Molecular Collisions

1. Collision Frequency

In the kinetic theory of gases, collision frequency refers to the number of collisions a gas molecule experiences per unit time. This concept is important for understanding both diffusion and the rates of chemical reactions.

The collision frequency depends on several factors:

• The number density of molecules \( \mathcal{N} \), which tells us how many molecules are present in a given volume.
• The relative speed \( v_{rel} \) between molecules.
• The collision cross-section \( \sigma \), which describes the effective area over which collisions occur between two molecules.

The collision cross-section \( \sigma \) is determined by the radii of the colliding molecules:

The collision volume is the space region within which two particles can collide during a time interval \( \Delta t \):

The number of particles in the collision volume is given by:

The number density \( \mathcal{N} \) is the number of particles per unit volume:

For an ideal gas, we use the ideal gas law to express the number density in terms of temperature and pressure:

2. Collision Frequency Formula

The collision frequency \( z \), or the number of collisions per unit time, is calculated by multiplying the number of particles in the collision volume by the relative speed:

Substituting the expression for the number density \( \mathcal{N} = \frac{p}{k_B T} \), we get:

Since \( v_{rel} \) is typically \( \sqrt{2} v_{avg} \), we can rewrite the collision frequency as:

3. Total Number of Collisions per Unit Time

When considering all particles in a gas, the total number of collisions per unit time accounts for the collisions between all molecules. Each molecule can collide with any other molecule in the gas, so the total number of collisions is based on the number density, collision cross-section, and the relative speed of the particles.

To calculate this, we multiply the collision frequency for one particle by the total number of particles \( N \). However, since each collision involves two particles, we must divide by 2 to avoid counting each collision twice:

Substituting the expression for \( z \), we get:

Using \( \mathcal{N} = \frac{N}{V} \), the total number of collisions becomes:

This gives the total number of collisions in the gas per unit time, considering all particles moving and colliding with each other.

Importance: The total number of collisions per unit time determines the rate of molecular interactions in a gas. It is crucial for understanding diffusion, viscosity, thermal conductivity, and other transport properties, as these properties depend on the frequency of intermolecular collisions.

4. Mean Free Path

The mean free path \( \lambda \) is the average distance that a molecule travels between collisions. It is related to the collision frequency and the average speed:

Substituting the expression for the collision frequency \( z \):

This simplifies to:

The mean free path depends on the temperature \( T \), pressure \( p \), and the collision cross-section \( \sigma \). Higher pressures and larger molecular sizes (larger \( \sigma \)) decrease the mean free path, while higher temperatures increase it.

Example: Mean Free Path for Different Gases

For helium (He), hydrogen (H₂), and argon (Ar), the mean free paths \( \lambda \) are calculated using their respective collision cross-sections \( \sigma \):

As the collision cross-section increases, the mean free path decreases, as shown in the table.