# How can point-like particles in an ideal gas reach thermodynamical equilibrium?

Having learned that the particles of an ideal gas must be point-like (for the gas to be ideal) I wonder how they can reach thermodynamical equilibrium (by "partially" exchanging momentum and energy). First the probability of two point-like particles to collide is literally zero, and second, they can only collide head-on which implies that they can only "swap" their momenta and energies.

How is this puzzle to be solved?

An ideal gas in equilibrium cannot be supposed to have reached its equilibrium from a non-equilibrium state by interaction of its particles, because by definition the particles of an ideal gas do not interact:

1. Hard spheres of radius $a$ that collide elastically do interact in the time average - even when they interact "almost never". (Among other things they induce a non-vanishing virial coefficient $B_2 = \frac{2}{3}\pi a^3$.) So there is no ideal gas with finite-sized particles.

2. A system of point-like particles which collide elastically cannot be distinguished from a system of point-like particles that don't interact at all.

Thus, an ideal gas is - eventually - just supposed to be in equilibrium state. It cannot have reached it by "internal mechanisms".

• The electron is not a point particle, nor is the proton, nor is a hydrogen atom, or a hydrogen molecule, and so on. – John Duffield Apr 15 '16 at 14:00

Ideal gasses do not exist. In real gasses, molecules do take up volume. This does affect the pressure-volume curve.

Still, an ideal gas model is close enough to reality to be useful. It is more useful if not taken so literally as to include its most unrealistic features.

Ideal gas does only mean that there are no forces between the particles. They do not have to be point-like. For example 2-atomic gasses could have 3 translatory and 2 rotational degrees of freedom in kinetic gas theory, while still no forces act between the molecules. So ideal but not point-like. For one-atomic gasses the atoms are often taken to be hard (frictionless) balls, having small diameters compared to their free path lengths in order to allow for collisions between the atoms. However for certain considerations (pressure on wall) it is sufficient to assume point-like particles, as it doesnt matter if they collide with each other or not due to consevation of momentum.

As far as I know, you can still suppose that ideal gas interacts with the wall of the box (particles have to bounce). So, particles with sightly different initial momentum will gain more and more different momentum. This process can redistribute the momentum in agreement with Statistical Mechanics.

This is what I remember when I attended my Statistical Mechanics lectures. I can't prove this.

As @mmesser314 said, ideal gas should be taken with the tongs.

• I tacitly assumed the collisions with the wall of the box are also elastic, so no transfer of energy takes place. – Hans-Peter Stricker Mar 31 '14 at 9:55

The notion of a point-like particle needs refinement. An electron is generally thought to be a point particle but still electrons interact through coulomb forces. What matters is that collision cross-sections between the particles constituting an ideal gas vanish.