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The barometric equation gives the pressure dependence of a perfect gas in a gravitational potential. In particular

$$ P(h) = P_0e^{-\frac{mgh}{k_BT}} $$

where $m$ is the mass of molecules and $T$ the temperature of the gas.

What would happen if we had an atmosphere composed of different species? I would expect it to be layered with heavier gases at the bottom and lighter ones on top with transitions between gases looking something like a $tanh(h)$. Can this be derived from statistical mechanics relatively easily?

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If we suppose ideal gases, remember that the pressure of an ideal gas depends on the number of molecules, however, it has an influence on the distribution of the gas with a gravitational potential. However, you can substitute in your equation the molar mass with the average molar mass. This molar mass $M$ can be found in the following equation, which you can find in Wikipedia: $$M = \sum _i x_i M_i$$ where $x_i$ is the mole fraction of the $i_{th}$ compound and $M_i$ the molecular mass of that $i_{th}$ compound. So, for example, air gives us a molar mass of $28.7 g/mol$. Substituting the formula you gave will give the result you wanted.

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