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
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.
Energy of air molecules in action - Stirling Engine
The Stirling engine was invented in 1816 by Rev. Robert Stirling of Scotland. Unlike steam engines, the Stirling is a very simple engine, that works on the principle of expansion / contraction of a trapped mass of gas. It became popular with the industrialists in the 19th century before being shadowed by the advent of electric motor.
These engines have become popular again due to the miniature models promoted by science enthusiasts.I bought one recently and it was just magical to watch - pretty fast and elegant working.
The energy transferred by the lamp makes air expanding with increased kinetic energy, which triggers off the rest of the process.
This is how it works:
You can get a quality working model from Amazon to learn and have fun with the Stirling engine. Please click the image below:
Don't use petrol as the fuel for the lamp - dangerous and inflammable; instead, you can use 90% strong rubbing alcohol.