The Kinetic Theory of Gases

The Kinetic Theory of Gases has put forward a series of assumptions in order to explain what has been observed experimentally in gases.

Although, their number may vary, the core message is the same. They are as follows:

  •   The molecules of a particular gas are identical and in random motion.
  •   The collisions between the molecules and the between them and the walls of the container are perfectly elastic.
  •   The volume of a molecule is negligible, compared with the volume of the container.
  •   There are no intermolecular attractions between the molecules.
  •   The time taken for a collision between two molecules is negligible compared with the time taken for the same between a molecule and the wall.

Based on these assumptions, a formula can be derived that connects the pressure, volume, the number of molecules, individual mass and of course, the mean velocity.

 

 

If an individual molecule collides with a wall, as shown in the animation, its momentum gets doubled.A gas molecule can move in any direction at a given time - in the x-direction, y-direction or z-direction.

Let's consider the motion in the x-direction, as shown in the animation; let the velocity be Ux

 

 

If the length of the cube, mass of the molecule and velocity are l, m and v respectively,
Momentum in the x-direction = mUx
Momentum in the -x-direction = -mUx
Change in momentum = 2mUx
Total time taken - from one end to the other and vice versa - = 2l / Ux
Rate of change in momentum = 2mu/(2l / Ux)
= mUx2/l
According to Newton's Second Law, the rate of change of momentum is the force exerted by the molecule on the wall.
Therefore, Force = [mUx / l]
Since pressure, P = force / area
Pressure on the wall, P = [mUx2/l] / l2
= mUx2 / l3
= mUx2 / V, where V is the volume of the container, THE cube.
If there are N molecules in the container,
P = m[U1x2 + U2x2 + U3x2 + ... + Unx2 ] / V
If the velocities are equal,
P = m[NUx2 ] / V P = NmUx2 / V

 

 

velocity components

 

Since velocity in each direction is equal,
U2 = Ux2 + Ux2 + Ux2 = 3Ux2
Ux2 = U2 / 3
U2 is called the mean square velocity. Therefore, it is written as c2̄
Ux2 = c2̄ / 3

So, P = Nmc2̄ / 3V

PV = 1/3 mNc2̄

Since mN = mass of air molecules, mN / V = density = ρ

ρ = 3P / c2̄

 

Kinetic Energy of a Gas Molecule

The ideal gas equation shows PV = nRT, where n and R are the number of moles and Universal Gas Constant respectively.

1/3 mNc2̄ = nRT
mc2̄ = 3nRT / N
1/2 mc2̄ = 3nRT / 2N
KEmolecule = (3nR/2N) T
KEmolecule = k T
KEmolecule ∝ T

So, the kinetic energy - KE- of a gas molecule is directly proportional to the absolute temperature of the gas.

The following animation shows the connection between the KE and absolute Temperature(T).

Please click on the canvas to start / stop the animation.

 

Canvas not supported
 Energize them:  

T  :  KE  :   P

 

Highly Recommended Reading:

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