Helium Volume And Temperature: A Physics Problem Solved

Hey guys! Today, we're diving into a classic physics problem involving helium gas, volume, and temperature. Specifically, we're tackling the question: How should the temperature change for a helium gas to triple its volume at constant pressure, given it initially occupies 100 L at 20°C? This is a super common type of problem in thermodynamics, and understanding the principles behind it can really solidify your grasp of gas laws. So, let's break it down step-by-step and make sure we all get it.

Understanding the Problem: Key Concepts and Initial Conditions

Before we jump into calculations, let's make sure we're all on the same page with the core concepts. This problem revolves around the relationship between volume and temperature of a gas when the pressure is kept constant. This relationship is described by Charles's Law, a fundamental principle in thermodynamics. Charles's Law states that the volume of a gas is directly proportional to its absolute temperature when the pressure and the amount of gas are kept constant. In simpler terms, if you increase the temperature of a gas while keeping the pressure the same, the volume will increase proportionally, and vice-versa. Think of it like a balloon: if you heat the air inside, the balloon expands. Conversely, if you cool it, the balloon shrinks. The initial conditions are also crucial for solving this problem. We know that the helium gas initially occupies a volume of 100 liters (L) at a temperature of 20°C. The goal is to figure out how the temperature needs to change so that the volume triples, meaning it becomes 300 L, while the pressure remains constant. We will need to convert the temperature from Celsius to Kelvin, as Kelvin is the absolute temperature scale used in gas law calculations. To convert from Celsius to Kelvin, we add 273.15 to the Celsius temperature. So, 20°C is equal to 20 + 273.15 = 293.15 K. This conversion is essential because using Celsius in these calculations will lead to incorrect results. The direct proportionality in Charles's Law holds true only when temperature is expressed in Kelvin.

Applying Charles's Law: The Formula and Calculation

Now that we have a solid grasp of the concepts and the initial conditions, let's put Charles's Law into action. Charles's Law can be expressed mathematically as: V1/T1 = V2/T2, where V1 is the initial volume, T1 is the initial absolute temperature, V2 is the final volume, and T2 is the final absolute temperature. This formula directly shows the proportional relationship between volume and temperature. When setting up our equation, it's crucial to identify each variable correctly. We know: V1 = 100 L, T1 = 293.15 K, and V2 = 300 L (since the volume triples). What we're trying to find is T2, the final temperature. Plugging the known values into the formula, we get: 100 L / 293.15 K = 300 L / T2. Now, it's just a matter of solving for T2. To isolate T2, we can cross-multiply and then divide. Cross-multiplying gives us: 100 L * T2 = 300 L * 293.15 K. Next, divide both sides by 100 L to solve for T2: T2 = (300 L * 293.15 K) / 100 L. Performing the calculation, we get: T2 = 879.45 K. So, the final temperature in Kelvin is 879.45 K. This is a significant increase in temperature, which makes sense given that the volume tripled. But remember, the problem might want the answer in Celsius, so we're not quite done yet.

Converting Back to Celsius: Final Answer and Interpretation

We've calculated the final temperature in Kelvin, but let's convert it back to Celsius to provide a more relatable answer. To convert from Kelvin to Celsius, we subtract 273.15 from the Kelvin temperature. Therefore, the final temperature in Celsius is: T2 (°C) = 879.45 K - 273.15 = 606.3 °C. So, to triple the volume of the helium gas at constant pressure, the temperature needs to increase to a whopping 606.3 °C! This result highlights the dramatic impact temperature can have on gas volume. It's also important to interpret this result in the context of real-world scenarios. Such a high temperature is not easily achievable and would require a significant amount of energy input. Moreover, at such extreme temperatures, the container holding the gas would need to be able to withstand the heat and pressure. This problem beautifully illustrates the practical applications of Charles's Law. Understanding how temperature and volume relate is crucial in various fields, from engineering and chemistry to meteorology and even cooking! For instance, understanding gas behavior is critical in designing engines, predicting weather patterns, and even ensuring the proper inflation of tires.

Common Mistakes and How to Avoid Them

Let's chat about some common pitfalls people encounter when tackling these kinds of problems. One of the biggest mistakes is failing to convert the temperature to Kelvin. As we discussed earlier, gas law calculations rely on the absolute temperature scale, and using Celsius will throw off your results completely. Always make that conversion a habit! Another frequent error is mixing up the variables or plugging them into the wrong places in the formula. It’s super helpful to write out what each variable represents (V1, T1, V2, T2) before plugging them into the equation. This simple step can prevent a lot of headaches. Pay close attention to the units as well. While liters (L) are commonly used for volume, you might encounter other units like milliliters (mL) or cubic meters (m³). Ensure you’re consistent with your units throughout the calculation, or convert them if necessary. Finally, sometimes students forget to answer the question that's actually being asked. In our problem, we calculated the final temperature, but the question might have asked for the change in temperature. Always double-check what the question is asking for and make sure your answer addresses it directly. By being mindful of these common errors, you can significantly boost your accuracy and confidence when solving gas law problems.

Practice Problems and Further Exploration

Okay, guys, to really master this stuff, practice is key! Let's try another similar problem. Suppose you have a gas in a container with a volume of 5 liters at a temperature of 25°C. If you heat the gas to 100°C while keeping the pressure constant, what will the new volume be? Try solving this on your own, remembering to convert temperatures to Kelvin first! And if you're feeling ambitious, try exploring other gas laws, like Boyle's Law (which relates pressure and volume) and the Ideal Gas Law (which combines pressure, volume, temperature, and the amount of gas). You can also investigate real-world applications of these laws, such as how they're used in hot air balloons or refrigeration systems. Understanding these concepts isn't just about acing your physics class; it's about understanding the world around you. Gases behave in predictable ways, and by grasping these principles, you're unlocking a deeper understanding of how things work. So, keep practicing, keep exploring, and most importantly, keep asking questions! Physics is awesome, and you've got this!

By understanding Charles's Law and the importance of absolute temperature, we've successfully solved this problem and gained valuable insights into the behavior of gases. Keep practicing, and you'll become a pro at these types of physics problems in no time!