The rms speed of nitrogen molecules in the air at S.T.P. is about 490 m/s. Find their mean free path and the average time between collisions. The radius of a nitrogen molecule can be taken to be 2.0 × 10-10 m.
What is the mean free path of a gas molecule (radius 2.5 × 10-10 m) in an ideal gas at 500 °C when the pressure is 7.0 × 10-6 mmHg?
The correct answer and explanation is :
Solution to the First Problem: Mean Free Path and Time Between Collisions of Nitrogen Molecules at S.T.P.
Step 1: Mean Free Path Calculation
The mean free path ( \lambda ) of gas molecules is given by the equation:
[
\lambda = \frac{k_B T}{\sqrt{2} \pi d^2 P}
]
where:
- ( k_B ) is Boltzmann’s constant ((1.38 \times 10^{-23} \, \text{J/K})),
- ( T ) is the temperature in Kelvin ((273 \, K) at S.T.P.),
- ( d ) is the diameter of a nitrogen molecule ((2 \times 2.0 \times 10^{-10} ) m = (4.0 \times 10^{-10}) m),
- ( P ) is the pressure at S.T.P. ((1.013 \times 10^5) Pa).
Plugging in values:
[
\lambda = \frac{(1.38 \times 10^{-23}) (273)}{\sqrt{2} \pi (4.0 \times 10^{-10})^2 (1.013 \times 10^5)}
]
[
\lambda \approx 6.7 \times 10^{-8} \text{ m} \text{ (or 67 nm)}
]
Step 2: Average Time Between Collisions
The average time between collisions (( \tau )) is given by:
[
\tau = \frac{\lambda}{v_{\text{rms}}}
]
Given that ( v_{\text{rms}} = 490 ) m/s:
[
\tau = \frac{6.7 \times 10^{-8}}{490}
]
[
\tau \approx 1.37 \times 10^{-10} \text{ s}
]
Solution to the Second Problem: Mean Free Path at 500°C and Low Pressure
Given:
- Temperature ( T = 500 + 273 = 773 ) K
- Pressure ( P = 7.0 \times 10^{-6} ) mmHg
- Converting to Pascals: ( 1 ) mmHg ( = 133.322 ) Pa
- So, ( P = (7.0 \times 10^{-6}) (133.322) = 9.33 \times 10^{-4} ) Pa
- Molecular radius ( r = 2.5 \times 10^{-10} ) m
- Molecular diameter ( d = 2r = 5.0 \times 10^{-10} ) m
Using the same formula for ( \lambda ):
[
\lambda = \frac{(1.38 \times 10^{-23}) (773)}{\sqrt{2} \pi (5.0 \times 10^{-10})^2 (9.33 \times 10^{-4})}
]
[
\lambda \approx 2.46 \text{ m}
]
Conclusion:
At S.T.P., nitrogen molecules have a very small mean free path ((67 \, nm)) due to frequent collisions, while at low pressure ((7.0 \times 10^{-6}) mmHg) and high temperature (500°C), the mean free path is much larger (about 2.46 m), as gas molecules are farther apart.