Barometric formulas: various derivations and comparisons to environmentally relevant observations

Three different lines of thinking (mechanical, mixed thermodynamical-mechanical, statistical thermodynamic) are presented to derive the noted barometric formula, which gives the altitude dependence of the pressure of a gas in a gravity field. It is shown that the first two methods can be extended to non-isothermal cases, whereas statistical thermodynamics relies on the concept of thermal equilibrium and its usefulness is limited to the isothermal barometric formula. The temperature changes in the gravity field are taken into account by two different methods: simple conservation of energy, and a more refined line of thought based on the adiabatic expansion of an ideal gas. The changes in gravitational acceleration are also considered in further refinements. Overall, six different formulas are derived and their usefulness is tested on the atmosphere of the Earth. It is found that none of the formulas is particularly useful above an altitude of 20 km because radiation effects make the temperature changes in the atmosphere difficult to predict by simple theories. Finally, the different components of air are also considered separately in the context of the barometric formula, and it is shown that the known composition changes of the atmosphere are primarily caused by photochemical processes and not by the gravity field.