Math Problem Statement
What is the mean free path for oxygen molecules at 300 K and atmospheric pressure (105 Pa)? Calculate the average frequency of collisions for a particular molecule. (The diameter of an oxygen molecule is 0.29 nm).
Solution
To calculate the mean free path () and average frequency of collisions () for oxygen molecules, we use the following formulas:
Step 1: Mean Free Path Calculation
The mean free path for gas molecules is given by: where:
- is the Boltzmann constant (),
- is the temperature in Kelvin (300 K in this case),
- is the diameter of an oxygen molecule (),
- is the pressure (given as ).
Calculation:
Substituting the values: Let's calculate this:
So, the mean free path .
Step 2: Collision Frequency Calculation
The average frequency of collisions for a molecule is given by: where:
- is the average speed of an oxygen molecule, given by:
- is the mass of an oxygen molecule. For , .
Calculate :
Calculate :
Summary
- Mean free path () ≈
- Collision frequency () ≈
Would you like further details or have any questions?
Related Questions for Exploration:
- How would increasing temperature affect the mean free path of oxygen molecules?
- What impact does reducing the pressure have on collision frequency?
- How is mean free path related to gas density?
- Can you derive the formula for mean free path from kinetic theory principles?
- What are real-world applications of understanding mean free path in gases?
Tip: Mean free path decreases with increased pressure, as molecules are packed closer together, increasing collision frequency.
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Math Problem Analysis
Mathematical Concepts
Kinetic Theory of Gases
Mean Free Path
Collision Frequency
Thermodynamics
Formulas
Mean free path formula: λ = (k_B T) / (sqrt(2) π d^2 P)
Average molecular speed formula: v̅ = sqrt((8 k_B T) / (π m))
Collision frequency formula: f = v̅ / λ
Theorems
Kinetic Theory of Gases
Suitable Grade Level
Undergraduate Physics
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