§ 12.5NCERT Class 11 · Physics · Chapter 12
Mean Free Path, Collision Dynamics & Brownian Motion
Microscopic Collision Dynamics: While gas molecules travel at speeds of hundreds of meters per second, an odor or smoke takes perceptible time to diffuse across a room. This is because molecules collide incessantly (billions of times per second), executing a tortuous zig-zag path.
1. Definition of Mean Free Path (λ)
The mean free path (λ) of a gas molecule is defined as the average distance traveled by the molecule between two successive collisions.
2. Derivation of the Mean Free Path Formula
- Assume gas molecules are hard spheres of diameter d.
- Two molecules collide if the distance between their centers becomes less than or equal to d.
- A moving molecule sweeps out an effective cylindrical collision volume of cross-sectional area σ = π d2.
- In time Δt, traveling at speed ⟨v⟩, it sweeps out volume Vcyl = π d2 ⟨v⟩ Δt.
- If n = N / V is the number density (molecules per unit volume), the number of collisions suffered in time Δt (assuming other molecules are stationary) is: Ncoll = n · π d2 ⟨v⟩ Δt.
- However, because all other molecules are also moving simultaneously with random velocities, the relative velocity between colliding pairs is on average vrel = √2 ⟨v⟩.
- Hence, collision rate = √2 n π d2 ⟨v⟩.
λ = Distance / Collisions = (⟨v⟩ Δt) / (√2 n π d2 ⟨v⟩ Δt) = 1 / (√2 · n · π · d2)
3. Dependencies of Mean Free Path (λ) on Temperature and Pressure
From the ideal gas equation P · V = N · kB · T, the number density n is:
n = N / V = P / (kB · T)
Substituting n into the mean free path equation:
λ = (kB · T) / (√2 · π · d2 · P)
| Condition | Relationship | Physical Consequence |
|---|---|---|
| At Constant Pressure (P = constant) | λ ∝ T | Heating a gas at constant pressure expands it, reducing number density n and increasing mean free path. |
| At Constant Temperature (T = constant) | λ ∝ 1 / P | Increasing pressure compresses the gas, packing molecules closer and decreasing mean free path. |
| At Constant Volume (V = constant) | λ = constant | Since number density n = N/V is fixed, λ does not depend on temperature or pressure! |
4. Collision Frequency and Relaxation Time
- Collision Frequency (Z or fcoll): Number of collisions made by a single molecule per second:
Z = vavg / λ = √2 · n · π · d2 · vavg
- Relaxation Time (τ): Average time interval between two successive collisions:
τ = 1 / Z = λ / vavg
At room temperature and atmospheric pressure, λ ≈ 10−7 m (several hundred molecular diameters), while τ ≈ 10−10 s.
5. Brownian Motion
Discovered by botanist Robert Brown in 1827 upon observing pollen grains suspended in water executing continuous, irregular, zig-zag motion.
- Physical Cause: Albert Einstein (1905) proved that Brownian motion is caused by the unbalanced bombardment of colloidal particles by surrounding fluid molecules.
- Significance: Provided direct experimental proof for the physical reality of molecules and atoms, confirming the kinetic molecular theory.
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