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 \( {\frac{1}{2}}\) (σ1+σ2) , where σ1 and σ2 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 = \( {\frac{1}{2}}\) (σ1+σ2)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 Nc
Mathematically, it is given by
Nc = \(\sqrt{2}\) лvσ²n
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:
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 vrel for a time Δt.
In doing so it sweeps out a ‘collision tube‘ of cross-sectional area A= 𝝅 r² and length vrelΔt and therefore volume = 𝝅 r² vrel Δt
• If there are n molecules per unit volume, then number of collision = 𝝅 σ² vrel Δt 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 \(\sqrt{2}\) лσ²vn
If there are n molecules per unit volume, then total number of collisions experienced by molecules = \(\sqrt{2}\) лσ²vn²
As a collisions involves 2 molecules,
Therefore the actual number of collisions per unit volume and per unit time = \( {\frac{1}{2}}\) \(\sqrt{2}\) 𝝅σ²vn²
∴ z= \( {\frac{1}{\sqrt{2}}}\) 𝝅σ²vn²
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= \( {\frac{1}{\sqrt{2}}}\) 𝝅σ²vn²
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 \( {\frac{1}{z}}\) (Vrel)
As each collision involves two molecules. So it follows that the mean free path (λ) is
λ = vrel/z
If we consider the average velocity then
λ = v/Nc
λ=v/√2 𝝅σ²vn
The expression λ=vrel/z can also be written as
λ= \( {\frac{kT}{σp}}\) …(1) ∵ [ z= \( {\frac{σp}{kT}}\)vrel ]
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.