An ideal gas in a force field Boltzmann distribution

2 DISTRIBUTION OF MOLECULES OF AN IDEAL GAS IN A FORCE FIELD (BOLTZMANN DISTRIBUTION) 

In an ideal gas, a large number of molecules are considered as noninteracting particles, i.e., the potential energy does not appear in this case. However, in an external force field this situation varies and molecules have a potential energy due to the influence of external forces. The question arises as to how molecules are distributed in this external field. 

In the absence of any external influence, because of chaotic thermal movement, the gas uniformly fills the given volume. If, however, an ideal gas is in an external force field, the distribution of molecules changes it becomes dependent upon the potential energy of the molecules in this field. 

Let us first find the distribution of a molecule concentration of an ideal gas in a uniform one-dimensional external force field. Consider the field acting on the molecules to be conservative and uniform. Choose a unique z-axis coinciding with the direction of a force vector, along which we shall examine the molecules distribution. 

The pressure difference dp is equal to the total force working on all molecules being of volume S dz acting on the plates areas  

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