§ 2. Pressure of a Gas.

§ 2. Calculation of the Pressure of a Gas.

We now wish to subject such gases to a closer examination. Since we assume that the molecules are subject to the general laws of mechanics, the principle of the conservation of living forceoriginal: "lebendige Kraft"; a historical term for kinetic energy ($mv^2$) and the motion of the center of mass must be satisfied both during the collisions of the molecules with one another and during impacts against the wall. We can still form the most varied conceptions regarding the internal nature of the molecules; as long as these two principles are satisfied, we will obtain a system that shows a certain mechanical analogy with real gases. The simplest such conception is that the molecules are perfectly elastic, infinitely little deformable spheres, and the vessel walls are perfectly smooth, equally elastic surfaces. However, where it is more convenient for us, we can assume a different law of interaction. Provided it is again in harmony with general mechanical principles, it will be no more, but also no less, justified than the assumption of elastic spheres, which we shall adopt for the time being.

We now imagine a vessel of volume $Ω$, of otherwise arbitrary shape, filled with a gas, at whose walls the gas molecules are to be reflected exactly like perfectly elastic spheres. A part of the vessel wall $AB$ with surface area $φ$ shall be flat. We place the positive abscissa axisThe horizontal or x-axis in a coordinate system perpendicular to this surface, directed from the inside toward the outside. The pressure on $AB$ is evidently not changed if we imagine a vertical cylinder behind this surface element with the base $AB$, in which the surface element $AB$ can be moved parallel to itself like a piston. This piston would then be driven into the cylinder by the molecular impacts. However, if a force $P$ acts upon it from the outside in the negative direction of the abscissa, its intensity can be chosen such that it maintains an equilibrium with the molecular impacts, and the piston makes only invisible fluctuations, now in one direction, now in the opposite.

During any moment of time $dt$, some molecules will perhaps be in the process of colliding with the piston $AB$—