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Blog: Boltzmann's So-called Constant: k

A New Thermodynamics

By Kent W. Mayhew

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, which was measured by the height of mercury. The height of mercury is 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) 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

 

 

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