B.Sc.
Part III: Chemistry Honours
Paper-V-A: (Physical Chemistry)
GROUP A: Gaseous State
Molecular Collisions in a gas
Collision diameter (𝛔) or (d)
Collision number
Collision frequency
Effect of temperature and pressure on collision frequency
MOLECULAR COLLISIONS IN A GAS
Assumptions:
(i) Molecules are rigid, non-interacting and spherical (with diameter 𝛔).
(ii) All molecules move with the same average velocity (vavg).
(iii) On an average, molecules collide while approaching each other
perpendicularly.
Collision diameter (𝛔) or (d) :
When two molecules approach each other for collision, a stage is reached
when strong repulsive forces start operating between them and thus the two
molecules start moving apart. The closest distance of approach between the
centers of the two molecules taking part in a collision is called collision
diameter. It is represented by 𝛔 or d.
A gaseous molecule can be regarded as a rigid (or hard) sphere of radius d.
the volume (4/3) 𝛑d3 is known as the effective volume of the molecule.
DR.KUMARI SEEMA, (HOD), CHEMISTRY, J. D. WOMEN’S
1
COLLEGE, PATNA
Collision number :
Let us consider the collisions of the gas molecules among themselves. A
moving molecule will collide with other molecules whose centers come
within a distance of 𝛔 or d from its center, the effective area of the target
is 𝛑𝛔2. The quantity 𝛑𝛔2 is called the collision cross-section for the
molecule and it is an area of an imaginary sphere of radius 𝛔 around the
molecule within which the center of another molecule cannot penetrate.
As the molecule is moving with an average velocity of v avg (m/sec), then
in one second the area swept out by a single molecule is 𝛑𝛔2 vavg (m3).
Let N⋆ = the number of molecules per unit volume (m3) of the gas.
DR.KUMARI SEEMA, (HOD), CHEMISTRY, J. D. WOMEN’S
2
COLLEGE, PATNA
Then the number of molecules within the area 𝛑𝛔2 vavg are 𝛑𝛔2 vavg N⋆. As
all the gas molecules are in a motion at a particular temperature, we must
consider the average velocity along the line of centers of the two colliding
molecules. Assuming that, on an average, molecules collide while
approaching each other perpendicularly, then the average velocity along
their centers is √2𝑣 as shown below:
Now number of collisions made by a single molecule with other molecules
per m3 per second (Z1) is given by
Z1 = 𝛑𝛔2 (vavg) N⋆
= √2 𝛑𝛔2 (vavg) N⋆ …………………(1)
= Collision Number
The number of collisions suffered by a single molecule per unit time per
unit volume of the gas, is known as the collision number.
Collision frequency :
Collision frequency is the number of molecular collisions occurring per
unit time per unit volume of the gas. Total number of bimolecular collisions
Z11 per m3 per sec is called as collision frequency is given as
Z11 = (Z1 N⋆)
= (√2 𝛑𝛔2vavg N⋆ ) N⋆
As collision frequency (Z11) = 𝛑𝛔2 (vavg) N⋆2 ……………..(2)
√
Hence, Z11 ∝ vavg
Z11 ∝ 𝛔2
Z11 ∝ N⋆2
𝟖𝑹𝑻
As vavg = …………………………(3)
𝝅𝑴
DR.KUMARI SEEMA, (HOD), CHEMISTRY, J. D. WOMEN’S
3
COLLEGE, PATNA
Moreover, PV=nRT ∴ N⋆ = = ⟹ (N⋆)2 =
Collision frequency (Z11) = 𝜋𝜎 = 2𝜎 …….(4)
√
Effect of temperature and pressure on collision frequency:
From the above equation, Z11 ∝ P2
√𝑻
Z11 ∝
𝑻𝟐
𝟏
Z11 ∝
𝑻𝟑∕𝟐
Thus, with decrease of temperature or with increase of pressure the gas
contracts. As the molecules come close to each other so frequency of
collisions also increase. If the collisions involve two unlike molecules of
masses M1 and M2 (reduced mass μ = ) and diameters 𝛔1 and 𝛔2.
Average diameter 𝛔12 = , then
collision frequency (Z12) = 𝜋𝜎 𝑁 𝑁 where, N1 and N2 are the
number of molecules per m3 of the two types of molecules.
DR.KUMARI SEEMA, (HOD), CHEMISTRY, J. D. WOMEN’S
4
COLLEGE, PATNA
