1. Meaning of Kinetic Energy for Gas Molecules
The kinetic energy of a gas molecule comes from its motion. Since gas molecules are always moving randomly at high speeds, each molecule has kinetic energy depending on how fast it is moving.
The faster the molecule moves, the greater its kinetic energy.
2. Average Kinetic Energy and Temperature
Not all molecules in a gas move at the same speed. Some move slowly, others move extremely fast. So instead of discussing individual molecules, we use the average kinetic energy of all molecules.
This average kinetic energy is directly related to the temperature of the gas.
2.1. Key Formula
The average kinetic energy of a molecule in an ideal gas is:
\( \dfrac{3}{2} k_B T \)
where:
- \( k_B \) = Boltzmann constant
- \( T \) = absolute temperature (Kelvin)
2.2. Meaning
Temperature is basically a measure of the average kinetic energy of gas molecules. Higher temperature → faster motion → higher kinetic energy.
3. Why Kinetic Energy Depends on Temperature Only
For an ideal gas, kinetic energy does not depend on the type of molecule or its mass. It depends only on temperature. This is because temperature controls how fast all molecules move on average.
3.1. Important Insight
If two gases are at the same temperature, their molecules have the same average kinetic energy, even if one gas is heavier than the other.
4. Relation Between Kinetic Energy and Pressure
Since pressure comes from molecules hitting the container walls, faster motion means stronger and more frequent collisions. This connects kinetic energy to pressure.
4.1. Useful Relation
Using the kinetic theory result:
\( P = \dfrac{1}{3} \rho \bar{c^2} \)
Higher kinetic energy → higher mean square speed \(\bar{c^2}\) → higher pressure (at constant volume).
5. Example of Temperature–Kinetic Energy Link
When a gas-filled balloon is warmed, the gas molecules move faster. Their kinetic energy increases, and they exert higher pressure on the balloon walls. As a result, the balloon expands.
5.1. Everyday Example
A pressure cooker whistle blows sooner when the flame is high because the high temperature increases molecular kinetic energy, leading to rapid pressure build-up.