How molecule-molecule collisions takes place?

Question

An extension of the kinetic molecular theory of gases recognizes that molecules have an appreciable size and deals with molecule-molecule collisions.

We begin studies of elementary reactions by investigating the collisions between the molecules of a gas. We are led to expression for the average distance that a molecule of a gas travels between collisions with other molecules and to two quantities that express the number of molecule-molecule collisions which occur in a unit time travel.

Consider a particular molecule A with diameter d, moving in the direction indicated. If the speed of molecule A is v, m remain stationary, molecule A will collide in 1 s with all the molecules that have remain centered within the cylinder. The volume of the cylinder whose radius is equal to the molecular diameter d is ∏d^2-vN*, is the diameter of molecules per unit volume. The mean free path, i.e. the distance traveled between collisions, is the free path length.

L = -v/∏d^2-vN* = 1/∏d²N*

A more detailed calculation shows that this result is not exactly correct. The assumption that only molecule A moves implies a relative speed of the colliding molecules of v. in fact if the molecules are all moving with speed v-, all types of collisions will occur, ranging from glancing collisions, where the relative angles to each other and the relative speed is √2v-. a correct result can be obtained in place of these recognitions that although molecule A moves a distance v- in 1 s, it collides with other molecules with a relative speed of √2v-. The mean path is then written as:

L = 1/ √2∏d²N*

How far a molecule travels between collisions has now been shown to depend on the number of molecules per unit volume and so on, the molecular diameter d.

The second matter to be investigated is the number of collisions per second that a molecule makes. This collision frequency is denoted by Z₁. In relation to the other molecules, the molecule A travels with an effective speed equals to the number of molecules in a cylinder of radius d and of length √2v. We therefore have:

Z₁ = 9√2u^-) (∏d²)N* = √2∏d²vN*

The last matter to be investigated is the number of collisions occurring in a unit volume per unit time. As can be imagined, this quantity is of considerable importance in understanding the rates of chemical reactions. The number of collisions per second per unit volume is called the collision rate, denoted Z₁₁.

The collision rate Z₁₁ is closely related to the collision frequency Z_t. Since there are N*molecules per unit volume and each of these molecules collided and not contacted twice. We therefore obtain 

Z₁₁ = ½ √2∏d²v^- (N*)² = 1/√2 ∏d²v^- (N*)

The mean free path, the collision frequency, and the collision have now been expressed in equations that involves the molecular diameter d. since the molecular speeds and the number of molecules per cubic meter of a particular gas can be determined, only molecular diameters need be known in order to evaluate l, Z¹ and Z¹¹. Many methods are available for determining the size of molecules.

Instance: use the collision diameter value of d = 374 pm to calculate the collision properties L, Z₁ and Z₁₁ for N² at 1 bar and 25 degree C.

Answer: the number of molecules in 1 m³ is:

N* = 6.022 Χ 10²³/ 0.0248 m³ = 2.43 Χ 10²⁵ m_-3

The mass of mole of N² molecules is:

M = 0.02802 kg

The average molecular speed form v- = [8kT/(∏m)]^½ = [8RT/∏M]^½ here we have;

v^- = [8(8.314 JK^-1 mol^-1) (298 K)/ ∏ (0.02802 kg mol^-1)] = 475 ms^-1