chapter 14: thermodynamic properties - berkeley · pdf filechapter 14: thermodynamic...

Notes on the Thermodynamics of Solids J.W. Morris, Jr.; Fall, 2008

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C h a p t e r C h a p t e r 1 41 4 :: T h e r m o d y n a m i c P r o p e r t i e s T h e r m o d y n a m i c P r o p e r t i e s Chapter 14: Thermodynamic Properties ......................................................................247

14.1 Introduction...............................................................................................247 14.2 Functional Relations: The Maxwell Relations..........................................250 14.3 Functional Relations between Properties..................................................251 14.4 Jacobian Determinants ..............................................................................253 14.5 Properties of a One-Component Fluid ......................................................258 14.7 The Local Condition of Stability ..............................................................261 14.8 Eigenvalues of the Property Matrix ..........................................................262

14.8.1 Coordinate transformations in state space ..................................262 14.8.2 The diagonal coordinates: eigenvectors......................................264

14.9 Stability Constraints on the Material Properties .......................................266 14.10 Local Stability of Interacting Systems.....................................................268 14.11 Behavior at the Stability Limit.................................................................270 14.12 Thermodynamic Properties at Low Temperature ....................................271 14.13 LeChatelier's Principle .............................................................................272

14.1 INTRODUCTION The equilibrium thermodynamic properties of a material are the quantities that de-termine the variation of the forces, {p}, with the geometric coordinates, {u}. Mathe-matically, they are the second and higher derivatives of its fundamental equation. Con-sider, for example, the force pk. The energy function determines the constitutive equation for pk:

pk =

∆¡E

∆uk = p¶

k({u}) 14.1

Assuming the continuity of the constitutive equation the value of pk in the state specified by a particular set of values of the coordinates {u} can be expressed in terms of its value, p0

k , when the coordinates have the reference values, {u0}, by the Taylor expansion:

pk = p0k +∑

i

∆p¶

k∆ui

∂ui + 12∑

ij

∆2p¶

k∆ui∆uj

∂ui∂uj + ... 14.2

where the derivatives are evaluated for the state {u0}, and {∂u} = {u-u0}. The coefficients that appear in the second term on the right hand side of equation 14.2 are the first partial derivatives of the constitutive equation, and are the first-order thermodynamic


48

properties, those in the third term are the second-order properties, etc. They are simple partial derivatives of the energy function:

∆p¶

k∆ui

= ∆

∆ui

∆¡E

∆uk =

∆2¡E

∆ui∆uk 14.3

∆2p¶

k∆ui∆uj

= ∆2

∆ui∆uj

∆¡E

∆uk =

∆3¡E

∆ui∆uj∆uk 14.4

and hence obey constitutive equations of the form ƒ = ƒ¶[{u}] 14.5 that are derived from the energy function. Unlike the forces and geometric coordinates, the thermodynamic properties cannot be controlled experimentally; they are properties of the system, or material properties. If there are n constitutive coordinates the energy func-tion determines n2 first order properties. Given the results of the previous chapter, we could equally well have determined the constitutive equations relating the forces and geometric quantities from any of the other valid thermodynamic potentials. These give new sets of thermodynamic properties that are defined with respect to alternate sets of operational variables. Given the operational variables {p1,...,pr,ur+1,...,un}, where p1,...,pr are the forces conjugate to the first r members of the set {u}, the appropriate thermodynamic potential is

Ïr = ¡Ïr[{u}',{p}'] = E - ∑k=1

r pkuk 14.6

The constitutive equations for the dependent forces and geometric coordinates are:

uk = u¶k [{u}',{p}'] = -

∆¡Ï

∆pk (k = 1,...,r) 14.7

pk = p¶k [{u}',{p}'] =

∆¡Ï

∆uk (k = r+1,...,n) 14.8

The potential ¡Ï determines n2 first-order thermodynamic properties by its second deriva-tives with respect to the variables {p1,...,pr,ur+1,...,un}. The fact that there are alternate sets of thermodynamic properties associated with different choices of the governing potential is familiar from the elementary thermody-namics of fluids, though the concept may not have been phrased in this way. For example, the specific heat of a fluid is sometimes measured at constant volume (Cv) and


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sometimes at constant pressure (CP). The isometric specific heat is determined by the second derivative of the energy function, E = ¡E(S,V,N) 14.9 with respect to the entropy

∆2¡E

∆S2 = ∆

∆S [T¶(S,V,N)] = TCv

14.10

while the isobaric specific heat is determined by the second derivative of the enthalpy function H = ¡E[S,P,N] 14.11

