Boltzmann’s So-Called Constant & Work
Consider a unit cube whose volume is “V” with surface area:” A”. If “M” represents the mass of overlying atmosphere, and “g” is gravitational constant. Then the pressure exerted by the Earth’s atmosphere on the top surface of the unit cube is:
Accordingly, eqn 1) can be rewritten:
Remember for a unit cube, then: V/A=h, wherein: “h” is the height of the unit cube. Therefore we can rewrite 3) as
Limit the volume change to only vertical expansion, i.e. along the y-axis and then differentiating both sides, we obtain the change in temperature with height as:
Eqn 7) similarly implies that Boltzmann’s constant (k) defines on a per molecule basis the proportionality for the work required to displace the overlying Earth’s atmosphere by a height of dh, per degree of temperature change.
The above means that Boltzmann’s constant is a function of the Earth’s gravitational field which is to say that it is only really a constant here on Earth. When universally applied it is only a so-called constant.
Of course for a mole of gaseous molecules what is said of Boltzmann's constant equally applies to the ideal ,gas constant (R)
Copyright Kent W. Mayhew