[Equation 2] § 2. Pressure of a Gas.

$\int\limits_0^t q dt = \int\limits_{t_1}^{t_1 + τ} q dt.$

During the time of the collision, however, the force which the molecule exerts on the piston is equal but oppositely directed to the force which, conversely, the piston exerts on the molecule; therefore:

$m \frac{du}{dt} = - q.$

If we therefore denote in the following by $ξ$ The Greek letter 'xi', representing the velocity component along the x-axis. the velocity component of the colliding molecule before the collision in the direction of the positive abscissa axis The horizontal or x-axis in a coordinate system., then this will be $-ξ$ after the collision, and we obtain:

$\int\limits_{t_1}^{t_1 + τ} q dt = 2 m ξ.$

Since the same applies to all other colliding molecules, it follows from Equation 1:

2) $P = \frac{2}{t} \sum m ξ,$

where the sum is to be extended over all molecules which strike the piston between the moments in time 0 and $t$. Only those which are in the process of colliding with the piston exactly at the moments 0 or $t$ are neglected, which is permissible if the entire time interval $t$ is very large compared to the duration of a single collision.

We shall see presently (§ 3) that, even if only a single gas is present in the vessel, by no means all of its molecules can have the same velocity. To encompass the greatest generality, we assume that there are different types of molecules in the vessel, all of which, however, are to bounce off the walls of the vessel like elastic spheres. $n_1 Ω$ molecules shall each have the mass $m_1$ and the velocity $c_1$ with the components $ξ_1, η_1, ζ_1$ in the coordinate directions. These shall be distributed on average uniformly within the interior space $Ω$ The Greek letter 'Omega', representing the total volume of the container. of the vessel, so that $n_1$ are allotted to the unit of volume. Furthermore, $n_2 Ω$ molecules shall be distributed in the same way, which in any case have a different ve—