∆2¡H

∆S2 = ∆

∆S [T¶(S,P,N)] = T

CP 14.12

The two properties are not numerically equal to one another. Given that there are n different choices of the operational variables for a given basic set {u}, there are n distinct thermodynamic potentials, a large number of first-order thermodynamic properties, and even larger number of properties of higher order. To impose order on the analysis of these properties it is important to establish the relations that minimize their number and restrict their values. Three types of relations are useful in simplifying the set of thermodynamic properties: 1. Mathematical relations set the number of independent first-order thermody-namic properties. If there are n constitutive coordinates then, as we shall show below, there are no more than n(n+1)/2 independent first-order properties. Given a specific choice of the independent properties all others can be expressed in terms of them. 2. Symmetry relations may restrict the number of independent properties still fur-ther. These affect only the tensor properties of materials, and insure that the thermody-namic behavior of a material is consistent with its physical symmetry. 3. The conditions of stability restrict the sign or magnitude of the thermodynamic properties of a homogeneous phase. Since material symmetry only affects tensor properties we shall defer a considera-tion of the symmetry relations until we are forced to deal with thermodynamic variables that are not simple scalars. In this chapter we discuss the results that follow from functional relations and from the conditions of stability.


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The important mathematical relations between the thermodynamic properties of a material are of two types: those that relate properties that are derived from a single potential, and those that relate properties that are associated with different potentials. The former follow from the mathematical requirement that the value of a second- or higher-order partial derivative be independent of the sequence of differentiation. This restriction leads to the familiar Maxwell relations between the first-order properties that are associated with a given potential. The latter are the functional relations that determine all the first-order thermodynamic properties from a minimal set. These relations can be found algebraically, but are most easily developed from the properties of the Jacobian determinants that govern changes in independent variables. 14.2 FUNCTIONAL RELATIONS: THE MAXWELL RELATIONS The first-order thermodynamic properties that are associated with a single thermo-dynamic potential are related by the requirement that the second partial derivatives of the fundamental equation be independent of the order of differentiation. That is, if ¡Ï = ¡Ï({x}) 14.13 then

∆2¡Ï

∆xi∆xj =

∆2¡Ï

∆xj∆xi 14.14

It follows that the matrix of thermodynamic properties, the nxn matrix whose elements are

¡Ïij =

∆2¡Ï

∆xi∆xj 14.15

is symmetric, ¡Ïij = ¡Ïji 14.16 A symmetric nxn matrix has n(n+1)/2 independent elements, n diagonal elements of the form ¡Ïii (i = 1,...,n), and n(n-1)/2 independent off-diagonal elements of the form ¡Ïij (i ≠ j = 1,...,n) that satisfy the symmetry relations 14.16. It follows that: there are a maximum of n(n+1)/2 independent first-order thermodynamic properties associated with a given thermodynamic potential. The equations 14.16 are called the Maxwell relations. In the particular case of the energy function they establish the result


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∆

∆ui [pj({u})] =

∆∆uj

[pi({u})] 14.17

Similar relations hold between the thermodynamic properties of higher order. 14.3 FUNCTIONAL RELATIONS BETWEEN PROPERTIES The various thermodynamic potentials are related to one another by Legendre transformations that change the independent variables in the fundamental equation. It follows that their second derivatives should also be related, so that it is possible to find all the first-order thermodynamic properties, and hence all the thermodynamic properties of any order, from the n(n+1)/2 properties that are determined by the energy function (or by any other thermodynamic potential). This result is easy to establish, provided that the second partial derivatives

¡Ekk =

∆2¡E

∆uk2 14.18

do not vanish (as we shall show below, this condition is only violated at critical points in the behavior of the system). Given the n constitutive equations for the forces that are de-termined by the energy function pk = p¶

k({u}) 14.19 it is possible to replace any coordinate, qk, by its conjugate force, pk, to generate the n constitutive functions pi = p¶

i (pk,{u}') (i ≠ k) 14.20 uk = u¶

k(pk,{u}') 14.21 where the set {u}' does not include uk. The thermodynamic properties associated with the set of variables {pk,{u}'} can then be found from the thermodynamic properties ¡Eij that are determined by the energy function. To show this, let k = 1 (this can be done without loss of generality by re-ordering the {u}). Then the differential of equation 14.19 for p1 is

