Kinetic Theory of Gases
Skills to develop
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 2Note that M / N is the mass of a single molecule. Thus,
These formulas correlate temperature, pressure and kinetic energy of molecules. The distribution of gas speed has been studied by Boltzmann and Maxwell as well, but this is beyond the scope of this course. However, you notice that at the same temperature, the average speed of hydrogen gas, H2, is 4 times more than that of oxygen, O2 in order to have the same average kinetic energy.
For two gases, at the same temperature, with molecular masses M1 and M2, and average speeds u1 and u2, Boltzmann and Maxwell theory implies the following relationship:
M1 u12 = M2 u22. Thus, M1 u2 -- = (---)2 M2 u1
The consequence of the above property is that the effusion rate, the root mean square speed, and the most probable speed, are all inversely proportional to the square root (SQRT) of the molar mass. Simply formulated, the Graham's law of effusion is
The theories covered here enable you to make many predictions. Apply these theories to solve the following problems.
Assume nitrogen behave as an ideal gas, then
At 300 K, any gas that behave like an ideal gas has the same energy per mol.
Gas Molar u (root-mean-squar speed) mass /(m/s) H 2 1966 He 4 1390 H2 28 525 O2 32 492 CO2 44 419
Molar masses are 349 and 352 for 235UF6 and 235UF6 respectively. Using the method above, their root-mean-square speeds are 149 and 148 m/s respectively.
The separation of these two isotopes of uranium was a necessity during the time of war for the US scientists. Gas diffusion was one of the methods employed for their separation.
Since the effusion rates are
The time required can be evaluated by