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Re: Entropy and states of matter



Regarding Don Polvani's question:

Thanks to David Bowman for his excellent post, but I am puzzled by the
difference between his requirement 1 (infinitely divisible) and requirement
2 (extensive). What does "extensive" mean here and why isn't this
requirement satisfied if the first requirement is satisfied?

These requirements are not necessarily intended to be mutually
orthogonal or mutually disjoint. It is true that the first 2
conditions are closely related (as it is also true that the 3rd & 4th
conditions are also closely related). In like manner one could also
argue that Newton's 1st law is already included in his 2nd one as the
special case of zero net force and zero acceleration.

Although being extensive (i.e. piecewise sumable over the various
bulk parts of a region of interest so that the total value of the
quantity scales directly proportional to the volume of the region
when that region has a uniform density of the quantity) pretty much
already entails the concept of infinite divisibility, it is not
necessarily the case that mere infinite divisibility also entails
extensiveness. Extensiveness goes beyond mere infinite divisibility.
For instance, consider the surface energy of each of the various
crystalline faces of a crystalline solid. If the solid is
macroscopic in size, not only is its internal bulk energy extensive
and approximately infinitely divisible, its facial surface energies
are also approximately infinitely divisible over partitions of the
faces into multiple subsets. But the overall surface energy of the
crystal is not an extensive quantity because the surface energy
scales proportional to the 2/3 power of the volume of the crystal,
rather than the 1st power as it would if it was truly extensive.

David Bowman
David_Bowman@georgetowncollege.edu