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Classical Gases

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.

The Classical Partition Function

The definition of partition function involves summation over quantized energy levels and one might assume that there is a corresponding classical expression for the partition function in which a classical description of the system energies is employed. For evaluating the classical partition function we take  the example of an ideal gas (monoatomic) at sufficiently high temperature such that the energy spacing are small compared to kT. The energy of the molecule is entirely translational kinetic theory,

$H (p_{x},p_{y},p_{z})$ = $\frac{p_{x}^{2}+p_{y}^{2}+p_{z}^{2}}{2m}$
Where px, py and pz are the momenta along x,y and z axes and m is the mass of the molecule.

From Quantum to Classical

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.

Ideal Gas

The ideal gas is one which obeys Boyle's and Joule's laws at all temperatures and pressures. Real gases do not obey these laws at all temperatures and pressures. In particular, an ideal gas does not condense to a liquid  because there are no attractive forces between its molecules to make them come together, whereas all real gases condense at sufficiently low temperatures. On the other hand, if the temperature is not too low and the pressure not too high, real gases do obey Boyle's and Joule's laws quite well. The ideal or perfect gas is a limiting behavior of a real gas, in which the interactions between particles are neglected. This approximation is valid when  the potential energy of interaction between particles is negligible compared with their kinetic energy of motion. In some physical situations a real gas can be well described as an ideal gas.

Maxwell Distribution

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