Thermodynamics of an lncompressible Solid and a Thermodynamic Functional Determinant R. J. Tykodi Southeastern Massachusetts University, North Dartmouth, Massachusetts 02747 Thermodynamics of an Incompressible Solid I n thermodynamic analysis, we often decide to treat a solid a s incompressible for simplicity. I t is useful to show just exactly what thermodynamic consequences we lock ourselves into when we do this. The incompressible solid is an especially simple thermodynamic system, but it does have afeature or twoofgeneral interest. Take a defmite amount of solid material (100 g of copper or 100 g of glass or whatever) and shape it into a solid slab. Place the slab in an environment of temperature T and uniform hydrostatic pressure p and treat it a s a closed thermodynamic system. Now increase (conceptually) the rigidity of the solid material until it becomes absolutely incompressible, that is, let (aVIJp), +0. Next, carry out the following change in state for the incompressible material
As any standard textbook in thermodynamics will show dG=SdT+Vdp
(2)
PZ
AG
=IVdp
= V(pZ-pl)
P1
(3)
Aithough the incompres.iihle material experiences no dcformation, it does undergo - a change - in Gihbsenerw -- for the process. We shall see, however, that for the process of eq I
posed of atoms with a structure composed of a regular (e.g., copper) or irregular (e.g., glass) lattice of interactive units. I shall refer to the whole argument a s the microsco~ic argument: Ifthrre is nochangein the milieu of an intehor lattse unit for a solid composed of Interactive lattice units. then there is no change inihose thermodynamic of the solid that depend on the volume, that is, no change in U,S , a n d A for the solid. Now we wme to a n interesting feature of the analysis: Incompressibility implies thermal inexpansibility.
We can see this in the following way. From the expression for dG (eq 2) we get Maxwell's equation
By the microscopic argument, it follows that (aSiapi, + 0 for a n incompressible solid. Thus, for a n incompressible solid, (aVlap), + 0 implies (JSlap)~+ 0 which in turn implies (aVlaTl,+ 0. The atomic nature of the microscopic argument is foreign to classical thermodynamics. I t would thus be satisfying to find a purely macroscopic thermodynamic basis for eq 4. Such a basis will appear in the second major section of this paper. We are now ready for a summary of the necessary thermodynamic characteristics of a n incompressible solid. For an incompressible solid (avlap),
=0
(avian,
=
o
V = constant
Cp=C,~C
Thus,
The fundamental differential forms are If the solid material is truly incompressible, an observer inside the solid material would find the lwal scene unaffected hy the change in state of eq 1.For example, distances from the observer to nearby lattice units would remain the same. The mean amplitude of vibration of those nearby units about their lattice points would remain the same, etc. Since the internal structure of the solid material is unaffected by the process of eq 1, the internal energy, the entropy, and thus the Helmholtz energy should remain constant during the change of state shown in eq 1. The expressions dl3 and dG, however, each contain a V d p contribution, so there are changes in H and G associated with the process of eq 1. The preceding observations assume a solid material com'Modell, M.; Reid, R. C. Thermodynamics and its Applications; Prentice-Hall: Englewood Cliffs, NJ. 1974; Section 7.4. 20sgood,W. F. Advanced Calculus; Macmillan: New York, 1925; pp 178-1 79.
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dU = CdT dS = (C1T)dT d.4 = S d T dH=CdT+Vdp dG=-SdT+Vdp
(10) (11) (12)
(13) (14)
Equations 6 1 4 are then the thermodynamic consequences we lock ourselves into when postulating incompressibility for a solid material. There is a n alternative route to eq 4. A Therrnodvnarnic Functional Determinant The usual thermodynamic conditions of stabihty Cv > 0, (31';fp,,< 0, etc., lncludr restrictionson some ofthe secondorder partial derivatives of appropriate thermodynamic potentials.' The conditions of stability are, in fact, consequences of t h e extremum (maximum or minimum) chracteristics of the thermodynamic potentials in terms of their preferred sets of variables.',' I shall use considera-
tions based on the stability-conditions restrictions on appropriate second-order partial derivatives to show eq 4 independently of the microscopic argument. For a function of two variables, F(Xi, X2), with continuous second derivatives, consider the functional determinant D(F):
-,
a"F a"F
D(q
ax2ax1
- 2%' 2% -
a 2 ~ a 2 ~ ax: ax: ax,& ax;
(1 ax2ax, &?
