Can gas at near-vacuum densities pool at the bottom of containers? Ordinarily, we assume that a gas expands to fill it's container, however, I've had the thought that at low enough densities, this may not be possible, because the force from pressure would not be enough to overcome gravity. Put another way, the particles of gas would not be fast enough to reach the top of the container. I have this idea that in such a situation, the gas would pool at the bottom of it's container, or at least be significantly more dense near the bottom.
I also think I might be entirely mistaken. I strongly suspect that I'm just describing a liquid.
Am I mistaken? If not, how low would the pressure/density need to be to observe this pooling effect, or a significant density gradient?
I understand that this effect is apparent in large containers, such as the Earth, which is why we have different pressures at different altitudes. I was more interested to know if there is a similar effect in small containers, due to extremely low pressures. (No bigger than say, a room in your house.)
 A: Pressure is caused by particles colliding with a surface. The accompanying change in momentum direction contributes to the force per surface unit, or pressure. The particle momentum and energy depends on the temperature. Pressure is also proportional to the number of particles per volume, the density. Fewer particles colliding results in a smaller pressure. This is all described by the ideal gas law.
The low pressure in your example is caused by a low density. For the particles to be gravitationally confined their energy would have to be extremely low and/or the gravity extremely strong. Such conditions are found in galaxies which contain gravitationally bound molecular clouds. A galaxy is gravitationally a very deep "container" and the molecules are at near absolute zero temperature.
A: For an ideal gas in equilibrium at temperature $T$ in a gravitational field with potential $\varphi$, the density goes like $n \sim e^{-m\varphi/kT}$ where $m$ is the mass of the individual gas particles. If the field is uniform so $\varphi = gz$ (with $z$ being the altitude), then $n \sim e^{-mgz/kT} \equiv e^{-z/z_0}$ where $z_0 \equiv \frac{kT}{mg}$.
$z_0$ is the characteristic altitude scale over which the density of the gas changes appreciably. Taking $m\approx 28\ \mathrm{amu}$ and $g=9.8\ \mathrm{m/s}^2$, we would find that $z_0 \approx \big(3030\ \mathrm{m}\big)\left(\frac{T}{100\  \mathrm{K}}\right)$; at room temperature, this is just short of $10\ \mathrm{km}$.
On the other hand, we could rearrange the equation to yield $T= mg z_0/k =\big(0.033 \mathrm K \big) \left(\frac{z_0}{1\ \mathrm m}\right)$, which tells us at what temperature we would see the gas density vary appreciably over the height of our box.  As you can see, for any realistically sized box we would have $T$ on the order of $1\ \mathrm K$ at the most, at which point any gas would have long since liquified (and almost any material will have solidified).

All of the preceding comments operate under the assumption that the gas can be approximated as a continuous medium with a smoothly varying density. In such a case, the motion of the individual gas particles is diffusive, and so watching a single particle bounce around would not give you any real insight into the presence of a gravitational field. If the gas density is extremely low, the motion of individual particles becomes ballistic, which is essentially parabolic motion under the influence of gravity, periodically interrupted by collisions.
We can estimate the density scale for which this occurs in the following way. First, if the average speed of each particle is $v$, the time scale over which gravity will appreciably change the particle's speed is $T_{grav} \sim v/g$.  Second, the average time between collisions is given by the mean free path $\lambda$ divided by $v$; since $\lambda = 1/n\sigma$ where $\sigma$ is the collision cross section, the time scale over which particles can be expected to collide is $T_{col} = 1/n\sigma v$.  Ballistic motion will occur when $T_{grav} \ll T_{col} \implies n \ll g/(\sigma v^2) \sim mg/\sigma kT$.  For atomic collisions, $\sigma \sim 10^{-20} \ \mathrm m^2$ and so $n \ll 10^{10} \ \mathrm{cm}^{-3} \left(\frac{100\ \mathrm K}{T}\right)$.
This is quite low density; at room temperature, it would correspond to a pressure on the order of $10^{-8}\ \mathrm{mbar}$, which is a pretty respectable vacuum level. For $n$ much less than this limit, we can imagine gravity to be the dominant influence on the motion of gas particles. In order for there to be a significant chance that particles will not reach the top of the box before gravity pulls them back down, we would need to have that their kinetic energy is on the order of $mgz$, with $g$ the height of the box.  That is,
$$\frac{1}{2}mv^2 \sim mgz \implies T \sim mgz/k \approx \big(0.033\ \mathrm K\big)\left(\frac{z}{1 \ \mathrm m}\right)$$
which is the same temperature scale which we found previously.

In summary, the answer to your question is essentially no for several reasons.

*

*For typical densities, the motion of individual particles is diffusive, with the particles scattering off each other too regularly for individual parabolic arcs to be identifiable. In such cases, the density varies as $e^{-z/z_0}$ with $z_0$ on the scale of kilometers.  While $z_0$ decreases with decreasing temperature, any real gas would liquify (and probably solidify) before $z_0$ was on the scale of a realistic box.

*For $n \ll 10^{10}\ \mathrm{cm}^{-3} \left(\frac{100 \ \mathrm K}{T}\right)$, the individual gas particles behave ballistically - but even then, in order for the trajectories of the particles to deviate substantially from linear ones (i.e. in order for gravity to have a noticeable effect on them), we would need $T \sim \big(0.033\ \mathrm K\big)\left(\frac{z}{1\ \mathrm m}\right)$, again too low for real gases and reasonably sized boxes.

Your intuition is essentially correct - for reasonable values of $T$ and $z$, as one decreases $n$ the gas in the box will always be more or less uniformly distributed. At a certain point, the gas will begin to condense in to a liquid which will sit at the bottom of the container, but the gas above it will not have a significant gradient in its density.
