# Barometric equation with different species [duplicate]

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?

• Or this? Commented Apr 12, 2022 at 11:51
• See also mixture settling. Commented Apr 12, 2022 at 11:52

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.