Kinetic Theory

The kinetic theory of gases is the study of the microscopic behavior of molecules and the interactions which lead to macroscopic relationships like the ideal gas law.

The study of the molecules of a gas is a good example of a physical situation where statistical methods give precise and dependable results for macroscopic manifestations of microscopic phenomena. For example, the pressure, volume and temperature calculations from the ideal gas law are very precise. The average energy associated with the molecular motion has its foundation in the Boltzmann distribution, a statistical distribution function. Yet the temperature and energy of a gas can be measured precisely.

Newton's Laws and Collisions

Applying Newton's Laws to an ideal gas under the assumptions of kinetic theory allows the determination of the average force on container walls. This treatment assumes that the collisions with the walls are perfectly elastic.

In this development, an overbar indicates an average quantity. In the expression for the average force from N molecules, it is important to note that it is the average of the square of the velocity which is used, and that this is distinctly different from the square of the average velocity.

Gas Pressure from Kinetic Theory

Under the assumptions of kinetic theory, the average force on container walls has been determined to be

The average force and pressure on a given wall depends only upon the components of velocity toward that wall. But it can be expressed in terms of the average of the entire translational kinetic energy using the assumption that the molecular motion is random.

and assuming random speeds in all directions Then the pressure in a container can be expressed as

Expressed in terms of average molecular kinetic energy:

This leads to a concept of kinetic temperature and to the ideal gas law.

Kinetic Temperature

The expression for gas pressure developed from kinetic theory relates pressure and volume to the average molecular kinetic energy. Comparison with the ideal gas law leads to an expression for temperature sometimes referred to is the kinetic temperature.

This leads to the expression

 

The more familiar form expresses the average molecular kinetic energy:

 

It is important to note that the average kinetic energy used here is limited to the translational kinetic energy of the molecules. That is, they are treated as point masses and no account is made of internal degrees of freedom such as molecular rotation and vibration. This distinction becomes quite important when you deal with subjects like the specific heats of gases. When you try to assess specific heat, you must account for all the energy possessed by the molecules, and the temperature as ordinarily measured does not account for molecular rotation and vibration. The kinetic temperature is the variable needed for subjects like heat transfer, because it is the translational kinetic energy which leads to energy transfer from a hot area (larger kinetic temperature, higher molecular speeds) to a cold area (lower molecular speeds) in direct collisional transfer.

Molecular Speeds

From the expression for kinetic temperature

 

substitution gives the root mean square (rms) molecular velocity:

From the Maxwell speed distribution this speed as well as the average and most probable speeds can be calculated.

Maxwell Speed Distribution

The speed distribution for the molecules of an ideal gas is given by

From this function can be calculated several characteristic molecular speeds and such things as what fraction of the molecules have speeds over a certain value at a given temperature. It is involved in many rates of phenomena.

Note that M is the molar mass and that the gas constant R is used in the expression. If the mass m of an individual molecule were used instead, the expression would be the same except that Boltzmann's constant k would be used instead of the molar gas constant R.