### 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 U_{x}

If the length of the cube, mass of the molecule and velocity are l, m and v respectively,

Momentum in the x-direction = mU_{x}

Momentum in the -x-direction = -mU_{x}

Change in momentum = 2mU_{x}

Total time taken - from one end to the other and vice versa - = 2l / U_{x}

Rate of change in momentum = 2mu/(2l / U_{x})

= mU_{x}^{2}/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 = [mU_{x} / l]

Since pressure, P = force / area

Pressure on the wall, P = [mU_{x}^{2}/l] / l^{2}

= mU_{x}^{2} / l^{3}

= mU_{x}^{2} / V, where V is the volume of the container, THE cube.

If there are N molecules in the container,

P = m[U1_{x}^{2} + U2_{x}^{2} + U3_{x}^{2} + ... + Un_{x}^{2} ] / V

If the velocities are equal,

P = m[NU_{x}^{2} ] / V
P = NmU_{x}^{2} / V

Since velocity in each direction is equal,

U^{2} = U_{x}^{2} + U_{x}^{2} + U_{x}^{2} = 3U_{x}^{2}

U_{x}^{2} = U^{2} / 3

U^{2} is called the **mean square velocity.** Therefore, it is written as c^{2}̄

U_{x}^{2} = c^{2}̄ / 3

So, P = Nmc^{2}̄ / 3V

**PV = 1/3 mNc ^{2}̄**

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

**ρ = 3P / c ^{2}̄**

#### 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 mNc^{2}̄ = nRT

mc^{2}̄ = 3nRT / N

1/2 mc^{2}̄ = 3nRT / 2N

KE_{molecule} = (3nR/2N) T

KE_{molecule} = k T

KE_{molecule} ∝ 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.

T : KE : P

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