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Units & Dimensions

I have borrowed from several authors the convention of expressing the ENTROPY  $\sigma$  in explicitly dimensionless form [the logarithm of a pure number is another pure number]. By the same token, the simple definition of TEMPERATURE  $\tau$  given by Eq. (10) automatically gives  $\tau$  dimensions of energy, just like  U. Thus  $\tau$  can be measured in joules or ergs or other more esoteric units like electron-volts; but we are accustomed to measuring TEMPERATURE in other, less ``physical'' units called degrees. What gives?

The story of how temperature units got invented is fascinating and sometimes hilarious; suffice it (for now) to say that these units were invented before anyone knew what temperature really was!^15.18 There are two types of ``degrees'' in common use: Fahrenheit degrees<sup>15.19</sup> and Celsius degrees (written $^\circ$C) which are moderately sensible in that the interval between the freezing point of water (0$^\circ$C) and the boiling point of water (100$^\circ$C) is divided up into 100 equal ``degrees'' [hence the alternate name ``Centigrade'']. However, in Physics there are only one kind of ``degrees'' in which we measure temperature: degrees absolute or ``Kelvin''<sup>15.20</sup> which are written just ``K'' without any $^\circ$ symbol. One K is the same size as one $^\circ$C, but the zero of the Kelvin scale is at absolute zero, the coldest temperature possible, which is itself an interesting concept. The freezing temperature of water is at 273.15 K, so to convert $^\circ$C into K you just add 273.15 degrees. Temperature measured in K is always written  T.

What relationship does  $\tau$  bear to  T?  The latter had been invented long before the development of Statistical Mechanics and the explanation of what temperature really was; but these clumsy units never go away once people have gotten used to them. The two types of units must, of course, differ by some constant conversion factor. The factor in this case is  $\kB$,   BOLTZMANN'S CONSTANT:

$$\tau \; \equiv \; \kB \, T \quad \hbox{\rm where}$$

$$\kB \; \equiv \; 1.38066 \times 10^{-23} \hbox{\rm ~J/K}$$ (15.12)

By the same token, the ``conventional entropy''  S  defined by the relationship

$${1 \over T} \; = \; {\partial S \over \partial U}$$ (15.13)

must differ from our dimensionless version  $\sigma$  by the same conversion factor:

$$S \; \equiv \; \kB \, \sigma$$ (15.14)

This equivalence completes the definition of the mysterious entities of classical thermodynamics in terms of the simple ``mechanical'' paradigms of Statistical Mechanics. I will continue to use  $\sigma$  and  $\tau$  here.