Calculating Oxygen Mass In A Container Ideal Gas Law Example
Hey guys! Ever wondered how to calculate the mass of a gas, like oxygen, inside a container? It might sound intimidating, but it's actually quite straightforward once you understand the ideal gas law. In this article, we're going to break down a classic physics problem step-by-step, making it super easy to grasp. We'll be focusing on how to determine the mass of oxygen (O2) in a container, given its volume, pressure, and temperature. So, let's dive in and unlock this fascinating concept together!
Understanding the Ideal Gas Law
Before we jump into the calculations, let's quickly recap the ideal gas law. This fundamental equation in chemistry and physics describes the state of a gas under ideal conditions. The ideal gas law is expressed as:
PV = nRT
Where:
- P is the pressure of the gas (in atmospheres, atm)
- V is the volume of the gas (in liters, L)
- n is the number of moles of the gas
- R is the ideal gas constant (0.0821 L·atm/mol·K)
- T is the temperature of the gas (in Kelvin, K)
This equation tells us how pressure, volume, the number of moles, and temperature are all related for an ideal gas. By understanding this relationship, we can solve for any one of these variables if we know the others. The ideal gas law is a cornerstone of thermodynamics and helps us predict the behavior of gases in various conditions. It's important to remember that the ideal gas law works best under conditions of relatively low pressure and high temperature, where gases behave more ideally. At high pressures or low temperatures, real gases may deviate from ideal gas law behavior due to intermolecular forces and molecular volume.
Problem Statement: Finding the Mass of Oxygen
Okay, let's tackle the problem at hand. We have a container with a volume of 8.2 liters. Inside this container, there's oxygen gas (O2) under a pressure of 6 atmospheres. The temperature inside the container is 127 degrees Celsius. We're also given the molar mass of oxygen (O2), which is 32 grams per mole. Our mission, should we choose to accept it, is to find the mass of oxygen gas present in the container. This is a classic application of the ideal gas law, and by breaking it down step-by-step, we'll see just how manageable it is. The key here is to first use the ideal gas law to find the number of moles of oxygen, and then use the molar mass to convert moles into grams. Remember, the ideal gas law provides a powerful tool for relating pressure, volume, temperature, and the amount of gas, allowing us to solve a wide range of problems in chemistry and physics. Understanding each component of the problem statement – the volume, pressure, temperature, and molar mass – is crucial for setting up the solution correctly.
Step 1: Convert Temperature to Kelvin
The ideal gas law requires the temperature to be in Kelvin (K). So, our first step is to convert the given temperature from Celsius to Kelvin. The conversion formula is:
K = °C + 273.15
In our case, the temperature is 127 °C. Adding 273.15 to it, we get:
K = 127 + 273.15 = 400.15 K
So, the temperature inside the container is approximately 400.15 Kelvin. It's super important to use Kelvin in gas law calculations because it's an absolute temperature scale, meaning it starts at absolute zero. Using Celsius would give us incorrect results because it's a relative scale with an arbitrary zero point. Converting to Kelvin ensures that our calculations are consistent with the physical principles underlying the ideal gas law. This simple conversion is a critical step in solving any gas law problem, and it's one that you should always remember to do first!
Step 2: Apply the Ideal Gas Law to Find Moles (n)
Now that we have the temperature in Kelvin, we can use the ideal gas law to find the number of moles (n) of oxygen gas. Recall the ideal gas law equation:
PV = nRT
We need to rearrange this equation to solve for n:
n = PV / RT
We have the following values:
- P = 6 atm
- V = 8.2 L
- R = 0.0821 L·atm/mol·K
- T = 400.15 K
Plugging these values into the equation, we get:
n = (6 atm * 8.2 L) / (0.0821 L·atm/mol·K * 400.15 K)
n ≈ 1.5 moles
So, there are approximately 1.5 moles of oxygen gas in the container. This calculation is the heart of the problem, as it uses the ideal gas law to directly relate the measurable properties (pressure, volume, and temperature) to the amount of gas present. Understanding how to rearrange the ideal gas law and plug in the correct values is essential for solving these types of problems. Once we know the number of moles, we're just one step away from finding the mass of oxygen!
Step 3: Convert Moles to Mass
We've found the number of moles of oxygen gas, but the question asks for the mass. To convert moles to mass, we use the molar mass of oxygen (O2), which is given as 32 g/mol. The formula for this conversion is:
Mass = n * Molar Mass
Where:
- Mass is the mass of the gas in grams
- n is the number of moles
- Molar Mass is the mass of one mole of the substance (in g/mol)
Plugging in the values, we get:
Mass = 1.5 moles * 32 g/mol
Mass ≈ 48 grams
Therefore, there are approximately 48 grams of oxygen gas in the container. This final step demonstrates the crucial link between moles and mass, a fundamental concept in stoichiometry and chemistry. By multiplying the number of moles by the molar mass, we can easily convert between these two units, allowing us to answer the original question. This conversion highlights the practical application of the mole concept in determining the amount of substance in a given sample. So, we've successfully navigated the problem from start to finish!
Final Answer
So, guys, we've successfully calculated that there are approximately 48 grams of oxygen gas (O2) in the 8.2-liter container, which is held at a pressure of 6 atmospheres and a temperature of 127 degrees Celsius. We used the ideal gas law to find the number of moles and then converted that to mass using the molar mass of oxygen. This whole process demonstrates how the ideal gas law can be used to solve real-world problems. Remember, the key is to break down the problem into smaller, manageable steps. First, convert the temperature to Kelvin. Then, use the ideal gas law to find the number of moles. Finally, convert moles to mass using the molar mass. By following these steps, you can confidently tackle similar problems in the future!
Practice Problems
To solidify your understanding, here are a couple of practice problems you can try:
- What mass of nitrogen gas (N2) is contained in a 10-liter container at a pressure of 5 atm and a temperature of 27 °C? (Molar mass of N2 = 28 g/mol)
- A container holds 5 grams of carbon dioxide (CO2) at a temperature of 25 °C. If the pressure is 2 atm, what is the volume of the container? (Molar mass of CO2 = 44 g/mol)
Working through these problems will help you become more comfortable with the ideal gas law and its applications. Remember to follow the same steps we used in the example problem, and don't hesitate to review the explanations if you get stuck. Practice makes perfect, and with a little effort, you'll be a pro at solving gas law problems in no time!
Conclusion
Calculating the mass of a gas in a container might seem like a daunting task at first, but as we've seen, it's quite manageable when you break it down step by step. The ideal gas law is your best friend in these situations, providing a powerful tool to relate pressure, volume, temperature, and the number of moles of a gas. By converting the temperature to Kelvin, using the ideal gas law to find moles, and then converting moles to mass using the molar mass, you can solve a wide range of gas-related problems. So, keep practicing, and don't be afraid to tackle those physics and chemistry challenges head-on. You've got this! Remember, understanding the fundamentals of gas behavior is crucial not only in academic settings but also in various real-world applications, from industrial processes to environmental science. So, mastering the ideal gas law is a valuable skill that will serve you well in your scientific endeavors.