Gas Pressure Calculation At Varying Temperatures

In the fascinating world of thermodynamics, understanding the behavior of gases under different conditions is crucial. Gas laws provide us with the framework to predict how gases will respond to changes in temperature, pressure, and volume. One of the fundamental gas laws is Gay-Lussac's Law, which states that the pressure of a gas is directly proportional to its temperature when the volume and the number of moles are kept constant. Let's dive into a practical problem where we'll apply this law to calculate the pressure of a gas at a different temperature while maintaining a constant volume. This understanding is not just academic; it has real-world applications in various fields, including engineering, chemistry, and even everyday life. Whether you're designing a pressure vessel or simply trying to understand how your car tires behave in different weather, grasping these concepts is incredibly beneficial.

Alright, guys, let's break down the problem we're tackling today. We have a gas sample that's currently occupying a volume of 44.8 liters. Now, this isn't just any old situation; it's under standard conditions. What does that mean? Well, it means we're talking about a temperature of 25°C (which is about 77°F for those of you who think in Fahrenheit) and a pressure of 1 atmosphere. These are our initial conditions, our starting point. Now, here's where it gets interesting. We're going to crank up the temperature to 34°C, which is a noticeable increase. But, and this is a big but, we're keeping the volume the same – it's staying fixed at 44.8 liters. The question we need to answer is: what's going to happen to the pressure? What will the new pressure be at this higher temperature? This is a classic gas law problem, and it's a perfect opportunity to see how temperature and pressure are related when volume is constant. We need to figure out how to apply the right gas law to solve this, and that's exactly what we're going to do.

So, how do we crack this problem? The key here is Gay-Lussac's Law. Remember, this law tells us that the pressure of a gas is directly proportional to its temperature, as long as the volume and the amount of gas stay the same. In simpler terms, if you heat a gas in a closed container (constant volume), the pressure is going to go up. This is because the gas molecules start moving faster and hitting the walls of the container more frequently and with greater force. The mathematical expression of Gay-Lussac's Law is quite straightforward: P1/T1 = P2/T2. Here, P1 is the initial pressure, T1 is the initial temperature, P2 is the final pressure (what we're trying to find), and T2 is the final temperature. It’s a neat little equation that allows us to directly relate pressure and temperature changes. But, and this is crucial, the temperature must be in Kelvin. Why Kelvin? Because it’s an absolute temperature scale, meaning zero Kelvin is absolute zero – the point where all molecular motion stops. Using Celsius or Fahrenheit can lead to incorrect results because they have arbitrary zero points. So, before we plug any numbers into our equation, we need to convert those Celsius temperatures to Kelvin. This is a critical step in any gas law calculation, and it's something we always need to keep in mind.

Alright, let's get down to the nitty-gritty and crunch some numbers! First things first, we need to convert our temperatures from Celsius to Kelvin. Remember, the formula for this is pretty simple: Kelvin = Celsius + 273.15. So, our initial temperature, 25°C, becomes 25 + 273.15 = 298.15 K. Our final temperature, 34°C, transforms into 34 + 273.15 = 307.15 K. Now that we have our temperatures in the correct units, we can plug them into Gay-Lussac's Law: P1/T1 = P2/T2. We know P1 is 1 atmosphere, T1 is 298.15 K, and T2 is 307.15 K. We're trying to find P2, the final pressure. Let's rearrange the equation to solve for P2: P2 = P1 * (T2 / T1). Now, it's just a matter of plugging in the values: P2 = 1 atm * (307.15 K / 298.15 K). When we do the math, we get P2 ≈ 1.03 atm. So, what does this tell us? It tells us that the pressure has increased slightly, from 1 atmosphere to approximately 1.03 atmospheres, as the temperature increased from 25°C to 34°C. This makes perfect sense according to Gay-Lussac's Law – when temperature goes up, pressure goes up, as long as the volume stays constant.

So, there you have it, guys! We've successfully calculated the final pressure of a gas sample after a temperature change, all while keeping the volume nice and constant. We started with a gas occupying 44.8 liters at standard conditions (25°C and 1 atm) and then heated it up to 34°C. By applying Gay-Lussac's Law, we were able to determine that the pressure increased to approximately 1.03 atmospheres. This exercise highlights the direct relationship between pressure and temperature when volume is held constant. Remember, the key to solving these types of problems is to correctly identify the relevant gas law, ensure your units are consistent (especially temperature in Kelvin!), and then carefully plug in your values. This understanding of gas laws isn't just about acing exams; it's about understanding the fundamental principles that govern the behavior of gases in our world. From the inflation of a tire to the workings of an engine, these principles are at play everywhere. So, keep practicing, keep exploring, and keep that curiosity burning! Who knows what other fascinating applications of gas laws you'll discover?