Kinetic theory of gas and Pressure of an ideal gas | L-1 | Thermodynamics

Kinetic theory of gas and Pressure of an ideal gas

Postulates of kinetic theory of gases

• A gas is made of tiny invisible, perfectly elastic particles. These particles are known as molecules.
• All the molecules of a pure gas are identical and they move continuously in all possible directions with all possible velocities.
• The gas exerts a pressure on the walls of chamber in which it is filled.
• The gaseous molecules collide with each other continuously and they traverse a straight line path between any two successive collisions.
• The size of the molecules is infinitely small as compared to the distance traversed by the molecules in between any two successive collisions.
• The collisions are instantaneous and there is no loss of kinetic energy in the collisions.
• The molecules exert no force on each other, except when they collide, and the whole molecular energy is kinetic.
• The volume of the gaseous molecules is negligible in comparison to the volume of the vessel in which gas is filled.
• The intermolecular distance in a gas is very large, so the molecules are free to move in the entire space available to them.

Mean free path

• The distance between any two successive collision is free path, and the mean of these free paths is mean free path.

Pressure of an ideal gas

• Let the components of c₁ in X, Y and Z directions are u₁, v₁ and w₁ respectively.

• Let the molecules strikes the wall ABCD with velocity u₁
• The momentum of the molecule along X-direction = mu₁
• Since the molecules and the walls are perfectly elastic, so during collision the velocity of the particles does not change, only its direction reverses.
• The momentum of the molecule along X-direction after collisions = – mu₁
• Therefore the change in momentum during one collision = mu₁ – (mu₁) = 2mu₁
• The molecule traverse 2l distance with the velocity u₁, before striking again to the same wall ABCD.
• So, the time taken by molecule between two successive collisions with the walls ABCD = 2l / u₁
• Now, the number of collisions per second with the walls ABCD = u₁ / 2l
• The change in momentum per second due to collision of this molecule with the wall ABCD = 2mu₁ × (u₁ / 2l) = mu₁² / l
• The net change in momentum per second due to all the n molecules striking the face ABCD, dp/dt = mu₁² / l + mu₂² / l + mu₃² / l + ... + mu_n² / l = (m / l ) Σu²
• According to Newton's second law force F = dp/dt
• Force exerted by gas on the face ABCD of the vessel = (m / l ) Σu²
• If A is the area, then the pressure P = F/A
• Pressure on the wall ABCD,   P_x = (m / l ) Σu² × (1 / l²) = (m / l³) Σu²
• Since volume of chamber,   V = l³
• ∴     P_x = (m / V) Σu²
• Similarly  P_y = (m / V) Σv²     and     P_z = (m / V) Σw²
• If the size of cube is very small, P_x = P_y =P_z = P
• ∴     3P = (m / V) Σ (u² + v² + w²)
• or    P = (m / 3V) Σ (u² + v² + w²)
• ∵     c² = u² + v² + w²)
• ∴     P = (m / 3V) Σ c² 

• To know about this lecture in more detail please visit on https://youtu.be/3tadOXResBo