Assuming the radius of diatomic molecules is approximately 3 ×10-10 m for what pressure in Pa will the mean free path in room-temperature (20°C) nitrogen be 0.014 m? The Boltzmann constant is 1.38 × 10-23 J/K, Avogadro’s number is 6.02 × 1023 molecules/mole, and the ideal gas constant is R = 8.315 J/mol•K = 0.0821 L ∙ atm/mol ∙ K.
Question
Assuming the radius of diatomic molecules is approximately 3 ×10-10 m for what pressure in Pa will the mean free path in room-temperature (20°C) nitrogen be 0.014 m? The Boltzmann constant is 1.38 × 10-23 J/K, Avogadro’s number is 6.02 × 1023 molecules/mole, and the ideal gas constant is R = 8.315 J/mol•K = 0.0821 L ∙ atm/mol ∙ K.
Solution
The mean free path (λ) of a gas molecule can be calculated using the formula:
λ = kT / √2πd²P
where:
- k is the Boltzmann constant (1.38 × 10^-23 J/K)
- T is the temperature in Kelvin (20°C = 293.15 K)
- d is the diameter of the molecule (2 * radius = 2 * 3 ×10^-10 m = 6 ×10^-10 m)
- P is the pressure we want to find
Rearranging the formula to solve for P gives:
P = kT / √2πd²λ
Substituting the given values into the formula gives:
P = (1.38 × 10^-23 J/K * 293.15 K) / √2π * (6 ×10^-10 m)² * 0.014 m
Solving this equation will give the pressure in Pa.
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