Kinetic Theory of Gases

Temperature and pressure are macroscopic properties of gases. These properties are related to molecular motion, which is a microscopic phenomenon. The kinetic theory of gases correlates between macroscopic properties and microscopic phenomena. Kinetics means the study of motion, and in this case motions of gas molecules.

At the same temperature and volume, the same numbers of moles of all gases exert the same pressure on the walls of their containers. This is known as Avogadros principle. His theory implies that same numbers of moles of gas have the same number of molecules.

Common sense tells us that the pressure is proportional to the average kinetic energy of all the gas molecules. Avogadros principle also implies that the kinetic energies of various gases are the same at the same temperature. The molecular masses are different from gas to gas, and if all gases have the same average kinetic energy, the average speed of a gas is unique.

Based on the above assumption or theory, Boltzmann (1844-1906) and Maxwell (1831-1879) extended the theory to imply that the average kinetic energy of a gas depends on its temperature.

They let u be the average or root-mean-square speed of a gas whose molar mass is M. Since N is the Avogadro's number, the average kinetic energy is (1/2) (M/N) u² or

          M        3R T     3
K.E.  = --- u2   = ---- = --- k T
         2 N        2 N     2

Note that M / N is the mass of a single molecule. Thus,

u = (3k N T / M)^1/2
= (3 R T / M)^1/2.

where k (= R/N) is the Boltzmann constant. Note that u so evaluated is based on the average energy of gas molecules being the same, and it is called the root-mean-square speed; u is not the average speed of gas molecules.

Calculate the kinetic energy of 1 mole of nitrogen molecules at 300 K?

Solution

Assume nitrogen behave as an ideal gas, then E_k = ³/₂ R T
= (³/₂) 8.3145 J/(mol K) * 300 K
= 3742 J / mol (or 3.74 kJ/mol)