Nitrogen Gas: RMS Speed Change Calculation
Hey guys! Let's dive into a physics problem involving nitrogen gas, its temperature, pressure, and the root-mean-square (rms) speed of its molecules. We'll be calculating how the rms speed changes when the gas is heated up in a tank. Sounds fun, right? This problem is a classic example of applying the ideal gas law and understanding how molecular motion relates to temperature. So, buckle up and let's get started!
Understanding the Problem: Nitrogen Gas Dynamics
Okay, so the setup is like this: we've got a tank with a volume of 5 × 10⁻³ m³. This tank is filled with nitrogen gas (N₂), and initially, the temperature is 27 °C, and the pressure inside is 1.2 atm. Now, the cool part: we heat the tank! As we crank up the heat, the pressure inside the tank increases to 2.5 atm. We know the molar mass of nitrogen is 28 g/mol. The big question is: How much did the rms speed of the nitrogen molecules change during this whole process? This is where our physics knowledge comes into play. We are going to use the gas laws and the concept of how temperature affects molecular motion to crack this problem. Remember, the rms speed is essentially the average speed of the gas molecules, and it’s directly related to the temperature of the gas. The higher the temperature, the faster the molecules are moving, and the higher the rms speed. So, logically, we anticipate the rms speed will increase as the temperature goes up. This problem is about figuring out how much that increase is. To solve this problem, we'll need to use a few key concepts and formulas. First, we need to understand the relationship between pressure, volume, and temperature (the ideal gas law). We'll also need the formula for rms speed, which connects molecular speed to temperature and molar mass. Let's not forget to convert the temperature from Celsius to Kelvin, because Kelvin is the absolute temperature scale, and that's what we need for our calculations. Now, let’s get down to the brass tacks and solve this problem step-by-step. Get your calculators ready!
We need to remember the relationship between the rms speed of gas molecules and their temperature. The problem gives us the initial and final pressures but doesn't explicitly state the final temperature. We'll need to figure out how the temperature changes when the pressure goes up, using the ideal gas law. Specifically, we'll use the ideal gas law to find the final temperature, then calculate the initial and final rms speeds using the formula that relates rms speed to temperature and molar mass. Finally, we'll subtract the initial rms speed from the final rms speed to find the change. This problem combines several key concepts from thermodynamics, including the ideal gas law, the relationship between temperature and molecular kinetic energy, and the concept of rms speed. Let’s tackle this problem in an organized, step-by-step manner. First, we'll find the initial temperature in Kelvin, then calculate the initial rms speed. Next, we will find the final temperature using the ideal gas law, and calculate the final rms speed. Finally, we will find the change in the rms speed by subtracting the initial rms speed from the final rms speed.
Step-by-Step Solution: Calculating the RMS Speed Change
Alright, let's break this down step by step to keep it clear and easy to follow. First things first, we need to convert the initial temperature from Celsius to Kelvin. Remember, the Kelvin scale is essential for gas law calculations. To convert, we just add 273.15 to the Celsius temperature. This is because the Kelvin scale is an absolute temperature scale, and we need to make sure we're using the correct units for our calculations. This will prevent any errors in our final answer and ensure we're following the laws of physics correctly. The initial temperature is 27 °C, so in Kelvin, it's 27 + 273.15 = 300.15 K. Now we know the initial temperature. Next, we have to find the final temperature. We can apply the ideal gas law to solve for the final temperature. The ideal gas law is expressed as PV = nRT, where P is pressure, V is volume, n is the number of moles, R is the ideal gas constant, and T is the temperature. Since the volume (V) and the number of moles (n) of the gas remain constant, we can write this relationship as P₁/T₁ = P₂/T₂, where P₁ and T₁ are the initial pressure and temperature, and P₂ and T₂ are the final pressure and temperature. Thus, we can rearrange the equation to solve for T₂, the final temperature: T₂ = (P₂/P₁) * T₁. We have all the values we need to solve for T₂: P₁ = 1.2 atm, P₂ = 2.5 atm, and T₁ = 300.15 K. Plugging in the values, we get T₂ = (2.5 atm / 1.2 atm) * 300.15 K = 625.31 K. Now that we know the initial and final temperatures, we can calculate the rms speeds. The rms speed (v_rms) of gas molecules is given by the formula v_rms = √(3RT/M), where R is the ideal gas constant (8.314 J/(mol·K)), T is the temperature in Kelvin, and M is the molar mass in kg/mol. It is crucial to use the molar mass in kilograms. The molar mass of nitrogen (N₂) is given as 28 g/mol, which is equal to 0.028 kg/mol. Now we have everything we need to calculate the initial and final rms speeds.
