Gas Behavior: Mole Fraction And Partial Pressure Calculation

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Gas Behavior: Mole Fraction and Partial Pressure Calculation

Hey guys! Let's dive into a chemistry problem involving ideal gases. We've got a 10.0 L tank and it's filled with two gases: carbon monoxide (CO) and dinitrogen difluoride (N2F2). The tank is at 11.2°C, and we know how much of each gas is present. Our mission? To calculate the mole fraction and partial pressure of each gas. It sounds complicated, but I promise we'll break it down step-by-step to make it super clear. This is a classic example of how the ideal gas law comes into play, and it's a fundamental concept in understanding gas behavior.

Understanding the Problem: Gases in a Tank

Alright, so here's the scenario: We have a sealed 10.0 L container. Inside, we have two different gases mingling together. Knowing how each gas behaves independently, and then figuring out how they behave when mixed, is a cornerstone of understanding physical chemistry. The temperature is constant, which simplifies our calculations. We're given the mass of each gas, which is the key to calculating mole fractions and partial pressures. Mole fraction tells us the proportion of each gas in the mixture, and partial pressure tells us the pressure each gas would exert if it were alone in the container. Before jumping into the calculations, let's make sure we have all the important definitions down. Ideal gas law, mole fraction and partial pressure are the most important for the questions. These concepts will help us grasp the core principles that govern gas behavior and the relationship between different gas properties. This understanding is key for anyone trying to master chemistry and understand how gases behave under different conditions. The goal is to figure out how much of each gas is present relative to the total number of gas molecules and how much pressure each gas contributes to the total pressure in the container. This knowledge is important for things like industrial processes and environmental studies where gas mixtures are common.

Step-by-Step Calculations: The Mole Fraction

First, let's calculate the mole fraction. This is the ratio of the number of moles of a component to the total number of moles in the mixture. To do this, we'll need to know the molar mass of each gas. Let's calculate the moles of each gas using the given masses and molar masses. We're given the mass of CO (8.02 g) and N2F2 (3.59 g). We need to calculate the number of moles of each gas.

  1. Carbon Monoxide (CO):

    • Molar mass of CO = 12.01 g/mol (C) + 16.00 g/mol (O) = 28.01 g/mol
    • Moles of CO = mass / molar mass = 8.02 g / 28.01 g/mol ≈ 0.286 mol

  2. Dinitrogen Difluoride (N2F2):

    • Molar mass of N2F2 = 2 * 14.01 g/mol (N) + 2 * 19.00 g/mol (F) = 66.02 g/mol
    • Moles of N2F2 = mass / molar mass = 3.59 g / 66.02 g/mol ≈ 0.054 mol

Now, let's calculate the mole fractions. The mole fraction (χ) of a gas is the number of moles of that gas divided by the total number of moles in the mixture.

  1. Total moles:

    • Total moles = moles of CO + moles of N2F2 = 0.286 mol + 0.054 mol = 0.340 mol

  2. Mole fraction of CO (χCO):

    • χCO = moles of CO / total moles = 0.286 mol / 0.340 mol ≈ 0.841

  3. Mole fraction of N2F2 (χN2F2):

    • χN2F2 = moles of N2F2 / total moles = 0.054 mol / 0.340 mol ≈ 0.159

As a check, the sum of the mole fractions should equal 1 (or very close, due to rounding errors). In this case, 0.841 + 0.159 = 1.000, which is perfect. The mole fraction gives a direct proportion of how much of each gas is there.

Step-by-Step Calculations: Partial Pressure

Now, let's calculate the partial pressures. The partial pressure of a gas is the pressure it would exert if it occupied the container alone. We'll use the ideal gas law to calculate the partial pressure of each gas: PV = nRT.

Where:

  • P = pressure (in atmospheres, atm)
  • V = volume (in liters, L)
  • n = number of moles (in moles, mol)
  • R = ideal gas constant (0.0821 L·atm/mol·K)
  • T = temperature (in Kelvin, K)

First, convert the temperature from Celsius to Kelvin:

  • T(K) = T(°C) + 273.15 = 11.2 °C + 273.15 = 284.35 K

Now, let's calculate the partial pressures:

  1. Partial pressure of CO (PCO):

    • PCO = (nCO * R * T) / V
    • PCO = (0.286 mol * 0.0821 L·atm/mol·K * 284.35 K) / 10.0 L ≈ 0.667 atm

  2. Partial pressure of N2F2 (PN2F2):

    • PN2F2 = (nN2F2 * R * T) / V
    • PN2F2 = (0.054 mol * 0.0821 L·atm/mol·K * 284.35 K) / 10.0 L ≈ 0.126 atm

  3. Total Pressure: The total pressure in the tank is the sum of the partial pressures. Total Pressure = PCO + PN2F2

    • Total Pressure = 0.667 atm + 0.126 atm ≈ 0.793 atm

So, there you have it! We've successfully calculated the mole fractions and partial pressures of both gases. This demonstrates how the ideal gas law can be applied to gas mixtures. The total pressure of the mixture is the sum of the partial pressures of the individual components, and the mole fraction gives us a measure of the relative amount of each gas in the mixture. Understanding these concepts is very important for many chemical reactions and applications.

Conclusion: Summarizing the Results

Alright, let's recap what we've found. We started with a tank containing carbon monoxide and dinitrogen difluoride gases at a specific temperature and volume. Through careful calculation, we used the ideal gas law to determine several key properties of the gas mixture. We calculated the mole fractions, which represent the proportions of each gas in the mixture, and we figured out the partial pressures, which tell us how much each gas contributes to the total pressure inside the tank. Calculating the mole fraction is the first step in understanding the composition of a gas mixture. Knowing the partial pressures helps us understand the individual contributions of each gas to the overall pressure. This is important in industrial processes, environmental monitoring, and in the study of chemical reactions. Being able to solve these types of problems is important for future chemistry studies. We found that the mole fraction of CO is approximately 0.841, and the mole fraction of N2F2 is approximately 0.159. The partial pressure of CO is about 0.667 atm, and the partial pressure of N2F2 is about 0.126 atm. These values give us a complete picture of the gas mixture within the tank. We now have a solid understanding of how to determine the mole fractions and partial pressures of gases in a mixture. We've used the ideal gas law, and now you can apply this knowledge to similar problems. This approach is fundamental to understanding gas behavior in a wide range of chemical and industrial applications.