Mean free path: The mean free path of an oxygen molecule is 2.0 × 10^-5 m, when the gas is at a pressure of 120 Pa and a temperature of 275 K. The molecular mass of oxygen is 32.0 g/mol and 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 = 8.314 J/mol ∙ K = 0.0821 L ∙ atm/mol ∙ K. Assuming that the molecules are moving at their root-mean-square speeds, the average time interval between collisions of an oxygen molecule is closest to A) 20 ns. B) 43 ns. C) 95 ns. D) 200 ns. E) 420 ns.

Question

Quizplus · Accepted Answer

The average time interval between collisions, or the mean time between collisions, can be calculated using the mean free path ($\lambda$) and the root-mean-square speed ($v_{rms}$) of the gas molecules. The formula for the root-mean-square speed is $v_{rms} = \sqrt{\frac{3kT}{m}}$, where $k$ is the Boltzmann constant, $T$ is the temperature in Kelvin, and $m$ is the mass of one molecule. The mass of one molecule can be found by dividing the molecular mass by Avogadro's number. Once $v_{rms}$ is calculated, the mean time between collisions ($\tau$) can be found using $\tau = \frac{\lambda}{v_{rms}}$.Given:- $\lambda = 2.0 \times 10^{-5} \, \text{m}$- $T = 275 \, \text{K}$- Molecular mass of oxygen = $32.0 \, \text{g/mol}$ = $32.0 \times 10^{-3} \, \text{kg/mol}$- Boltzmann constant ($k$) = $1.38 \times 10^{-23} \, \text{J/K}$- Avogadro's number = $6.02 \times 10^{23} \, \text{molecules/mol}$First, find the mass of one oxygen molecule:$$m = \frac{32.0 \times 10^{-3} \, \text{kg/mol}}{6.02 \times 10^{23} \, \text{molecules/mol}}$$Then, calculate $v_{rms}$:$$v_{rms} = \sqrt{\frac{3kT}{m}}$$Finally, calculate $\tau$:$$\tau = \frac{\lambda}{v_{rms}}$$Without going through the detailed calculation due to the format of this response, the correct answer is found by applying these principles and calculations, which leads to an average time interval between collisions closest to the value given in option B, 43 ns.

Mean Free Path: the Mean Free Path of an Oxygen