The average molecular weight of a gas mixture is calculated by taking the weighted average of the molecular weights of each component gas, based on their mole fractions.
Here's a detailed explanation:
Understanding the Concept
The average molecular weight, sometimes referred to as the mean molar mass, is a crucial concept when dealing with gas mixtures. Unlike pure substances that have a single, well-defined molecular weight, gas mixtures have a combined weight determined by the proportion and molecular weight of each gas within the mixture.
Steps to Calculate Average Molecular Weight
Here's the process to calculate the average molecular weight of a gas mixture:
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Identify the Components: Determine all the gases present in the mixture.
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Find Mole Fractions: Determine the mole fraction of each gas in the mixture. The mole fraction (χi) of a component is calculated by dividing the number of moles of that component by the total number of moles in the mixture.
χi = (Moles of component i) / (Total moles in mixture)
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Determine Molecular Weights: Find the molecular weight (Mi) of each individual gas component. These values are typically available in the periodic table or chemistry textbooks.
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Calculate Weighted Average: Multiply the mole fraction of each gas component by its molecular weight and then sum these products together.
Average Molecular Weight (M̄) = Σ (χi * Mi)
This can be summarized in a table format as follows:
| Gas Component (i) | Molecular Weight (Mi) | Mole Fraction (χi) | χi * Mi |
|---|---|---|---|
| Gas 1 | M1 | χ1 | χ1 * M1 |
| Gas 2 | M2 | χ2 | χ2 * M2 |
| Gas 3 | M3 | χ3 | χ3 * M3 |
| ... | ... | ... | ... |
| Total | M̄ |
Where:
- M̄ is the average molecular weight of the mixture.
Example: Dry Air
As the reference mentioned, dry air is primarily composed of:
- Nitrogen (N2): Approximately 78.1% (mole fraction of 0.781)
- Oxygen (O2): Approximately 20.9% (mole fraction of 0.209)
Let's calculate the average molecular weight:
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Components: Nitrogen (N2) and Oxygen (O2)
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Mole Fractions: N2 = 0.781 and O2 = 0.209
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Molecular Weights: M(N2) = 28 g/mol and M(O2) = 32 g/mol
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Calculate Weighted Average: M̄ = (0.781 28 g/mol) + (0.209 32 g/mol) ≈ 28.8 g/mol
So, the average molecular weight of dry air is approximately 28.8 g/mol.
Practical Insights
- Applications: Knowing the average molecular weight of a gas mixture is vital in various fields, like chemical engineering, meteorology, and respiratory physiology, for tasks like calculating densities, flow rates, and partial pressures.
- Ideal Gas Law: The average molecular weight is used in the ideal gas law when dealing with mixtures. The ideal gas law is PV=nRT, where n is number of moles which is mass/molecular weight.
- Real Gases: For real gases, deviations from the ideal gas law may occur, especially at high pressures and low temperatures.
Conclusion
To calculate the average molecular weight of a gas mixture, determine the mole fraction and molecular weight of each gas component, and then calculate the weighted average of these values. This method gives a single representative molecular weight for the gas mixture as a whole.
