6 The Ideal Gas Law
Even with a primitive thermometer one can very accurately check if temperature is constant or not. For arbitrary gases or gas mixtures the following is true with high accuracy:
P1 · V1 j= P2 · V2 mmmmif T jjis constant
Amazingly, experiments further show that all gases and gas mixtures have the same coefficient of volume expansion! Extrapolation leads to absolute zero at -273.15° C and by using a clever translation of the temperature scale, we obtain the simple relation
V1 / T1 j= V2 / T2 mmmmif P jjis constant
Putting these two insights together, we get the following equation for any two equilibrium states of the same amount of gas:
( * )
Clausius and Boltzmann have shown with the model of the Ideal Gas, that this equation holds exactly when the gas particles are much smaller than their average distance travelled between two collisions, and if the particles can be regarded as perfectly elastic colliding balls, which exercise no forces on each other at distance. Finally these insights will provide our definition of temperature. But first we want to point out that this equation is also valid from the perspective of a fast-moving observer of these two states of equilibrium of the gas:
In section 3 we have already pointed out that the fast-moving observer also sees two equilibrium states of the gas. His values P1' jand P2' jdiffer from the values P1 jand P2 nby the same factor jfPj(v2j)j. This factor jcan be easily added to both sides of the gas equation (*). The same applies to the volumes and temperatures. The equation (*) therefore applies if one replaces all of the values with their prime partners. We need to know nothing concrete about the nature of the transformations, the assumption in 2 of their mere existence is sufficient.
The invariance of the ideal gas law shows up as formula 385 in Pauli's paper [3, p.700]. Pauli derives the formula using a rather complicated line of arguments.