Molecular Mean Free Path

Molecular Mean Free Path

In a dilute gas, molecules travel along nearly straight free-flight segments between binary collisions. Modeling the gas as identical hard spheres of diameter d gives a collision cross-section σ = πd². Accounting for the relative motion of target and test molecules in a Maxwellian velocity distribution yields the mean free path λ = 1 / (√2 n σ), where n is the molecular number density.

At absolute temperature T, the Maxwell-Boltzmann thermal mean speed is v_mean = √(8 k_B T / (π m)), giving a mean collision time estimate t_c = λ / v_mean, collision rate 1 / t_c, and ideal-gas pressure P = n k_B T.

Worked Example

Calculate the mean free path, thermal mean speed, collision time, collision rate, and pressure for nitrogen molecules (m = 4.65 × 10⁻²⁶ kg, k_B = 1.380649 × 10⁻²³ J/K) at density n = 1 × 10²⁰ m⁻³, diameter d = 2/5 nm = 0.4 × 10⁻⁹ m, and temperature T = 300 K:

1. Collision Cross-Section: σ = π d² = π (0.4 × 10⁻⁹ m)² ≈ 5.02654824574 × 10⁻¹⁹ m².
2. Mean Free Path: λ = 1 / (√2 × 10²⁰ × σ) ≈ 0.01406744244 m = 14.067442439955 mm.
3. Thermal Mean Speed: v_mean = √(8 × 1.380649 × 10⁻²³ × 300 / (π × 4.65 × 10⁻²⁶)) ≈ 476.26191008040 m/s.
4. Mean Collision Time and Rate: t_c = λ / v_mean ≈ 29.537198214277 μs, corresponding to a collision rate of 33.855614630254 × 10³ s⁻¹.
5. Ideal-Gas Pressure: P = n k_B T = 10²⁰ × 1.380649 × 10⁻²³ × 300 = 0.4141947 Pa.

Experiment

Use the Molecular Mean Free Path interactive module to explore how number density, hard-sphere diameter, and temperature control collision kinematics:

1. Start at Density units1e20 = 1, Diameter nm = 2/5, and Temperature K = 300, then double Density units1e20 to 2. Notice that Mean free path mm halves from 14.067442439955 mm to 7.0337212199774 mm while thermal mean speed stays constant.
2. Increase Diameter nm from 2/5 to 4/5. Because σ ∝ d², doubling the molecular diameter quarters the mean free path.
3. Set Density units1e20 to 0 to inspect the vacuum collision limit, where pressure and collision rate drop to 0 and mean free path/time become Unavailable.

Limitations

This model evaluates the standard dilute hard-sphere kinetic estimate using t_c = λ / v_mean with a fixed approximate nitrogen molecular mass (4.65 × 10⁻²⁶ kg). Across the positive grid the molecular volume fraction is below 10⁻⁶; at high pressures or low temperatures, intermolecular attractions (van der Waals forces), quantum rotational-vibrational effects, and excluded-volume corrections become important.

References

For the kinetic theory derivations of ideal-gas pressure, Maxwell-Boltzmann speed distributions, and hard-sphere mean free path, consult:

• OpenStax University Physics Volume 2, Section 2.2: Pressure, Temperature, and RMS Speed
• OpenStax University Physics Volume 2, Section 2.4: Distribution of Molecular Speeds