Introduction to the Kinetic-Molecular Theory of Gases

Any law is an empirical generalization which describes the results of several experiments. A law, however, only describes results; it does not explain why they have been obtained. A theory is a description which explains the results of experiments. The kinetic-molecular theory of gases is a theory of great explanatory power. We shall see how it explains the ideal gas law, which includes the laws of Boyle and of Charles; Dalton's law of partial pressures; and the law of combining volumes.
 
The kinetic-molecular theory of gases can be stated as four postulates:

    1.  A gas consists of molecules in constant random motion.
    2.  Gas molecules influence each other only by collision; they exert no other forces on each other.
    3.  All collisions between gas molecules are perfectly elastic; all kinetic energy is conserved.
    4.  The volume actually occupied by the molecules of a gas is negligibly small; the vast majority of the volume of the gas is empty space through which the gas molecules are moving.
 

These postulates, which correspond to a physical model of a gas much like a group of billiard balls moving around on a billiard table, describe the behavior of an ideal gas. At room temperatures and pressures at or below normal atmospheric pressure, real gases seem to be accurately described by these postulates, and the consequences of this model correspond to the empirical gas laws in a quantitative way.
 
Gas Pressure Explained
In the KMT, pressure is the force exerted against the wall of a container by the continual collision of molecules against it.  Newton's second law of motion tells us that the force exerted on a wall by a single gas molecule is equal to the mass of the molecule multiplied by the velocity of the molecule.  The molecule rebounds elastically and no kinetic energy is lost in a collision.  All the molecules in a gas do not have the same velocity. The average velocity is used to describe the overall energy in a container of gas.
 
The pressure-volume product of the ideal gas equation is directly proportional to the average velocity of the gas molecules.  If the velocity of the molecules is a function only of the temperature, then the KMT gives us Boyle's Law.
 
The statement that the pressure-volume product of an ideal gas is directly proportional to the total kinetic energy of the gas is also a statement of Boyle's Law, since the total kinetic energy of an ideal gas depends only upon the temperature.
 
Gas Temperature Explained
If you look at the ideal gas law equation carefully you will see that it shows that the total kinetic energy of a collection of gas molecules is directly proportional to the absolute temeprature of the gas.  The ideal gas law can be rearranged to give an explicit expression for temperature. Temperature is a function only of the mean kinetic energy, the mean velocity and the mean molar mass.
 
As the absolute temperature decreases, the kinetic energy must decrease and thus the mean velocity of the molecules must decrease also. At T=0, the absolute zero of temperautre, all motion of gas molecules would cease and the pressure would then also be zero  No molecules would be moving. Experimentally, the absolute zero of temperature has never been attained, although modern experiments have made it to temperatures as low as 0.01 K.
 
It has been necessary to use the average velocity of the molecules of a gas because the actual velocities are distributed over a very wide range.   This distribution, can be described by Maxwell's law of distribution of velocities, or you can think of a superhighway.  There is an average speed on the highway.  Some cars are travelling slower, some faster, some at exactly the right speed.  Even those on cruise control are never exactly on the right speed because of the discrepencies in the speed control on each individual engine and the ground geometry over which the car must be moved.
 
It is not necesary to use a Maxwell-Boltzmann distribution of velocities to explain either the nature of temperature or Charles' law, although it is the correct expression of the distribution. Charles' law can be obtained for any distribution in which the velocities of the gas molecules are a function of  the nature of the gas and the absolute temperature only.
 
Partial-Pressure Explained
Dalton's law of partial pressures follows from the KMT of gases.  If the gas molecules in a mixture are in constant and random motion and if there are no forces operating between the molecules except collisions, then on the average, the net effect of collisions with other molecules must be zero.  For this reason each gas acts as if it were present alone.
 
Velocity Explained
The root mean square velocity of a molecule can be obtained by using the formula

         vrms = (3RT/M)1/2
 

Example: Calculate the root-mean-square velocity of oxygen moleules at room temperature, 25oC.  M is the molarcular mass of oxygen which is 31.9998 g/mol; the molar gas constant is 8.314 J/mol K, and the temperature is 298.15 K.  The molecular mass must be divideved by 1000 to convert it into a usable form, therefore

vrms = (3(8.314)(298.15)/(0.0319998))1/2   = 481.2 m/s

So an oxygen molecule travels through the air at 481.2 m/s which is 1726 km/h, much faster than a jetliner can fly and faster than that of most rifle bullets.
 

The very high speed of gas molecules under normal room conditions would indicate that a gas molecule would travel across a room almost instantly.  In fact, gas molecules do not do so.   If a small sample of a very odorous (and poisonous) gas, H2S is released in one corner of a room, our noses will not detect it in another corner of the room for several minutes unless the air is vigorously stirred by a mechanical fan.  The slow diffusion of gas molecules which are moving very quickly occurs because the gas molecules travel only short distances in straight lines before they are deflected in a new direction by collision with other gas molecules.
 
The distance any single molecule travels between collisions will vary from very short to very long distances, but the average distance that a molecule travels between collisions in a gas can be calculated. This distance is called the mean free path of the gas molecules. If the root-mean-square velocity is divided by the mean free path of the gas molecules, the result will be the number of collisions one molecule undergoes per second.  This number is called the collision frequency of the gas molecules.
 
The postulates of the KMT of gases permit the calculation of the mean free path of gas molecules. The gas molecules are visualized as small hard spheres.  Without going into the mathmatical detail; as the temperature rises the mean free path increases; it also rises as the pressure decreases, and as the size of the molecules decrease.   Taking all this into account, the oxygen molecules from above have a mean free path of 67 nm.   Diffusion takes place slowly because even though molecules are moving very fast, they travel only short distances in any one straight line.