If you assume a perfectly stationary and spherical planet with a point like gravity source at its center you can use relatively simple math (this amounts to evaluating a single integral in one variable) to come to the conclusion that an ideal gas atmosphere with constant temperature will extend infinitely far into space, with its pressure tapering off exponentially.

Of coarse, as far out as the moon and all these restrictive assumptions loose their convenience pretty quickly. In our previous construction the gas is bound by both gravity and pressure fields, but it is obvious that the moon’s gravity would dominate once you get close enough. Moreover, quantum mechanical effects cannot be ignored when you acknowledge that our “gas” is now in an untrained low pressure environment while being bombarded by cosmic radiation. The molecules will behave as individuals moving only under the influence of gravity, since their their Gaussian motion can now be mostly ignored, so the fact that they are moving well past Earth’s escape velocity is gonna mean they will simply leave forever. Lastly, the unstable ones (like diatomic oxygen) will decay, and molecules from other parts of the solar system will occasionally be captured into Earth’s upper atmosphere.

In conclusion, the surface Earth’s atmosphere is an unstable and dense gas maintained by an ecosystem. Far into space you’ll find some sparse molecules, atoms, leptons, etc. moving at high velocities through a sea of ionizing radiation. These conditions don’t resemble an ideal gas at all, since these particles don’t really constitute a classical continuum of matter.