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Blog: Boltzmann's So-called Constant: k

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 and truly a constant here on
Earth. When universally applied it is only a so-called constant at best. This further alienates our new thermodynamics from
tradiational 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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Sommerfield quote:"Thermodynamics is a funny subject. The first time you go through it, you don't understand it at all. The second time you go through it, you think you understand it, except for one or two small points. The third time you go through it, you know you don't understand it, but by that time you are so used to it, so it doesn't bother you any more."

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