**Collision Diameter, Collision Number, Collision Frequency and Mean Free Path **

**1. Collision diameter :**

** **

**collision diameter is defined as **the **closet distance of approach** **between the centres of the two molecules ****taking part in a collision.** **It is denoted by ****σ****.**

- Although the interactions of molecules in a gas are very complicated but here we
**consider the simple molecular model,**i.e.,**molecules are considered as hard spheres.**

• Here, we assume that the P.E. is zero at distances between centres greater than \( {\frac{1}{2}}\) (σ_{1}+σ_{2}) , where σ_{1 }and σ_{2}are diameters of 2 molecules.

• The**P.E**. is**very high (infinite) at short distances.**

• If the molecules are**alike**then they will**collide****if their centres come within distance****σ**,i.e.;if they are different then

**σ = \( {\frac{1}{2}}\) (σ**_{1}+σ_{2})**The distance****σ****is called the****collision****diameter.**

**2. Collision frequency and Collision number:**

**According to the kinetic model**, the molecules of a gas are constantly moving in different directions with different velocities and hence keep on colliding with each other.

**The number of collisions which a single molecule makes with other molecules per second****is called**__collision number.__

It is denoted by**N**_{c}

Mathematically, it is given by

**N _{c} = \(\sqrt{2}\) лvσ**

**²n**

**where,**

**v = average velocity of the gas molecule in cms**

^{-1}**σ = molecular diameter (in cm) and**

**n = number of molecules per cm³ of the gas.**

__Derivation of expression for N:__

__Derivation of expression for N:__

__ __

**Consider the position of all the molecules except one to be frozen. **

Then note what happens as ** one mobile molecule travels through the gas** with a

**mean relative speed**

**v**for a

_{rel}**time**

**Δ**

**t**.

In doing so it

**sweeps out**a ‘

**collision tube**‘ of

**cross-sectional area**

**A=**

**𝝅 r²**and length

**v**

_{rel}**Δ**

**t**and therefore

**volume =**

**𝝅 r² v**

_{rel }**Δ**

**t**

**• If there are n molecules per unit volume, then**

**number of collision**

**=**

**𝝅**

**σ**

**² v**

_{rel }**Δ**

**t n**

**Note:** here we have taken **mean relative speed **but this relative speed when the **relative velocities of two molecules** **make an angle of ****45°**** to each other, i.e., ****\( {\sqrt{2}}\)****v.**

• Thus the **number of collisions which one molecule ****will experience**** per unit time** will be **\(\sqrt{2}\) л****σ****²****v****n**

** If there are n molecules per unit volume**, then

**total number of collisions experienced by molecules =**

**\(\sqrt{2}\) л****σ**²**v**n²**As a collisions involves 2 molecules**,

Therefore the **actual number of collisions per unit volume and per unit time = ****\( {\frac{1}{2}}\) \(\sqrt{2}\) ****𝝅****σ²vn²**

**∴ z= \( {\frac{1}{\sqrt{2}}}\) ****𝝅****σ²vn²**

**Collision frequency**

It is defined as the **total number of collisions which takes place per second among the molecules present in one cm³ of the gas**.

It is denoted by **z**

** z= \( {\frac{1}{\sqrt{2}}}\) ****𝝅****σ²vn²**

__Effect of temperature and pressure on collision frequency:__

__Effect of temperature and pressure on collision frequency:__

**• At constant volume, ****Collision frequency** **increases with increasing temperature. **

**• At constant temperature,** **the collision frequency is proportional to the pressure. **

It is because, the more is the pressure, the more is the **number density **of the molecules in the sample and the more is the **rate of encounter,** i.e., the **more is the collision frequency.**

**3.The mean free path: **

It is defined as the **average distance a molecule travels between collisions.**

• **If a molecule collides with a frequency z**, it spends a **time** 1/z **in free flight between collisions**, and therefore travels a distance **\( {\frac{1}{z}}\)** **(V _{rel})**

**As each collision involves two molecules.** So it follows that the **mean free path (λ) is**

**λ = v _{rel}/z **

**If we consider the average velocity then**

**λ = v/N _{c}**

λ=

**v/**

**√2**

**𝝅**

**σ**

**²vn**

**The expression λ=v _{rel}/z can also be written as**

**λ= \( {\frac{kT}{σp}}\)****…(1)**

**∵ [**z= \( {\frac{σp}{kT}}\)v

_{rel }]

**From Above equation (1), we can say that **

**Mean****Free Path(λ) is inversely proportional to the pressure****i.e.,Doubling the pressure reduces the value of λ by half.****λ is inversely proportional to σ² (σ= collision diameter)****λ does not depend upon velocity of gas molecules****λ is independent of the temperature at constant volume.**

**T/p remains constant when temperature is increased** [**∵ temp. appeared in above equation, in a sample of constant volume, the pressure is proportional to T ]**

Therefore **Mean Free Path is independent of the temperature** in a sample of gas **in a container of fixed volume.**

The **distance between collisions** is determined by the **number of molecules present in the given volume**, **not by the speed at which they travel.**