At 12:50 AM 2001/06/23 +0200, Falk Koenemann wrote:
>...
>The dimensions of a physical term do not tell you which function it follows.
>This we find from considering boundary conditions or other constraints; this
>discussion is very much about finding out which are the proper constraints.
>...
>At any given scale you have the choice (a) to consider some given mass with a
>given surface, then to consider a part of that surface _at constant mass_; in
>this case f/A holds. Or (b) you consider a smaller mass with a smaller
surface
>which is then the _entire_ surface of the mass; in that case f/A does not
>hold. ...
With trepidation, I swallow the bait that Rob Twiss has declined.:-) Here
are the data Falk used to illustrate case (b) for a spherical system with
decreasing volume and mass at constant pressure:
>...
>Assume: U/V = 5 is scale-independent. Calculate f and f/A.
>From the divergence theorem it follows that f and r are proportional
>if V is allowed to approach zero at constant P.
>
> P = U/V V U r A f f/r f/A
> 5 1,0 5,0 0,239 0,716 3,581 15,00 5,00
> 5 0,9 4,5 0,230 0,668 3,581 5,36
> 5 0,8 4,0 0,222 0,617 3,457 5,60
> 5 0,7 3,5 0,212 0,565 3,324 5,89
> 5 0,6 3,0 0,201 0,509 3,180 6,24
> 5 0,5 2,5 0,189 0,451 3,020 6,69
> 5 0,4 2,0 0,176 0,389 2,842 7,31
> 5 0,3 1,5 0,160 0,321 2,638 8,22
> 5 0,2 1,0 0,140 0,245 2,397 9,79
> 5 0,1 0,5 0,111 0,154 2,094 13,57
>
>Hence f/A approaches infinity as V approaches zero.
>It follows that f/A and U/V are not equivalent if A is a closed surface.
>...
Accepting the values in the V column, the values in the "r" column appear
to be r^3 rather than r, and this affects the values in the A column.
Correcting these values and working backwards from my contention that P =
f/A holds for every equilibrium state, two possibilities appear: (1) the
divergence theorem is not consistent with the first law of thermodynamics,
or (2) the divergence theorem stipulates that f and r^2 are proportional
(rather than f and r):
P = U/V V U r A f f/r^2 f/A
5 1,0 5,0 0,62 4,836 24,18 62,832 5
5 0,9 4,5 0,599 4,508 22,54 5
5 0,8 4,0 0,576 4,168 20,838 5
5 0,7 3,5 0,551 3,813 19,063 5
5 0,6 3,0 0,523 3,44 17,201 5
5 0,5 2,5 0,492 3,046 15,232 5
5 0,4 2,0 0,457 2,625 13,127 5
5 0,3 1,5 0,415 2,167 10,836 5
5 0,2 1,0 0,363 1,654 8,269 5
5 0,1 0,5 0,288 1,042 5,209 5
Dugald M Carmichael Phone/V-mail: 613-533-6182
Dept of Geological Sciences and Geological Engineering
Queen's University FAX: 613-533-6592
Kingston ON K7L3N6 E-mail: [log in to unmask]
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