= V(26.2 x
where .32FiaX2.3Xl = a2FlaXlaxz.We m u s t investigate whether D O always has one sign when the function F is one of the thermodynamic potentials U, S , A, H, or G, expressed in its preferred set of variables for a closed one-phase system. I n other words, a s the representative point for the closed one-phase system moves about in the thermodynamic state-space appropriate to the thermodynamic potential F (i.e., U, S, A, H, or G), does D(F) always have the same sign (perhaps going to zero a t one or more singular points)? I n answering the question posed, we shall provide ourselves with the macroscopic tools needed for estab1ishi.g eq 4. We shall find that when F = U, S, A, or H, the values o f D O have only one sign. When F = G, the values ofD(G) will have to be investigated. For U = U(S,V) and for S = S(U,V)
lo-"-
13.6x lo-") > 0
(21)
I n the neighborhood of 0 K, a s T -+ 0, D(G) -to*
because CpIT+O and(JV/Jr),=-(&Slap)?+ Om T - t 0 So, if D(G) is to be of one sign only, then it must satisfy a n equation such as
H. B. CalIen4.5has
shown that [G = G(T,p ) and U =
U(S,V)I J2G -=--
a?
$G -=---
because a t equilibrium U is a minimum (JzU/aSz > 0) for constant S and V, and S is a maximum (a2S/aU2< 0) for constant U and V.',2 For H = H(S, p ) and D(fC < 0
JpJT a2G -=--
1
Dm
a2u av2
1 8% D(u) JIGS
(24) (25)
1
Jp2 D m
as2
(26)
and thus 1 D(G)= D(U)'O
(27)
for all substances. In other words, the matrix that has D(U) a s its determinant and the matrix that has D(G) a s its determinant are inverses of each other, and thus eq 27 holds. I summarize what we have learned about D(F) so far. because (ap/aV), = 1/(aViap), < 0. For G = G(T, p), D(G) needs exploring.
-
where a (l/V)(aVldT)p and p E -(lIV)(aVIap),. Since i t is not immediately clear whether D(G) is of fured sign or not, let us look a t some special cases. For a n ideal gas: pV = nRT, a = UT, p = llp, and
because Cp 2 (512)nR for a n ideal gas. For benzene3a t 1atm and 20 'C
For all substances,D(U)> 0 and D(S)> 0. For any substance,neitherDWl nor D(S)ever changes sign,
although each may possibly go to zera at a singular point. For all substances,D(H)< 0 and D(A) < 0. Forany substance,neither D(H)norD(A)everchanges sign, although each may possibly go to zera at a singular point. For all substances by Callen's r e s ~ l t sD(G) ~ . ~ 2 0, and for any substance, D(G)never changes sign, although D(G) may possibly go to zero at a singular point (such as 0 K). This is what we need for a macroscopic approach to eq 4. For a n incompressible solid, wemay play offD(G)and the
%can. J. A,, Ed. Langes Handbook of Chemistry, 12th ed.; McGraw-Hill: New York. 1979. %a en, 4 B Tnermodynarnrcs.W ey hew York, 1960,pp 362363, eqs (G16) and (G 17) There are typograpn cal errors n ooln e uations. 'Callen. ti. B. Thermodynamics and an introduction to Thermostatistics, 2nd ed.; Wiley: New York, 1985; Section 8.2 and problem 8.2-3. (Thepublisher's list of errata showscorrectionsforpp 203-2101. Volume 68 Number 10 October 1991
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microscopic argument against each other. By the microscopic argument, we can say that for a n incompressible solid, D(G)= 0 and each state of an incompressible solid is a singular state with ( a v l a p )=~ O, @via% = 0, and D(G) = 0.On the other hand, since eq 23 is generally ~ a l i d ,it~can .~ supply a macroscopic thermodynamic underpinning for the microscopic argument: as (aviaph. + 0, so that D(G) not change sign, it follows that ( a v i a n , -t 0 also. Conclusion We have looked a t a rather simple thermodynamic system: the incompressible solid. We found that incompress-
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ibility implies thermal inexpansibility as well. We first demonstrated the connection between incompressibility and thermal inexpansibility by the microscopic argument. seeking a macroscopic thermodynamic that would do the same work as the microscopic argument, we were led to consider the hnctional determinant D O for F = U, S, A, H, G; and, we came upon the little-known general thermodynamic inequality D(G) 2 0. Thus, the idealized model of the incompressible solid has brought us in contact with some equations of a very general nature.
