The simplest proof of the ideal gas law

Prerequisite: the section Entropy in statistical physics (needed for the definition of temperature !)

Let us keep natural physical units : amounts of substance n are counted in molecules (avogadro constant = 1) and entropy is counted in natural logarithms (k=R=1).

Between the phase spaces of a given system of n gas molecules in 2 volumes V and V', the space of position coordinates of each of the n gas molecules is expanded by a volume factor (V'/V), while the rest of dimensions (momentum, internal rotations and vibrations) with both energy functions in these dimensions (kinetic energies and intra-molecular potential energies) remain fixed. Thus, the condition of thermal equilibrium at the same temperature T is preserved from one volume to the other just by diluting the phase space along position coordinates, thus dividing the probability of each elementary state by (V'/V)n. This leads to a variation of entropy
S'S = n.ln(V'/V).
For a small expansion (V'=V+dV), dS = n.dV/V

V = n.dV/dS.

As the energies are preserved in this isothermal expansion, the received heat energy TdS balances the released work of pressure PdV:
PV = n.PdV/dS = nT

When do gases depart from the ideal gas law, and how much ?

It happens when the dilution of the probability density of positions of each molecule of gas does not follow proportionality to volume, because what the variation of volume brings to the system (roughly an additional empty space for each molecule), consists in positions with probabilities that may differ from the average of probabilities in the space, as the latter includes the case of positions near another molecule, which can differ.

For example if each molecule is occupying space in such a way that it forbids a volume b for the positions of other molecules then in the first order of correction for almost perfect gases, the formula becomes (V/n)−b= T/P.
On the other hand, some molecules such as water (vapor), attract each other in some ways, presenting positions at intermediate distances (between collision and separation) with a lower potential energy, so that an excess amount of probability can be found concentrated at such distances, which may exceed the amount which was missing by the previous effect (due to the volume forbidden at shorter distances, or rather partially forbidden by a higher potential energy, so that these effects are of the same nature with different signs). The effect of these bounds can be expressed by the above formula with a negative value for b. This b is independent of volume but depends on temperature, especially its negative terms are very sensitive to it (becoming important at low temperatures).

Another cause of deviation from the ideal gas law, also expressible by the same formula in the same approximation (when V/n is much larger than |b|) but happening without interaction, is a quantum effect, for gases of light atoms or molecules at low temperature: the movement of each molecule occupies a "position" with volume h3 in phase space. In each space dimension, molecules with mass m at temperature T have a thermal uncertainty (rough interval) of momentum of about 2mT, giving an interval of positions h/2mT (this formula is approximative ! would need a difficult calculation for the precise coefficient), whose cube is a sort of space volume |b| occupied by each molecule. The sign of b depends on the type of the gas molecules, and has similar effects between the classical and quantum cases: Example of the dihydrogen molecule at 2 Kelvin: k*2K=2.761 . 10-23 J and pi*hbar/sqrt(2*(proton mass)*2.761*10^-23 J)= 1.09 nm.

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