H.2 Euler-Maclaur in Sum Formula
7.1 Stability Requirements for Entropy
For simplicity, we consider a homogeneous system having entropyS(U,V,N)and assume that constant values ofU, V, and N will guarantee isolation. We first follow Callen [2, p. 203] based on an analysis by Griffiths [20] and pose the question of whether this system is stable with respect to breakup into two homogeneous subsystems, each having a volumeV/2 and number of molesN/2, one having energy(U −U)/2 and the other having energy(U+U)/2. The energy of the combined subsystems is(1/2)(U−U)+ (1/2)(U+U)=U. SinceSis a homogeneous function of degree one in these extensive variables, the corresponding entropies of the subsystems are(1/2)S(U−U,V,N)and
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(1/2)S(U+U,V,N). Therefore, the homogeneous system will be stable with respect to this breakup by an irreversible process if
(1/2)S(U−U,V,N)+(1/2)S(U+U,V,N)≤S(U,V,N). (7.1) This requirement is represented graphically inFigure 7–1. By rewriting the left-hand side of Eq. (7.1) in the form
S(U−U,V,N)+(1/2)[S(U+U,V,N)−S(U−U,V,N)] ≤S(U,V,N), (7.2) we verify that the entropy of the composite system lies on the straight line (chord) joining (1/2)S(U−U,V,N)and(1/2)S(U+U,V,N)at the valueU, midway betweenU−U andU+U. Thus, stability for all values ofUrequiresSto be aconcave functionofU(as viewed from below). Thus, the situation inFigure 7–1a is stable, and that inFigure 7–1b is unstable. The equal sign in Eq. (7.2) would correspond to a situation of neutral stability that would involve a hypothetical reversible process. We will discuss this possibility in Chapters 9 and 10 in connection with phase transformations.
For infinitesimal changesU→δU, we can expand the entropies in Eq. (7.1) to obtain S(U±δU,V,N)=S(U,V,N)±SU(U,V,N)δU+(1/2)SUU(δU)2+ ã ã ã, (7.3) where the subscripts U represent partial differentiation.1Then neglecting terms of the third order and higher, Eq. (7.1) becomes, after division by(δU)2/2,
SUU≡ ∂2S
∂U2
V,N ≤0. (7.4)
S
U U + ΔU U − ΔU
(a)
S
U U + ΔU U − ΔU
(b)
FIGURE 7–1 Conditions forS(U,V,N), represented by the solid curves, for stability (a) or instability (b). To be stable, S(U,V,N)must be a concave function ofUat fixedVandN. A composite system having the same values ofU,V, andNwould have an entropy represented by the intersection of the chord with the vertical line atU. (a) Stable (concave) and (b) Unstable (convex).
1In this chapter, subscripts that indicate partial derivatives imply the natural variable sets for each function, explicitlyS(U,V,N),U(S,V,N),H(S,p,N),F(T,V,N), andG(T,p,N).
Equation (7.4) is a requirement forlocalstability because it corresponds to infinitesimal changes. IfSUU=0, we could examine higher derivatives. For example, we would need SUUU=0 andSUUUU <0, but such a requirement would still be local.
The situation depicted inFigure 7–2is more complicated because the second derivative SUUchanges sign at the so-called spinodal pointsUS1andUS2. The region between points US1andUS2is clearly unstable with respect to infinitesimal variationsδU. The remainder of the curve is stable with respect to infinitesimal variations. The states betweenU1 and US1and betweenUS2, andU2, whereU1andU2are points of common tangency, are more difficult to analyze because the above analysis requires values ofU−UandU+Uthat are symmetrically situated and can span distant portions of the curve.