To find the initial rms speed (v₁), plug in the initial temperature (300.15 K): v₁ = √(3 * 8.314 J/(mol·K) * 300.15 K / 0.028 kg/mol) = √(266515.71) ≈ 516.25 m/s. Then, calculate the final rms speed (v₂) using the final temperature (625.31 K): v₂ = √(3 * 8.314 J/(mol·K) * 625.31 K / 0.028 kg/mol) = √(554988.45) ≈ 745.00 m/s. Finally, the change in rms speed (Δv) is the difference between the final and initial rms speeds: Δv = v₂ - v₁ = 745.00 m/s - 516.25 m/s = 228.75 m/s.
Therefore, the change in the rms speed of the nitrogen molecules is approximately 228.75 m/s. Wasn't that fun? We have successfully calculated the change in the rms speed of the nitrogen molecules by using the ideal gas law, converting units, and using the rms speed formula. This problem illustrates the relationship between temperature and molecular motion.
Key Takeaways: Understanding the Results
So, what does all this mean, in a nutshell? We've found that when we heat the nitrogen gas, the rms speed of the molecules increases. That makes perfect sense, right? When you increase the temperature, you're essentially giving the molecules more kinetic energy, which translates to them moving faster. The change in rms speed is 228.75 m/s. This increase in speed is a direct consequence of the temperature increase. Remember, the rms speed is a measure of the average speed of the gas molecules. This also implies that the average kinetic energy of the gas molecules increases with temperature. This is a fundamental concept in thermodynamics. Think of it this way: the hotter the gas, the more energetic the molecules, and the faster they're zipping around! This is a classic example of how the microscopic properties of a gas (molecular motion) are related to its macroscopic properties (temperature and pressure). This highlights the importance of understanding the ideal gas law and the kinetic theory of gases. The ideal gas law helps us relate the pressure, volume, and temperature of a gas, while the kinetic theory gives us the insight into the relationship between temperature and the motion of gas molecules. The problem reinforces the understanding of the relationship between temperature, molecular motion, and the rms speed of gas molecules. It's a great example of how to apply physical laws to real-world scenarios. We've seen how a change in temperature directly impacts the molecular motion within the gas, leading to a change in the rms speed. This is a fundamental concept in thermodynamics, and it's essential for understanding how gases behave under different conditions.
The increase in the rms speed is due to the increase in temperature. Since the volume is constant, the increase in pressure is directly related to the increase in temperature, as described by the ideal gas law. As the temperature rises, the average kinetic energy of the gas molecules increases. This increase in kinetic energy is what causes the rms speed to increase. The molecules move faster, leading to a higher rms speed. So, the heat provides the energy for the molecules to move faster. Understanding these relationships allows us to predict the behavior of gases under varying conditions, like changes in temperature or pressure. This is crucial in many applications, from designing engines to understanding weather patterns.
Conclusion: Wrapping Things Up
Well, guys, we've successfully worked through the problem! We started with a tank of nitrogen gas at a specific temperature and pressure, then heated it up and observed a pressure increase. We then calculated the change in the rms speed of the nitrogen molecules. This involved using the ideal gas law, the rms speed formula, and a bit of unit conversion. We saw that as the temperature increased, the rms speed of the molecules increased as well. This exercise demonstrates the fundamental relationship between temperature and molecular motion, a core concept in thermodynamics. Understanding these relationships is crucial in many scientific and engineering fields. This problem is a great example of how you can apply physics principles to understand the behavior of gases. We used the ideal gas law to relate pressure, volume, and temperature, and then applied the formula for rms speed to calculate the change. Remember, the key is to understand the relationships between the different properties of a gas, such as temperature, pressure, and the speed of its molecules. Always double-check your units and conversions, and you'll be well on your way to mastering these kinds of problems!
So, the next time you encounter a problem involving gas behavior, remember the steps we've taken here: Identify the knowns, apply the relevant formulas (ideal gas law, rms speed formula), and do the calculations systematically.
Thanks for sticking around, and I hope this helped you understand the concepts better! Keep practicing, and you'll become a pro at these problems in no time. If you have any questions or want to try another problem, just let me know. Peace out, and keep learning! We've seen how the average kinetic energy of the molecules increases with temperature, which is directly related to the rms speed. This problem is a perfect demonstration of the kinetic theory of gases, which explains how the motion of gas molecules relates to the gas's temperature and pressure. Finally, we calculated the change in the rms speed, which gave us a clear picture of how much faster the molecules were moving after the gas was heated.
