Suppose that the atmosphere is composed of 21% $O_2$ and 78% $Kr$ (instead of $N_2$). Since the density of $Kr$ is greater than the density of $O_2$, the lower atmosphere (where we live) should be deprived of $O_2$. Am I right? I know that diffusion has a role here, but is there a temperature where diffusion can't overcome the fact that $O_2$ should be in the upper atmosphere?
If you assume a constant temperature atmosphere and constant homogeneous gravitational field, you get the height dependence of the different molecular or atomic gas species concentrations n easily using the equilibrium Boltzmann energy distribution $n∝\exp(-E/kT)$, where $E=mgh$ is the potential energy of a gas molecule with mass $m$, $g$ is the gravitational acceleration, $h$ is the height above ground, $k$ is the Bolzmann constant, and $T$ is the absolute temperature. So, in this idealized picture, you need no consideration of (thermo-)diffusion processes, convection, turbulence, and the like, which would be a scientific major project to take into account. This shows that the heavier the molecule the faster is the exponential decrease in concentration with height. This means that also the concentration ratio of $Kr$ to $O_2$ decreases exponentially with height. If you assume that the percentages given for $O_2$ and $Kr$ are those of the total atmospheric content, then it follows by integration of these distributions that the ratio of $Kr$ to $O_2$ concentrations at ground level is higher than the ratio following from the total atmospheric percentages.