Barometric formula

For constant gravitational acceleration and temperature,

For a dilute gas with a floor at $z=0$, constant $g$, uniform cross-section and no relevant ceiling, $P(z)=\beta mg{e}^{-\beta mgz}$. Its mean gravitational energy is $k_{B}T$, and $P(z>h)=e^{-\beta mgh}$. A linear potential contributes $k_{B}T$, not the $k_{B}T/2$ of a quadratic degree of freedom.

This is the constant scale height of an isothermal atmosphere. The pressure and density decay length in an isothermal atmosphere is $H=k_{B}T/(mg)$.