15. Calculate this, and you'll find that T2 is approximately 60.03 K. So, the final temperature of the gas after the transformation is approximately 60.03 Kelvin. Wow! That’s a significant drop in temperature. This result makes sense because both the volume and pressure decreased, and the gas law tells us that temperature is directly proportional to pressure and volume when all other factors are constant. This application of the combined gas law is a classic example of how physics uses math to describe the real world. By understanding these principles and practicing them, you’ll be much better prepared to deal with similar problems, whether in an exam or in real-world scenarios. Remember the key takeaways: convert temperatures to Kelvin, write down your known values and unknowns, simplify the equation, and double-check your work to avoid common mistakes. This step-by-step approach not only solves the problem but also strengthens your understanding of the underlying physics concepts. It's a great exercise in problem-solving that combines knowledge of the behavior of gases with some basic algebraic skills. The process itself is as important as the final answer because it builds your capacity for logical and step-by-step thinking.

Calculando a Temperatura Final

Let’s go through the final steps to make sure we’re crystal clear. After applying and simplifying the combined gas law, we arrived at the equation T2 = (3/16) * 320.15. Doing the math, we find that T2 is about 60.03 Kelvin. Because the problem initially provided the temperature in Celsius, it is often useful to convert the answer back to Celsius to make it easier to understand. To convert Kelvin back to Celsius, subtract 273.15 from the Kelvin value. So, T2 in Celsius is approximately 60.03 - 273.15 = -213.12 °C. This negative value indicates a significant drop in temperature. It is a critical reminder that a decrease in both volume and pressure leads to a decrease in the gas temperature. Always, always check to make sure your answer makes sense in the context of the problem.

Does the answer make sense? Yes, it does. We knew that the volume decreased and the pressure decreased. The combined gas law tells us that, if the volume and pressure of a gas are reduced, the temperature will also be reduced. Because we calculated a significant decrease in temperature, it aligns with our understanding of the behavior of gases. What does the answer mean in the real world? It shows how changes in a gas's environment can drastically affect its temperature. This principle is applied in various technologies, from refrigeration systems to the inner workings of engines. This problem has shown us how the properties of gases are interconnected. By altering the volume and pressure, we have also altered the temperature. It is a fundamental concept in thermodynamics that has broad implications in science and engineering. This kind of problem is also valuable for your exam preparation. It tests your ability to apply the combined gas law, your understanding of temperature scales, and your problem-solving skills. By going through these steps, you've not only solved the problem, but you've also reinforced your understanding of how gases behave under pressure and temperature changes. Congratulations, guys, you have successfully completed the problem!