At what pressure will be particles in a medium be unable to form a sound wave when disturbed? How can this pressure be described mathematically?
My guess is that this would correspond to the point at which the restoring force due to pressure is unable to create a transverse wave and the disturbed particles travel infinitely far away before the hypothetical wave reaches it's amplitude. But I have no idea how you would even begin to start finding a quantitative value for this point.
It's obviously not a sharp cut-off, but as a general guide sound waves cannot propagate if their wavelength is equal to or less than the mean free path of the gas molecules. This means that even for arbitrarily low pressures sound will still propagate provided the wavelength is long enough. Possibly this is stretching a point, but even in interstellar gas clouds sound waves (more precisely shock waves) will propagate, but their length scale is on the order of light years.
I expect that there is no minimal pressure.
A sound wave is a density wave. If the particles are close to each other they will interact due to strong forces like van der Waals force and Coulomb force. Reducing the pressure in constant volume leads to long distances between the particles.
Lets assume the particles have a huge distance and we ignore even the still remaining weaker forces like the gravity force. The gas is now in a thermodynamic equilibrium. Then one creates a shock wave by pressing two particles together. This shock wave will move through the media and decay as known from particles at higher pressure. The sound-propagation velocity will increase with the pressure.
It will be difficult to measure, because the effect is overlapped by random thermal movement of the particles. But that is rather a technical problem.
If you are interested such questions, you should have a look at the Maxwell's demon thought experiment too.
Effectively zero, but it takes a mental stretch to get there.
When you're dealing with a gas, lower pressure means that there is a longer mean free path, meaning the atoms/molecules can be expected to go longer and longer between collisions. You can get this either by spacing out the particles more (lower density) or by slowing them down (lower temperature). As this time between collisions increases, your system looks more like a bunch of particles moving in one direction, rather than a group of particles "sloshing" back and forth. In this limit of zero pressure, you don't get interesting sound wave propagation any more, and I'm pretty sure this is what your intuition is guiding.
Liquids, on the other hand, have great promise. Liquid helium-II (superfluid helium) is very weird--it can flow uphill to escape a container, and it will conduct heat better than any known material. This heat conduction is considered to be carried by sound, although you're more likely to encounter the term "phonon" in the literature. Phonons can be observed at very low fluid pressures, meaning that sound can propagate at very low pressure in this system.
Empirically the Altitude/Pressure where sound stops propagating can be determined in a Hypobaric chamber. Simply place a sound emitting device in the chamber with a microphone. A mercury manometer on the chamber can determine the 'altitude' where sound stops being transmitted to the microphone.
