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A gas enclosed in a container is characterized by its P, V and T. This is the macroscopic description of the gas. Particles of the gas (molecules) are in constant motion. The number of particles in the gas is itself so large that any attempt to relate the macroscopic parameters P, V, T and E with the motion of individual particles would be futile. Hence, certain assumptions are made regarding the particles (molecules) of a gas, averages of physical quantities over the large number of particles involved are obtained and these averages are finally related to the macroscopic parameters of the gas. This is the approach of kinetic theory of gases.
Ideal gas and real gas:
A gas obeying ideal gas equation at all pressures and temperatures is an ideal gas. In an ideal gas intermolecular interactions are absent. Real gases are composed of atoms or molecules which do interact with each other. Hence, no real gas is truly ideal as defined here. If the atoms/molecules of a real gas are so far apart that there is practically no interatomic/intermolecular interaction, the real gas is said to be in the ideal state. This can happen at sufficiently low density of the real gas. At low pressures or high temperatures, the molecules are far apart and therefore molecular interactions are negligible. Under these conditions, behavior of real gases is close to that of an ideal gas.
Mean free path:
When a molecule of a gas approaches another molecule, there is a repulsive force between them, due to which the molecules behave as small hard spherical particles. This leads to elastic collisions between the molecules.
It is convenient and useful to define mean free path, as the average distance traversed by a molecule with constant velocity between two successive collisions. The mean free path is expected to vary inversely with the density of the gas = N/V , where N is the number of molecules enclosed in a volume V. It is also seen that mean free path is inversely proportional to the size of the molecule, say the diameter d.
Therefore,
Pressure of ideal gas:
We now express pressure of an ideal gas as a kinetic theory problem. Let there be n moles of an ideal gas enclosed in a cubical box of volume V with sides of the box parallel to the coordinate axes. The walls of the box are kept at a constant temperature T. Here we will use the word molecular speed rather than molecular velocity since the kinetic energy of a molecule depends on the velocity irrespective of its direction.
Gas molecules are continuously moving randomly in various directions, colliding with each other and hitting the walls of the box and bouncing back. As a first approximation, we neglect intermolecular collisions and consider only elastic collisions with the walls. A typical molecule is shown in the figure moving with the velocity v, about to collide with the shaded wall of the cube. The wall is parallel to yz-plane. As the collision is assumed to be elastic, during collision, the component vx of the velocity will get reversed, keeping vy and vz components unaltered.
Considering all the molecules, their average y and z components of the velocities are not changed by collisions with the shaded wall. Thus the y and z components remain unchanged during collision with the wall parallel to the yz-plane. Hence the change in momentum of the particle is only in the x component of the momentum, Δpx is given by Δpx = final momentum - initial momentum = (-mvx ) - (mvx ) = -2mvx
We now set the average force exerted by one molecule on the wall equal to the average rate of change of momentum during the time for one collision.
Root mean square speed:
From above, we get
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