dp1 = ¡E11du1 + ∑i=2

n ¡E1idui 14.22

where the summation does not include du1. Solving this equation for du1,


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du1 = dp1¡E11

- ∑i=2

n

¡E1i

¡E11 dui 14.23

Equation 14.23 is just the differential of the constitutive equation

u1 = u¶1(p1,{u}') = -

∆∆p1

[¡Ï(p1,{u}')] 14.24

and shows that the thermodynamic properties, ¡Ï1j, associated with ¡Ï are

¡Ï11 = - ∆

∆p1 [u¶

1(p1,{u}') ] = - 1

¡E11 14.25

¡Ï1i = ¡Ïi1 = ¡E1i¡E11

14.26

To obtain the rest of the thermodynamic properties, ¡Ïij (i,j ≠ 1), in the transformed representation we write the differential of the function p¶

i ({u}) , i ≠ 1:

dpi = ¡Ei1du1 + ∑j=2

n ¡Eijduj 14.27

On substituting equation 14.23, equation 14.27 becomes

dpi = ¡Ïi1dp1 + ∑j=2

n

¡ Ïijduj 14.28

which is the differential of the function

pi = p¶i (p1,{u}') =

∆∆ui

[¡Ï(p1,{u}')] 14.29

with Ïi1 given by equation 14.26 and

¡Ïij = ¡Ïji = ¡Eij - ¡Ei1¡Ej1

¡E11 (i,j ≠ 1) 14.30

By iterating this procedure we can find the thermodynamic properties associated with any thermodynamic potential from the set ¡Eij. It follows that:


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if there are n independent constitutive variables there are a total of n(n+1)/2 independent thermodynamic properties, which can be taken to be the set ¡Eij that follows from the energy function.

14.4 JACOBIAN DETERMINANTS The procedure outlined in section 14.3 is general, and could be used to find all first-order thermodynamic properties from a basic set. However, it is cumbersome. There is a much more efficient procedure for relating the thermodynamic properties that are associated with different choices of the thermodynamic potential. It begins from the realization that a change in the thermodynamic potential is achieved by changing the independent variables, or coordinates, that govern the thermodynamics of the system. For example, the energy function is based on the coordinates {u}. The forces {p} are determined by the constitutive equations

pk = p¶k({u}) =

∆¡E

∆uk 14.31

and the first-order thermodynamic properties are given by the partial derivatives

¡Eij = ∆

∆uj p¶

i ({u}) 14.32

The function ¡ Ïr is based on the coordinates {p1,...,pr,ur+1,...,un}. Its derivatives are

uk = u¶k({p1,...,pr,ur+1,...,un}) = -

∆¡Ïr

∆pk (k = 1,...,r) 14.33

pk = p¶k({p1,...,pr,ur+1,...,un}) =

∆¡Ïr

∆uk (k = r+1,...,n) 14.34

and the first-order properties associated with it are

¡Ïij = - ∆

∆pj u¶

i ({p1,...,pr,ur+1,...,un}) (i,j = 1,...,r) 14.35

¡Ïij = ∆

∆uj p¶

i ({p1,...,pr,ur+1,...,un}) (i,j = r+1,n) 14.36

¡Ïij = - ∆

∆uj u¶

i ({p1,...,pr,ur+1,...,un})

(otherwise)


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= ∆

∆pj p¶

i ({p1,...,pr,ur+1,...,un}) 14.37

The problem is to determine the partial derivatives given in 14.35-14.37 from those given in equation 14.32 after a coordinate transformation that involves substituting r of the pk for the conjugate uk. The pk are related to the original coordinates, {u}, by equations 14.31. There is a general method for relating partial derivatives of equivalent functions that are expressed in terms of different sets of independent variables, or coordinates, that is based on the Jacobian determinant of the transformation. The Jacobian is simply the determinant of the partial derivatives of the new variables with respect to the original ones. Suppose that the set of independent variables (x1,...,xn) is replaced by the set (y1,...,yn). If the set {y} is equivalent to the set {x} then each of the coordinates, yi, must be expressible as a function of the {x}: yi = y¶

i ({x}) 14.38 The Jacobian of the transformation is the determinant of the partial derivatives of the {y} with respect to the {x}

∆(y1,...,yn)∆(x1,...,xn) = det

∆yi

∆xj 14.39

where it is understood that the quantities in the numerator of the Jacobian are expressed as functions of the quantities in the denominator. The matrix ∆yi/∆xj is an nxn matrix. Recall that the determinant of an nxn matrix, Aij, is det(Aij) = ∑

P (-1) PA1iA2j...Ank 14.40

where the series i,j,...,k is initially the numbers 1,2,...,n, the summation is over all permu-tations of the numbers 1,2,...,n, and the order of the Pth permutation, P, is the number of interchanges that must be made in the sequence 1,2,...,n to achieve it. The Jacobian has the mathematical property that it gives the relative change in the differential volume element of the coordinate frame when the transformation of coordinates is legitimate, that is, when the functions of (x1,...,x2) can also be written as functions of (y1,...,yn).