We therefore resort to the following modified analysis. We represent the entropy, internal energy, and volume per mole by the lower case letterss,u, andv, respectively. The original system hasN moles, entropyS(U,V,N)=Ns(u,v), internal energyNu, and vol- umeNv. We consider breakup onto a composite system consisting of two homogeneous systems, one having(1−f)Nmoles and intensive parametersu1,v,s(u1,v), and the other havingfNmoles and intensive parametersu2,v,s(u2,v), where
1−f = u2−u
u2−u1; f = u−u1
u2−u1. (7.5)
Without loss of generality we take u2 > u1. The volume of the composite system is N(1−f)v+Nfv =Nv=V and its number of moles isN(1−f)+Nf =N. It has energy
N(1−f)u1+Nfu2= N
u2−u1[(u2−u)u1+(u−u1)u2] =Nu=U. (7.6) The entropies of the subsystems are(1−f)Ns(u1,v)andfNs(u2,v). After division byN, the requirement for stability becomes
(1−f)s(u1,v)+fs(u2,v)≤s(u,v), (7.7)
S
US1 US2
U1 U2
FIGURE 7–2 S(U,V,N)versusUunder conditions for which some states are locally stable and others are locally unstable. The states between the spinodal pointsUS1andUS2are locally unstable and states outside these points are locally stable. But states betweenU1andUS1and betweenUS2andU2are globally unstable, so they are metastable.
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which can be rewritten
s(u1,v)+ u−u1
u2−u1[s(u2,v)−s(u1,v)] ≤s(u,v). (7.8) The requirement represented by Eq. (7.8) is shown inFigure 7–3, from which we see that the entropy per mole of the composite system is represented by the intersection of a vertical line atuwith a chord joininganypointss(u1,v)ands(u2,v), as long asu2>u>u1
is satisfied. This criterion shows that the general requirement for stability is concavity of s(u,v)as a function ofuat fixedv. SinceS(U,V,N)=Ns(u,v)=Ns(U/N,V/N), we see for stability thatS(U,V,N)is a concave function ofU at fixedV andN. Thus the states inFigure 7–2betweenU1 andUS1 and betweenUS2andU2, although locally stable, are globally unstable and are termed metastable. By lettingu1 = u−δu,u2 = u+δuand expanding Eq. (7.8) for smallδu, one obtains∂2s/∂u2 ≤ 0 as a local stability condition, consistent with Eq. (7.4).
Returning to the general analysis ofS(U,V,N), we can inquire about stability against breakup into two homogeneous subsystems, each having the same energyU/2 and mole numbersN/2, but different volumes(V −V)/2 and(V+V)/2. By the same reasoning as above, stability requires
(1/2)S(U,V−V,N)+(1/2)S(U,V+U,N)≤S(U,V,N). (7.9) For infinitesimal changesδV
SVV ≡ ∂2S
∂V2
U,N
≤0. (7.10)
The same reasoning applies to changes ofNor to any other extensive variables on which Scould depend.
s
u u2
u1
FIGURE 7–3 s(u,v)versusuunder conditions for which some states are locally stable and others are locally unstable.
At constantv, we test the state atuagainst breakup into a composite system consisting of states having molar energiesu1andu2that are not equidistant fromu. The entropy per mole of the composite system lies on the straight line at positionuand exceedss(u,v)which lies on the curve. Therefore, the state atuis globally unstable, even though it is locally stable.
If bothUandVare different for the subsystems, we obtain
(1/2)S(U−U,V−V,N)+(1/2)S(U+U,V+U,N)≤S(U,V,N). (7.11) For infinitesimal changes inUandV, Eq. (7.11) becomes
SUU(δU)2+2SUVδUδV+SVV(δV)2≤0, (7.12) where the derivatives are evaluated atU,V,N. Testing Eq. (7.12) forδV = 0 orδU = 0 recoversSUU ≤ 0 andSVV ≤ 0 as above. But for generalδV andδU, a new condition emerges. We can write Eq. (7.12) in the matrix form
δU δV SUU SUV
SUV SVV δU δV
≤0, (7.13)
which involves a real symmetric matrix that can be diagonalized. Its eigenvaluesλsatisfy det
SUU−λ SUV SUV SVV −λ
=0, (7.14)
which leads to a quadratic equation with roots λ±=SUU+SVV
2 ±
SUU+SVV 2
2
+SUV2 −SUUSVV (7.15)
=SUU+SVV
2 ±
SUU−SVV 2
2
+SUV2 .
