Kinetic theory





In physical chemistry, the kinetic theory of gases is a theory that explains
the macroscopic properties of gases by consideration of their composition at
a molecular level.

Postulates

The fundamental principles of the kinetic theory are given in the form of
several postulates:

   * Gases are composed of molecules in constant random, motion. The moving
     particles constantly collide with each other and with the walls of the
     container.
   * The collisions between gas molecules are elastic.
   * The total volume of the gas molecules is neglible compared to the
     volume of the container.
   * The forces of attraction between the molecules are negligible.

The above postulates accurately describe the behavior of ideal gases. Real
gases approach ideality under conditions of low density and high temperature.

Pressure

Pressure is explained by the kinetic theory as arising from the force
exerted by collisions of gas molecules with the walls of the container. The
derivation of the mathematical expression for pressure is given below:

Consider a gas with N molecules, each of mass m, enclosed in a cuboidal
container of volume V. Suppose that a gas molecule collides with a wall of
the container which is perpendicular to the x co-ordinate axis. Then the
momentum lost by the particle and gained by the wall is given by

     2mvx

where vx is the x-component of the initial velocity of the particle.
Now, force is the rate of change of momentum. The particle under
consideration impacts with the wall once every 2l/vx time units, where l is
the length of the container. Therefore the force due to this particle is

     [mv_x *v_x \over l]

and the total force on the wall is

     [m\sum_j v_{jx}^2 \over l]

where the summation is over all the gas molecules in the container. Since
the particles are moving randomly in all directions, and since v2 = vx2 +
vy2 + vz2 for each particle, the expression for the total force becomes

     [m\sum_j v_j^2 \over 3l]

This can be written as

     [Nmv_{rms}^2 \over 3l]

where vrms is the root mean square velocity of the gas. Therefore, pressure,
the force per unit area, equals

     [Nmv_{rms}^2 \over 3Al]

where A is the area of the wall. Thus, we have the following expression for
the pressure

     [P = {Nmv_{rms}^2 \over 3V} ]

This result is interesting and significant because it relates pressure, a
macroscopic property, to the average (translational) kinetic energy per
molecule (1/2 mvrms2), which is a microscopic property.

Temperature

The above equation tells us that the product of pressure and volume is
proportional to the average molecular kinetic energy. Further, the ideal gas
equation tells us that this product is proportional to the absolute
temperature. Putting the two together, we arrive at one important result of
the kinetic theory: average molecular kinetic energy is proportional to the
absolute temperature. The constant of proportionality is 3/2 times
Boltzmann's constant, which is the ratio of the gas constant to Avogadro's
number. This result is related to the equipartition theorem.