Mean Free Path
For hard-sphere molecules in a dilute equilibrium gas, lambda = 1/(sqrt(2) pi d^2 n_number) = k_BT/(sqrt(2) pi d^2 P).
Why this shows up in the exam
Estimating molecular collision spacing · Comparing gases at different pressure and temperature · Relating transport scales to molecular size
Learn the idea
Mean free path is the average distance a molecule travels between successive intermolecular collisions. Crowding molecules or making their collision diameter larger shortens the unobstructed distance available to each molecule.
🧠 Memory hook: Mean path shrinks with crowding and with diameter squared.
Get this one clearly and it pays off every single time it shows up in the paper. 🎯
Formulas & facts to keep ready
- lambda = 1/(sqrt(2) pi d² n_number) — hard-sphere mean free path at molecular number density n_number
- lambda = k_B T/(sqrt(2) pi d² P) — ideal-gas pressure-temperature form
- lambda proportional to T/P — scaling for fixed molecular diameter
How to approach it
- 1Choose number-density or P-T form
- 2Convert diameter to metres
- 3Check that the result has length units
Common slip-ups that cost marks
- •Using molar density in the molecular formula
- •Forgetting the square on diameter
- •Omitting the relative-motion factor sqrt(2)
🌟 That's the whole idea — you've got this. Try the practice set below; every question you attempt makes it stick a little harder.
Original chapter practice
Original questions for this chapter, not past-paper questions or an exact mapping to this individual concept.
A gas has rms molecular speed 300 m/s at 300 K. What is its rms speed at 1200 K, assuming ideal behavior?
More from Kinetic Theory of Gases
Ideal gas law and gas laws
The ideal gas law and related gas laws describe the relationships between pressure, volume, temperature, and number of moles for ideal gases.
Degrees of freedom and thermal properties
Degrees of freedom determine the distribution of energy among molecules, affecting internal energy, specific heats, and the ratio of specific heats (γ).
Kinetic theory and molecular motion
The kinetic theory explains the behavior of gases in terms of the motion and collisions of their molecules, relating properties like pressure, temperature, and kinetic energy.
RMS speed and temperature dependence
The root mean square (rms) speed of gas molecules depends on temperature and molar mass, and is a key measure of molecular motion in gases.
Mean free path and collisions
Mean free path is the average distance a molecule travels between collisions, and depends on molecular size and number density.
Ideal-Gas Equation and Molecular Form
For a dilute ideal gas in thermal equilibrium, the state variables satisfy PV = nRT = Nk_B T, where intermolecular potential energy and molecular volume are neglected.