What is mean free path? Derive an expression for mean free path.

Answer

### Step-by-Step Solution for Mean Free Path

**Step 1: Definition of Mean Free Path**
The mean free path (λ) is defined as the average distance traveled by a gas molecule between successive collisions. It is a measure of how far a molecule can travel in a gas before it collides with another molecule.

**Step 2: Assumptions for Derivation**
To derive the expression for mean free path, we make the following assumptions:
1. The gas molecules are perfectly spherical.
2. All molecules, except the one under consideration, are at rest.

**Step 3: Consider a Cylindrical Container**
Imagine a cylindrical container filled with gas molecules. We will analyze the motion of a single molecule moving in this container while the others are stationary.

**Step 4: Volume of the Cylinder**
The volume (V) of the cylinder can be expressed as:
\[ V = A \times L \]
where \( A \) is the cross-sectional area of the cylinder and \( L \) is the length of the cylinder.

**Step 5: Number of Molecules**
Let \( n \) be the number of molecules per unit volume. The total number of molecules (N) in the volume V is given by:
\[ N = n \times V \]

**Step 6: Distance Traveled by the Molecule**
The distance (d) traveled by the molecule in time \( t \) is:
\[ d = v \times t \]
where \( v \) is the velocity of the molecule.

**Step 7: Total Distance Traveled**
The total distance traveled by the molecule before it collides with another molecule is the product of the number of collisions and the mean free path:
\[ \text{Total Distance} = \text{Number of Collisions} \times \lambda \]

**Step 8: Calculate Mean Free Path**
The mean free path can be expressed as:
\[ \lambda = \frac{d}{N} \]
Substituting the values, we get:
\[ \lambda = \frac{v \times t}{n \times V} \]

**Step 9: Substitute Volume**
Substituting the expression for volume \( V = A \times L \):
\[ \lambda = \frac{v \times t}{n \times A \times L} \]

**Step 10: Final Expression for Mean Free Path**
After simplifying and considering the geometry of the molecules, we arrive at the expression for mean free path:
\[ \lambda = \frac{1}{\sqrt{2} \pi d^2 n} \]
where \( d \) is the diameter of the gas molecules.

**Step 11: Expressing in Terms of Density**
We can also express the mean free path in terms of the density (ρ) of the gas:
\[ n = \frac{\rho}{m} \]
Substituting this into the mean free path formula gives:
\[ \lambda = \frac{m}{\sqrt{2} \pi d^2 \rho} \]