I'm a laywoman in physics and recently found myself pondering about the matter reflected in the title of this post.
To make my question more precise from the mathematical standpoint, let's suppose you are given a 3D image of the momentary positions of the nuclei of all atoms of an unknown monoatomic substance in a certain volume at a certain moment of time. Rotating the image in a 3D visualization program, you see that the positions look pretty chaotic from any angle, unlike a crystalline structure. You know neither the image's scale nor any of the parameters such as the pressure or temperature. The only information you are given is that the substance is not ionized and is in a thermodynamic equilibrium and either in the liquid state or in the gaseous state and that the pressure and the temperature are below the critical pressure and the critical temperature, respectively. You can extract the numerical XYZ positions and do any calculations with them, but, as stated above, you don't know the scale. How can you tell whether it's a liquid or a gas? What criterion can be used to reach that end?
My first guess was that whilst a gas doesn't have any correlation between the positions, a liquid does, but then I realized it's a wrong answer because a gas is not necessarily an ideal gas, so it's unclear to me how I can tell whether it's a liquid or a gas if there's some correlation between the positions in the image. I tried to find the answer on the Internet and this SE, but did not succeed and humbly hope that physics experts on this SE can tell me the answer.
UPDATE: Sure, the limiting cases of an ideal gas and a tightly packed liquid are easy, but what do I do in the general case? In other words, how can I deduce whether it's a liquid or a gas if the spread of distances between neighboring nuclei is moderate, that is, neither very small nor very large?
Best Answer
Everything you've said is correct, which is why the conclusion is: there is no fundamental difference! Under the modern classification, they're just the same fluid phase of matter.
For example, consider the phase diagram of water. If you take water vapor, slowly heat it up, then pressurize it, and then slowly cool it down, you'll end up with liquid water. This entire process is completely smooth. There isn't any sharp point, like a phase transition, where the behavior qualitatively changes; thus we can't make a sharp distinction between liquids and gases.
There are fluids that are "liquid-like" (densely packed, strong interactions between neighbors) and fluids that are "gas-like" (sparse, weak interactions between neighbors) but no dividing line, just like how there's no moment where a shade of grey changes from white to black.
By contrast, ice really can be distinguished from liquid water or water vapor. You can't turn either of the two into ice without crossing a phase transition. At that point, the atoms will suddenly become ordered, and you can see this from a snapshot of their positions.
Edit: in response to the 25 comments, I'm not saying there's no difference between liquids and gases, I'm saying that there are clearly liquid-like things, and clearly gas-like things, but a continuous spectrum between them. Here are some properties that characterize gases:
The opposite properties characterize liquids. In the easy cases, you could use any of these to make the call. But all of these properties change continuously as you go from one to the other, as long as you go around the critical point. This isn't true for a solid/liquid or solid/gas phase transition.