By applying simple physical principles we can obtain an expression for the density of Earth’s atmosphere at a certain altitude \(h\)
\begin{equation} \rho(h) = \rho_0 e^{-h/\alpha} \end{equation}
where \(\rho_0\) is the air density at sea level, and \(\alpha = RT/(Mg) \approx 7500\) m is a constant. Integrating the above formula from sea level to the height of Mount Everest (\(\approx 9000\) m) gives
\begin{equation} \int_0^{9000}\rho_0 e^{-\frac{h}{\alpha}} dh = \rho_0\alpha\left(1-e^{-9000/7500}\right) \approx \rho_0\cdot 5200\;\mathrm{m} \end{equation}
More advanced research finds a neat interpretation of this relation: the accessible height for air is 5200 m. Above this limit air is excluded, probably due to repulsion from the bedrock at these altitudes — there are reasons to believe that such rock has significantly different properties compared with rock at sea level (e.g. positive gravitational potential). In fact, both experimental work and theoretical modelling — even at the atomistic level! — have given strong evidence for the air exclusion effect: best fitting to available data is achieved with so-called air-free models.
As an example of the success of this research, one has been able to explain the existence of life in the highest regions of the Tibetan Plateu: air exists in these regions in hidden valleys (also called interpeak volumes) below the 5200 m-level, which consequently have air density \(\rho_0\). Much of present day air exclusion research is actually devoted to quantifying the amount of hidden valleys, given measurements of air density in various regions around the world (valleys that otherwise would be very difficult to discover).
Even if this research field lately has progressed heavily, there is still a lot of exciting work waiting to be done. Of the many topics can be mentioned so-called partial air exclusion on the outer borders to certain high plains, air transport between hidden valleys (which typically are connected), and the possibility of having different accessible heights for different types of air.
A future potential application of the air exclusion effect is to build storage e.g. for food at high altitudes. With no air around, food is expected to stay fresh forever!