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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
The fundamental principles of the kinetic theory are given in the form of several postulates:
The above postulates accurately describe the behavior of ideal gases. Real gases approach ideality under conditions of low density and high temperature.
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:
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
and the total force on the wall is
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
This can be written as
where vrms is the root mean square velocity of the gas.
Therefore, pressure, the force per unit area, equals
where A is the area of the wall.
Thus, we have the following expression for the pressure
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.
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.
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