Boltzmann’s So-Called Constant & Work

Consider some constant volume thermometer where the temperature change is measured by the pressure change of the ideal gas in the thermometer’s glass bulb, as measured by the height of mercury in a tube. The height of mercury is equated to the displacement of its mass against gravity, which is really what work is all about. Although we are now starting to entertain circular arguments the point remains, if we know how many gaseous molecules (N) are in the thermometer’s glass bulb, and the pressure (P), volume (V), and temperature (T), then by rearranging the ideal gas law (PV=NkT), k can be calculated:

      k=PV/NT             1)

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:

    P=Mg/A       2)

Accordingly, eqn 1) can be rewritten:

k=(Mg/A)(V/NT)   3)

Remember for a unit cube, then: V/A=h, wherein: “h” is the height of the unit cube. Therefore we can rewrite 3) as

          k=(Mg/T)(h/N)     4)                      

Which becomes:

kTN=Mgh          5)                  

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:

   NkdT=Mgdh       6)                                   

 

Thus:

  k=(Mg/N)(dh/dT)      7)                      

 

Eqn 7) simiply implies that Boltzmann’s constant (k) defines on a per molecule basis, the proportionality for the work by an ideal gas that is required to displace the overlying Earth’s atmosphere by a height of dh, per degree of temperature change.

The above implies that Boltzmann’s constant (k) is a function of the Earth’s gravitational field, which is to say that it is only really and truly only a constant here on Earth. When universally applied it is only a so-called constant at best, one too often misused in sunjects like cosmology. This further alienates our new thermodynamics from traditional considerations. 

 

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

 

 

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