MHT-CET : Physics Entrance Exam

MHT - CET : Physics - Kinetic Theory of Gases Page 1

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Fundamental Assumptions of the Kinetic Theory of Gases

 A gas consists of a large number of very small particles called molecules. The molecules of the gases are hard, elastic spheres of small diameters (~ 10- 10 m). All the molecules of the same gas are identical. The actual volume of the molecules is negligible compared to the total volume of the gas.

• Since the molecules are widely separated, the intermolecular forces of attraction
are negligible.
• The molecules are always in a state of random motion and they move in all possible directions with all possible velocities (molecular chaos).
• Due to their random motion, the molecules constantly collide with each other. These collisions are perfectly elastic, i.e., there is no loss of kinetic energy during the collisions.
• The collisions of the molecules with the walls of the container give rise to the pressure exerted by the gas.
• Between two successive collisions, the molecules travel in a straight line with uniform speed. The average distance between the successive collisions is called the Mean Free Path.
• The time of collision is negligible as compared to the time taken to traverse the mean free path.
• The average number of molecules per unit volume (molecular density) remains constant.
• At constant temperature, the average kinetic energy of the molecules remains constant. This energy is directly proportional to the absolute temperature.

 The Free Path The distance covered by a molecule between two successive collisions is called the free path. This value does not remain constant.

 Mean Free Path The average distance covered by a molecule between any two successive collisions is called the mean free path. If S is the total distance covered by the molecule and N is the number of collisions it has suffered, ,

 Mean Square Velocity: It is defined as the mean (average) of the squares of the velocities of all gas molecules. Where C1, C2, …, CN are the velocities of respective molecule.

 Root Mean Square Velocity (R.M.S. Velocity) (C) It is defined as the square root of the mean square root of the mean velocity of the gas molecules.

 Mean Velocity: It is defined as the average velocity of all gas molecules.

Pressure Exerted by Gas:

Consider a perfect gas in a cube of side 'l'. Let total number of molecules of gas = N
Mass of each molecule = m0
Area of each face of cube A = l2
Volume of the cube V = l3
Total mass of the gas m = Nm0

 Density of the gas r =

Pressure exerted by gas is same along X, Y, Z directions.
\ Number of molecules travelling parallel to each axis =

 \ Velocities of the molecules travelling parallel to X-axis.

C1, C2, …, C N/3.
Consider the motion of first typical molecule along x-direction.
Momentum of the molecule (before colliding with face ABCD) = m0C1
Momentum after collision =
-m0C1
\ Change in momentum of molecule = - 2m0C1
\ According to the principle of conservation of linear momentum
Change in momentum of face ABCD = 2m0C1
Between two successive collisions with the same face ABCD, the molecule travels distance 2l with velocity C1.

 Time interval between the two collisions =

 \ Number of collisions per unit time =

\ Change in momentum of surface per second = Change in momentum per collision × Number of collisions per second

\ Total rate of change of momentum of the surface due to N/3 molecules =

According to Newton's 2nd law, rate of change of momentum
= force creating on the surface (F)

\ Pressure exerted by the gas molecules on the surface

Kinetic Energy per Unit Volume

Since, m0C2 = K.E. of gas molecule

\ N. m0C2 = Total K.E. of gas

To show that Average K.E. per Molecule is kT

 Total K.E. of gas = PV (From (ii))

But, PV = RT (n = 1) for an ideal gas

But gas constant R = N0k
Where k = Boltzmann constant

Þ K.E. per molecule is directly proportional to its absolute temperature.

If number of molecules in 1 mole of a gas = N0

 If M is mass of 1 mole of gas, then K.E. per unit mass =

RMS Velocity

 \ PV = MC2 (for one mole gas, mass of the gas (m) = molecular weight (M))

But, PV = RT (n = 1)

 \ MC2 = RT Þ \ C2 =

R and m are constant, C
µ or C2 µT

 P = rC2 \ C2= ,C=

Since, Cµ

C0 = RMS velocity at 0° C.
C = RMS velocity at t° C.
T = t + 273

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