From the second form, we see that both roots are real, which is a general property for the eigenvalues of any real symmetric matrix. From the first form, and recalling thatSUU ≤0 andSVV ≤0, we see that there are no positive roots provided that
SUUSVV−S2UV ≥0. (7.16)
After diagonalization, Eq. (7.13) can be rewritten in the form
λ+(δX1)2+λ−(δX2)2≤0, (7.17) whereλ± ≤ 0 andδX1andδX2are linear combinations ofδUandδV that can be found by calculating the eigenvectors of the matrix. Thus,SUU ≤ 0 andSVV ≤ 0 together with Eq. (7.16) guarantee that Eq. (7.12) is satisfied.2They insurelocallythat the surfaceSwill not lie above its local tangent plane. Callen [2, p. 206] refers to Eq. (7.16) as a “fluting condition.”
By a procedure similar to that used to derive Eq. (7.8), we can test a system with entropy Ns(u,v) with respect to breakup into a composite of three systems having entropies Nf1s(u1,v1),Nf2s(u2,v2), andNf3s(u3,v3), wheref1,f2, andf3 are positive fractions that sum to unity, chosen such that total energy and total volume are conserved. This leads to a stability criterion of the form
f1(u,v)s(u1,v1)+f2(u,v)s(u2,v2)+f3(u,v)s(u3,v3)≤s(u,v), (7.18)
2For an alternative procedure that would lead to Eq. (7.16), seeSection 7.2.
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where thefisatisfy the following linear equations:
f1+f2+f3 =1, f1u1+f2u2+f3u3=u, f1v1+f2v2+f3v3 =v.
(7.19) We could use Cramer’s rule to solve Eq. (7.19) by means of determinants, but the actual expressions are cumbersome and not needed as long as we note the following properties.
A solution is only possible if the determinant of the coefficients of thefiis not zero, which will be true if the points(u1,v1),(u2,v2), and(u3,v3)lie at the vertices of a non-degenerate triangle in theu,vplane. We shall refer to these vertices as 1, 2, and 3, respectively, in which case that determinant is equal to 2A123, whereA123>0 is the area of that triangle. As shown below, the point(u,v)wheres(u,v)is to be tested for stability must be chosen within or on that triangle. With(u1,v1),(u2,v2), and(u3,v3)fixed, thefiwill be linear functions ofuand v that can be written in the formfi(u,v), as already indicated in Eq. (7.19); furthermore, they will satisfy
fi(uj,vj)=δij; fi(u0,v0)=A0jk/A123, (7.20) whereδijis the Kronecker delta,i,j,k, are cyclic permutations of 123, and the quantities A0jk are areas of triangles defined below. The first member of Eq. (7.20) follows from Cramer’s rule because the determinant of a matrix having two identical columns is zero. If the point(u0,v0)is referred to as point zero, Cramer’s rule can also be used to show that A0jkis the area of triangle 0jk. Consistent withA123 >0, the areasA0jk ≥0 are positive as long as(u0,v0)lies inside or on triangle 123. If(u0,v0)were to lie outside triangle 123, at least one of thefiwill be negative, which is unacceptable. See Figure 8–11 that pertains to an isomorphous problem.
From these properties of thefi(u,v), it follows that the left-hand side of Eq. (7.18) rep- resents a plane that passes through the pointss(u1,v1),s(u2,v2), ands(u3,v3). Therefore, geometrically, the global stability criterion represented by Eq. (7.18) states that s(u,v) lies above or onany such plane. In other words, for stabilitys(u,v)must be a concave functionofuandv. Ifs(u,v)violates Eq. (7.18) for any such plane, that state will be globally unstable, but would be locally stable if Eq. (7.12) were satisfied.