Ideal Gas - Classical Thermodynamic Ideal Gas

Classical Thermodynamic Ideal Gas

The thermodynamic properties of an ideal gas can be described by two equations: The equation of state of a classical ideal gas is the ideal gas law

This equation is derived from Boyle's Law: (at constant T and n); Charles's Law: (at constant P and n); and Avogadro's Law: (at constant T and P). By combining the three laws, it would demonstrate that which would mean that . Under ideal conditions, or rather . The internal energy of an ideal gas given by:

where

• P is the pressure
• V is the volume
• n is the amount of substance of the gas (in moles)
• R is the gas constant (8.314 J·K−1mol-1)
• T is the absolute temperature
• k is a constant used in Boyle's Law
• b is the proportionality constant; equals V/T
• a is the proportionality constant; equals V/n
• U is the internal energy
• is the dimensionless specific heat capacity at constant volume, ≈ 3/2 for monatomic gas, 5/2 for diatomic gas and 3 for more complex molecules.

In order to switch from macroscopic quantities (left hand side of the following equation) to microscopic ones (right hand: side), we use

where

• N is the number of gas particles
• is the Boltzmann constant (1.381×10−23J·K−1).

The probability distribution of particles by velocity or energy is given by the Boltzmann distribution.

The ideal gas law is an extension of experimentally discovered gas laws. Real fluids at low density and high temperature approximate the behavior of a classical ideal gas. However, at lower temperatures or a higher density, a real fluid deviates strongly from the behavior of an ideal gas, particularly as it condenses from a gas into a liquid or solid. The deviation is expressed as a compressibility factor.

The ideal gas model depends on the following assumptions:

• The molecules of the gas are indistinguishable, small, hard spheres
• All collisions are elastic and all motion is frictionless (no energy loss in motion or collision)
• Newton's laws apply
• The average distance between molecules is much larger than the size of the molecules
• The molecules are constantly moving in random directions with a distribution of speeds
• There are no attractive or repulsive forces between the molecules or the surroundings

The assumption of spherical particles is necessary so that there are no rotational modes allowed, unlike in a diatomic gas. The following three assumptions are very related: molecules are hard, collisions are elastic, and there are no inter-molecular forces. The assumption that the space between particles is much larger than the particles themselves is of paramount importance, and explains why the ideal gas approximation fails at high pressures.