The diameter of a nitrogen molecule is about 0.29 nm. In a tank of nitrogen at a pressure of 2.5 atm and temperature 273 K, what is the mean free path of a nitrogen molecule?
Question
The diameter of a nitrogen molecule is about 0.29 nm. In a tank of nitrogen at a pressure of 2.5 atm and temperature 273 K, what is the mean free path of a nitrogen molecule?
Solution
To calculate the mean free path of a nitrogen molecule, we can use the formula:
λ = k*T / (sqrt(2) * π * d² * P)
where:
- λ is the mean free path
- k is the Boltzmann constant (1.38 * 10^-23 J/K)
- T is the temperature in Kelvin
- d is the diameter of the molecule
- P is the pressure
First, we need to convert the diameter from nm to m:
d = 0.29 nm = 0.29 * 10^-9 m
Then, we can substitute the values into the formula:
λ = (1.38 * 10^-23 J/K * 273 K) / (sqrt(2) * π * (0.29 * 10^-9 m)² * 2.5 atm)
But we need to convert the pressure from atm to Pa (Pascal), because the Boltzmann constant is in J/K, and 1 J = 1 Pa*m³. The conversion is 1 atm = 1.01325 * 10^5 Pa.
So, the pressure is:
P = 2.5 atm = 2.5 * 1.01325 * 10^5 Pa
Substituting the pressure back into the formula, we get:
λ = (1.38 * 10^-23 J/K * 273 K) / (sqrt(2) * π * (0.29 * 10^-9 m)² * 2.5 * 1.01325 * 10^5 Pa)
Now, you can calculate the value of λ, which will be the mean free path of a nitrogen molecule in the tank.
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