Does free molecular flow have a newtonian force in between 2 chambers? If a vacuum chamber is in free molecular flow pressure say $10^{-7}$ Pascal, and another container is added to it that is at $10^{-9}$ Pascal, will they reach pressure equilibrium eventually? At higher pressures, say between $101$ kPa (1 atm) and $50$ kPa, the flow between them is laminar flow. Therefore there is the Newtonian equation of (101 kPa - 50 kPa)/m^2 which gives 1x51E^7 newtons of suction force from 1 chamber to the next. Does the same happen during free molecular flow?
 A: Like the continuum flow case, the two connected chambers will eventually reach equilibrium and there will be a net force pushing gas from the high pressure chamber to the low pressure chamber until that occurs.
Let's consider flow from one chamber to another. The momentum carried by an average individual molecule* is
$$p = m u_{\perp} = \sqrt{m k_B T} ,$$
where $m$ is its mass, $u_{\perp}$ is velocity perpendicular to the opening, $k_B$ is Boltzmann's constant, and $T$ is temperature in the source chamber. The number of molecules passing through the opening per unit time is
$$Q = N A u_{\perp} = N A \sqrt{\frac{k_B T}{m}} , $$
where $N$ is molecule number density. Therefore, the force associated with the flow in one direction will be
$$F = p Q = k_B N T A = P A , $$
where $P$ is pressure in the source chamber. This is the expression for the continuum case. However, none of this presupposes collisions between molecules, so it is also valid for free molecular flow. Ultimately, this force will be supplied by the walls of the chambers.
*A more careful derivation would integrate over the Maxwell-Boltzmann distribution.
A: The mean free path of a gas molecule is given by:
$$ \ell = \frac{kT}{\sqrt{2}\pi d^2 P} $$
For nitrogen (the main component of air) the molecular diameter is $d \approx 0.36$ nm so at $10^{-7}$ Pa the mean free path is about 70 kilometers, which is orders of magnitude larger than the average vacuum chamber. That means the gas does not flow in the usual sense. What you have is effectively a Joule expansion. The gas molecules will rapidly equilibrate, but by collisions with the chamber walls rather than collisions with each other.
