19.6:
Molecular Kinetic Energy
Consider a gas in a container; the random motion of molecules causes several collisions with the walls.
During collisions, each molecule exerts a force on the container's walls, which is the source of the pressure exerted by the gas.
If all gas molecules have the same magnitude of velocity, the interaction with the wall in the x-direction for small time intervals results in a change of x-component of velocity and momentum.
If the number of molecules per unit volume remains uniform, the number of collisions for certain area and change in the total momentum during the small interval can be determined.
Applying Newton's second law, the pressure exerted by the gas depends on the molecule's mass, speed, and number of molecules per volume.
Considering the isotropic condition, the average velocity of a gas molecule is related to the velocity components and is equal to three times the velocity of a component.
By substituting and comparing with the ideal gas equation, the average translational kinetic energy of n mole of gas is determined.
19.6:
Molecular Kinetic Energy
The word "gas" comes from the Flemish word meaning "chaos," first used to describe vapors by the chemist J. B. van Helmont. Consider a container filled with gas, with a continuous and random motion of molecules. During collisions, the velocity component parallel to the wall is unchanged, and the component perpendicular to the wall reverses direction but does not change in magnitude. If the molecule’s velocity changes in the x-direction, then its momentum is changed. During the short time of the collision, each molecule exerts a force on the container's walls, which is the source of the pressure exerted by the gas. The exerted force can be expressed in terms of velocity, where the total velocity squared is the sum of the squares of its components (x-, y-, and z-directions). Further, by substituting the values and comparing them with the ideal gas equation, the average translational kinetic energy of n moles of an ideal gas can be determined. The obtained result shows that average translational kinetic energy is directly proportional to the absolute temperature.
The root-mean-square speed of the particles in a gas is defined as the square root of the average velocity squared of the molecules in a gas, while the average distance traveled and time between collisions is called the mean free path and mean free time, respectively. The mean free path is inversely proportional to the number of molecules per unit volume, and also inversely proportional to the cross-sectional area of a molecule; the larger the molecules or the higher the number of molecules, the shorter the mean distance between collisions.
Suggested Reading
This text is adapted from Openstax, University Physics Volume 2, Section 2.2: Pressure, Temperature, and RMS Speed.