What distinguishes an ideal gas from a real gas is the absence of interactions between the particles. Although an ideal gas might seem to be an unrealistic model, its properties are experimentally accessible by studying real gases at low densities. Since even the molecules in the air you are breathing are separated by an average distance of about ten times their diameter, nearly ideal gases are easy to find. The most important features that is missing from a classical ideal gas is that it does not exhibit any phase transitions. Other than that, its properties are qualitatively the same as those of real gases, which makes it valuable for developing intuition about statistical mechanics and the thermodynamics. The great advantage of the ideal gas model is that all of its properties can be calculated exactly and nothing is obscured by mathematical complexity.

Where p_{x}, p_{y} and p_{z} are the momenta along x,y and z axes and m is the mass of the molecule.

Classical mechanics is the oldest branch of physics. Some of the greatest minds of all times, such as Sir Isaac Newton, Joseph Lagrange, Leonhard Euler, Simon Laplace, Henry Poincare, Sir William Hamilton and Carl Jacobi, laid the foundation and built the theoretical structure of the subject. The well formulated structure of classical mechanics has in fact provided an ideal paradigm for the structural development of the relatively new branches of physics, such as electrodynamics, relativistic mechanics, quantum mechanics and statistical mechanics. In the second half of the nineteenth century it seemed that the laws of classical mechanics, developed by Newton, Hamilton, Lagrange, Poincare and Jacobi, the Maxwell theory of electromagnetic phenomena and the laws of classical statistical mechanics could account for all known physical phenomena. Hence, a new mechanics is introduced which is termed as quantum mechanics.

It is assumed that the distribution of molecular velocities is isotropic and that the distribution of molecular speeds for any one component of velocity is independent of that in any other component-that is, the motion of the molecules is chaotic. Further, it is assumed that the gas is so dilute that only binary encounters are important. On the basis of these assumptions it follows that, for each change in velocity in a binary molecular encounter, there is a restoration of the original velocities by an inverse encounter.

**f(c) = $A\ exp(-\beta c^{2})$**

This equation gives the Maxwell velocity distribution for the molecules of the model considered. Frequently, however, it is more important to know the fraction of the molecules of a gas that move with a particular speed, irrespective of direction. Let N(c) dc be the number of molecules per unit volume whose speeds lie in the range c + dc, irrespective of direction. The velocity distribution has spherical symmetry with respect to the three spatial directions, so that N(c) dc is equal to the number of velocity vectors whose tips end in the volume shell lying between radii c and c + dc

**N(c) dc = 4$\pi c^{2}$f(c)dc**

Substituting the value of f(c)

This equation gives the Maxwell velocity distribution for the molecules of the model considered. Frequently, however, it is more important to know the fraction of the molecules of a gas that move with a particular speed, irrespective of direction. Let N(c) dc be the number of molecules per unit volume whose speeds lie in the range c + dc, irrespective of direction. The velocity distribution has spherical symmetry with respect to the three spatial directions, so that N(c) dc is equal to the number of velocity vectors whose tips end in the volume shell lying between radii c and c + dc

N(c) dc = 4$\pi c^{2}A\ exp(-\beta c^{2})$

A molecule of a diatomic gas can be imagined to be a tiny dumbbell, with two tiny hard or rigid spheres located at the two ends of a spring. This dumbbell model of the molecule of a diatomic gas has the spring representing the force restoring the mean separation between the bonded atoms which are represented as the spheres. A dumbbell shaped diatomic molecule can itself move, as a whole, from one to another place and thereby can have the kinetic energy of translational motion. The spheres of the dumbbell may vibrate about their mean separation, with the restoring force of the spring bringing them closer if moving away from each other and pushing them away from each other if moving closer to each other. A dumbbell shaped diatomic molecule can then possess the energy of such molecular vibrations. *A diatomic molecule can then possess three modes of storing energy within it: translational, vibrational and rotational.*

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