H.2 Euler-Maclaur in Sum Formula
7.3 Stability Requirements for Other Potentials
We can also obtain stability requirements for other potentials, such asH,F, andG, which are Legendre transforms ofU. An important distinction arises, however, because some of the natural variables on which these functions depend are intensive.
7.3.1 Enthalpy
For the enthalpyH(S,p,N), stability requires
(1/2)H(S−S,p,N)+(1/2)H(S+S,p,N)≥H(S,p,N). (7.29) For infinitesimal changesδS, the local stability requirement is
HSS:= ∂2H
∂S2
p,N≥0. (7.30)
But there is no equation analogous to Eq. (7.29) involving changes p because p is intensive and therefore must be the same in each member of the composite system that we compare toH(S,p,N). We therefore deduce an inequality forHppby relating to a partial derivative of its Legendre transformU. As shown in Section 5.5, we have
Hpp:= ∂2H
∂p2
S,N
= − 1
UVV ≤0. (7.31)
Thus for local stability, H is a locally convex function of the extensive variableS but a locally concave function of the intensive variablep. As a result of this, the fluting condition HSSHpp−HSp2 ≤0 is true by default because both terms are non-positive. The fact that this inequality has the correct sense can also be seen as follows. We suppressNfor simplicity of notation. Then(∂U/∂S)V =T =(∂H/∂S)p, so
∂2U
∂S2
V
= ∂
∂S ∂H
∂S
p
V
=HSS+HSp ∂p
∂S
V
. (7.32)
But
∂p
∂S
V
= −(∂V/∂S)p
∂V/∂p
S
= −HSp
Hpp
. (7.33)
Therefore
USS=HSSHpp−HSp2 Hpp
. (7.34)
SinceUSS≥0,HSS≥0, andHpp ≤0, we see consistently thatHSSHpp−HSp2 ≤0.
In a similar manner, we can show that
HSS=USSUVV−USV2 UVV = D
UVV, (7.35)
so the fact thatD≥0 could have been deduced fromHSS ≥0 andUVV ≥0. It is generally the case that all fluting conditions can be deduced from conditions on non-mixed second derivatives provided that appropriate Legendre transforms are considered.
7.3.2 Helmholtz Free Energy
For the Helmholtz free energyF(T,V,N), we have an equation analogous to Eq. (7.29) but involvingV and this leads directly to the local requirementFVV ≥ 0. We also have FTT = −1/USS ≤ 0. So for local stability,Fis a locally convex function of the extensive variableV and a locally concave function of the intensive variableT. By methods similar to those discussed for the enthalpy, we have the local stability requirement
UVV =FVVFTT−FVT2 FTT
(7.36) soFVVFTT−FVT2 ≤0, which is no contest becauseFTT ≤0 so both terms are non-positive.
We also have
FVV = D
USS ≥0, (7.37)
another redundancy.
7.3.3 Gibbs Free Energy
For the Gibbs free energy G(T,p,N), both T and p are intensive, so local stability re- quirements involving their derivatives must be obtained indirectly from their Legendre transforms. We haveGTT = −1/HSS ≤ 0 andGpp = −1/FVV ≤ 0 as anticipated for both principal second partial derivatives with respect to intensive variables. In this case, the fluting condition is not trivial. It is most easily related to derivatives ofForH, which differ from it by a single Legendre transform. Thus we can use either
FTT= GTTGpp−G2Tp Gpp
(7.38) or
Hpp=GTTGpp−G2Tp GTT
, (7.39)
104 THERMAL PHYSICS
either of which shows that
GTTGpp−GTp2 ≥0. (7.40)
A somewhat more involved calculation3shows thatGpp = −USS/D,GTT = −UVV/D, and GTp= −USV/Dwhich results in
GTTGpp−G2Tp= 1
USSUVV−USV2 ≥0, (7.41)
so the two non-trivial fluting conditions are just reciprocals of one another.
7.3.4 Summary of Stability Requirements
By means similar to those discussed above, we can extend the stability requirements to any number of variables. For stability of a homogeneous system:
• The entropy,S, must be a concave function of its natural extensive variables.
• The internal energy,U, must be a convex function of its natural extensive variables.
• Legendre transforms of U, such as H,F, and G, must be convex functions of their natural extensive variables and concave functions of their natural intensive variables.
We did not discuss the Massieu functions, which are Legendre transforms of the entropy, but they must be concave functions of their extensive variables and convex functions of their intensive variables.
Fluting conditions involve mixed partial derivatives, but are always redundant with requirements on non-mixed second partial derivatives ofS,U, or some Legendre trans- form ofU.
It is possible to consider thermodynamic functions, perhaps derived from some model, for which the requirements for local stability are true for some range of variables but for which the requirements for global stability are violated. Such situations can occur when different phases of a composite system are in equilibrium but in which phase transitions can occur. We shall illustrate this in Chapter 9 by means of the van der Waals model.
In applying the above requirements, it is extremely important to note that they only apply to the extensive thermodynamic functions and the natural variables, extensive and intensive, on which they depend. Moreover, if one uses a “density” of some extensive variable, such as the Helmholtz free energy per mole,f = F/N, one finds that df = −s dT −pdv wherev = V/N is also a “density,” namely the volume per mole. Althoughf andv are certainly intensive, they still behave from the point of view of stability like the extensive variablesF andV from which they originate. In other words,
∂2f/∂v2
T ≥ 0 for local stability, corresponding to f being a convex function ofv, just asFis a convex function ofV. ButTis not a “density” so
∂2f/∂T2
v ≤ 0 for local stability, meaning that f is a concave function ofT. This peculiarity arises because the local stability condition
3For instance,GTT= −1/(∂US/∂S)p,(∂US/∂S)p=USS+USV(∂V/∂S)p, and(∂V/∂S)p= −UVS/UVV.
for an intensive variable such asTis derived from a Legendre transformation, rather than splitting a system into parts having different values ofT, as was done forV.