The Kinetic Theory of Gasses

 

Consider a single molecule moving about inside a cubic vessel, of side (l), that continually collides with the six faces of the cube. The molecule is considered infinitesimal in size and the collisions with the wall are perfectly elastic so that no energy is lost. Thus, the velocity (c) will be constant throughout all the collisions. It is also assumed that the striking angle and the reflecting angle of the molecule to and from the wall are the same. Thus, there can be a succession of paths traced by the molecule such as those depicted in the left hand side cube in figure 7.

 

 

Figure 7. Locus of the Movement a Gas Molecule in a Cubic Vessel

 

Thus, a succession of paths depicted as a, b, c, d, e, ….. are followed. On the right hand side of figure 7 is shown a nest of identical cubes and the same path of the gas molecule is depicted but instead of striking the wall and bouncing back the gas molecule is shown passing from one cube to another in the same straight line. It is clear that a, b, c, d…. etc. will be exactly the same in the two diagrams.

 

Let the rectangular co-ordinates of the nest parallel to the three directions of the edges of the cube be Ox, Oy and Oz and let (), (), () be the direction cosines of the line abcde. Let a molecule travel a distance (c) in unit time and, thus, it will travel a distance (c) parallel to Ox, (c) parallel to Oy and (c) parallel to Oz. Now in traveling the distance (c) parallel to Ox it will encounter faces of the nest perpendicular to Ox at equal distances (l) apart. Each encounter will coincide with a collision between the molecule and a face perpendicular to Ox.

The number of collisions in unit will consequently be, . Now at each collision the momentum is reversed. Thus, if (m) is the mass of a molecule, then the total change in momentum on impact will be (2mcl). It follows that the total momentum change for all impacts between the molecules and the two faces perpendicular to Ox must be, The change in momentum per impact times the number of impacts.

 

 

There will be similar impacts with the other faces, i.e.,  and 

 

Thus, the total of all the impacts exerted by the molecule on all the six faces in unit time will be

 

The total impact exerted in unit time is also the total pressure exerted on the six faces of the cube. If there are a great number of molecules inside the vessel having masses of m₁, m₂, m₃, . …  moving with different velocities c₁, c₂, c₃ ….. and are so small that they never collide with one another the total impact they exert on all six faces will be,

 

 

If these molecules move in all directions at random they will obviously exert equal pressures on the six faces of the cube. As the total area of the six faces is (6l²), the pressure (P) per unit area is given by,

 

                        

 

The numerator in this fraction is twice the total kinetic energy of motion of all the gas molecules while the denominator is three times the volume of the gas,

 

Consequently,                                (2)

 

And the pressure is equal to two-thirds of the kinetic energy per unit volume.

 

There are a number of interesting conclusions from the arguments put forward so far. Since the kinetic energies are additive, pressures must also be additive,

 

Thus, the pressure exerted by a mixture of gasses is the sum of the pressures exerted by the constituents of the mixture. This is Dalton’s Law of Partial Pressures.

 

If the volume of the vessel is allowed to change, while the molecules and their motion are kept the same energy, then

 

The pressure will vary inversely as the pressure. This is Boyle’s Law.

 

If (C²) is defined as the average value of (c²) throughout the gas, the average being taken by mass, then

 

   

                   and as          

 

                      where (v) is the volume and () is the density of  the gas

 

    Consequently, from equation (2),

 

 

Thus the speed of the molecular motion of a gas can be calculated for any physical condition. At room temperature and at one atmosphere pressure, air has a density of 1.203 gram per liter thus the molecular velocity can be calculated to be about 500 m per second (roughly the speed of a rifle bullet)

 

The kinetic theory of gasses can be extended much further but sufficient discussion is given here to demonstrate its scope and use. Those wishing to pursue the subject further are recommended to read “An Introduction to the Kinetic Theory of Gasses” by Sir James Jeans.