Dugald Carmichael schrieb:
Hi Dugald,
I was out of town for a few days.
> I would argue that the two definitions of pressure, P = f/A and P = dU/dV,
> are fully compatible and equally
> valid. P = dU/dV may be theoretically more fundamental, but in practice
> only P and V are directly measureable, and devices that measure P depend on
> P = f/A. Only by measuring/monitoring/controlling P and V can we
> experimentally keep track of changes in U (i.e., dU = PdV).
Dugal's comment contains two problems: equivalence of the two definitions of P,
and measurability.
1. Equivalence of f/A and dU/dV. Assume shape of V is spherical.
Assume: f = 5, f/A = P is scale-independent.
Assume that f/A and U/V are equivalent for the case r = 1.
Calculate U/V and f as V approaches zero at constant P.
r V A P = f/A U/V f = PA f/r
1,0 4,189 12,566 5 1,19 62,83 62,83
0,9 3,054 9,161 1,64 45,80 50,89
0,8 2,145 6,434 2,33 32,17 40,21
0,7 1,437 4,310 3,48 21,55 30,79
0,6 0,905 2,714 5,53 13,57 22,62
0,5 0,524 1,571 9,55 7,85 15,71
0,4 0,268 0,804 18,65 4,02 10,05
0,3 0,113 0,339 44,21 1,70 5,65
0,2 0,034 0,101 149,21 0,50 2,51
0,1 0,004 0,013 1193,66 0,06 0,63
Hence U/V is scale-dependent. This is physically impossible.
Furthermore, as V vanishes, so does f, but f vanishes faster than r.
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.
I can send the Excel-file which I used for calculations if desired.
2. Measurability. Some standard unit volume must be assumed to which the
measurements refer. This may be the volume of some unit mass, or a unit volume
in the standard state. Then V is no longer a variable, and then f/A works.
The problem is also present in the theory of gases: in the kinetics of gases the
reference volume is a volume not in Euclidean space, but in a velocity space.
The volume must be proportional to the average velocity of the gas atoms, ie. it
is a natural property which cannot be fiddled with.
> >By assuming a priori that your unit volume is a cube, you thereby constrain
> >the shape so that you have identical r in the principal directions. In
> >effect you give away one degree of freedom in continuum mechanics - which
> >you do have in discrete mechanics; but why is this done? What is the
> >theoretical justification?
> >...
>
> A thermodynamic system is any part of the universe we choose to consider.
> Let our system be a one-unit-of-volume part of a large solid body in a
> state of constant, uniform, homogeneous elastic strain. I would contend
> that the shape of the system is arbitrary.
Potential theory says: the shape can be arbitrary if the system contains n
discrete bodies such that the surface of the system does not pass through mass.
But this implies that the system does not interact with its surrounding, in fact
a surrounding may not be existent (as in the case of a planetary system). Any
fluxes (gravity, heat, light) are due to fields which have radius-parallel
components (ie. normal components) only.
The shape of the system is not arbitrary if rotational/shear forces can do
mechanical work. There is no shape-independent equilibrium condition for the
rotational momentum in Newtonian mechanics.
If we consider a thermodynamic system within a solid, ie. within a larger volume
that is continuously filled with mass, we get one more problem: not only does
the system surface run through mass - it does so too in an ideal gas - but in
addition, system and surrounding are permanently bonded to one another. This
point is of utmost importance for the equilibrium conditions; however it can
only be considered if the existence of bonds is indeed taken into account, which
is not done in continuum mechanics.
A standard assumption in potential theory is that the shape of the system is
simply connnected (any loop within the system may approach zero without
touching the surface; this condition certifies that the system does not have
holes) and convex (the surface has only one curvature). Furthermore, the surface
must not have cusps (places where the surface runs from opposite sides into the
same tangent plane because in such a case one side's inside is the other's
outside). These conditions are necessary under all conditions. In general, they
imply that any two points within the system can be connected through a line that
must not touch or penetrate the surface.
A condition not necessary, but often made, is that the surface may not have
discontinuities (corners), ie. the function describing the surface must be twice
differentiable.
Thus I concede that the shape of the system may be a cube, but it is certainly
not arbitrary. And whether it is permitted to be a cube is a question that
touches the rotational equilibrium condition f x r.
Falk Koenemann
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