dV({y})dV({x}) =

∆(y1,...,yn)∆(x1,...,xn) = det

∆yi

∆xj 14.41

which is the Jacobian of the coordinate transformation. When the transformation cannot legitimately be made, for example, when the new variables {y} are functions of only


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some of the original variables {x}, then the Jacobian vanishes. The existence of a non-zero Jacobian is a criterion for the linear independence of the new set of variables that is widely used in mathematics. The utility of the Jacobian in thermodynamics follows ultimately from the fact that the different thermodynamic potentials are related to one another by coordinate transformations. If the new set of coordinates is equivalent to the old set then the Jacobian of the transformation does not vanish. The relations between the Jacobians for the various coordinate transformations then establish relations between the values of the equilibrium material properties. Moreover, the properties of the Jacobian make it possible to relate the thermodynamic properties in two coordinate systems without ever actually knowing the constitutive equations or the fundamental equation in either frame. The useful properties of the Jacobian follow from the general properties of partial derivatives and determinants. Five of the most useful are given below for the two-dimen-sional case. Their generalization to an arbitrary number of coordinates is straightforward.

∆(y1,y2)∆(x1,x2) = det

∆yi

∆xj =

∆y1

∆x1

∆y2

∆x2 -

∆y1

∆x2

∆y2

∆x1 14.42

∆(y1,y2)∆(x1,x2) = -

∆(y2,y1)∆(x1,x2) 14.43

∆(y1,z)∆(x1,z) =

∆y1

∆x1 z 14.44

∆(y1,y2)∆(x1,x2) =

∆(y1,y2)

∆(z1,z2)

∆(z1,z2)

∆(x1,x2) 14.45

∆(y1,y2)∆(x1,x2) =

∆(y1,y2[u,v])∆(x1,x2) =

∆(y1,u)∆(x1,x2)

∆y2

∆u v +

∆(y1,v)∆(x1,x2)

∆y2

∆v u 14.46

where the subscript on the partial derivative (∆y/∆x)v has the meaning that the derivative is taken with the value of v held constant. The last form of equation 14.46 follows from the differential relation

dy2(u,v) =

∆y2

∆u v du +

∆y2

∆v u dv 14.47

and the algebraic relation

∆(a,b+c)

∆(f,g) = ∆(a,b)∆(f,g) +

∆(a,c)∆(f,g) 14.48


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The utility of the Jacobian determinant for relating the values of partial derivatives that are referred to different independent coordinates can be illustrated by a simple example. Let the functions u = u(x,y) 14.49 v = v(x,y) 14.50 be given, and assume that the partial derivatives

ux = ∆u∆x 14.51

uy, vx and vy do not vanish. We wish to find the partial derivatives of x and y with respect to u and v when the functions are transformed to x = x(u,v) 14.52 y = y(u,v) 14.53 The problem can be solved algebraically with a method that is sufficient, but te-dious. The partial derivative

xu = ∆∆u [x(u,v)] 14.54

refers to the variation of x with u when dv is zero, and can be found by simultaneous solution of the linear differential relations du = uxdx + uydy 14.55 dv = 0 = vxdx + vydy 14.56 The solution is vydu = (uxvy-uyvx)dx 14.57 from which

xu = vy

∆(u,v)

∆(x,y)-1

= vy

uxvy - uyvx 14.58

since all of the terms on the right-hand side are known from equations 14.49-50, the partial derivative xu is determined.


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Equation 14.58 can be found much more simply and directly by using the Jacobian of the transformation. It follows from equation 14.44 that xu is just the value of the Jacobian determinant that governs a transformation of coordinates from the set (u,v) to the set (x,v). Hence,

xu = ∆(x,v)∆(u,v) =

∆(x,v)

∆(x,y)

∆(x,y)∆(u,v) = vy

∆(u,v)

∆(x,y)-1

14.59

The procedure that obtains equation 14.59 can be readily adapted to evaluate the other partial derivatives, xv, yu and yv in terms of the known derivatives ux, uy, vx, and vy. The constitutive equations that govern thermodynamic properties are usually functions of several variables. To establish relations between the thermodynamic properties it is usually sufficient to be able to interchange these variables (i.e., replace coordinates by their conjugates) two at a time. To determine the influence of the remaining variables, consider the following problem. Given u = u(x,y,z) 14.60 v = v(x,y,z) 14.61 or, more restrictively, given only the partial derivatives ux, uy, uz, vx, vy, vz, we wish to find the partial derivatives xu, xv, xz, yu, yv, yz of the transformed functions x = x(u,v,z) 14.62 y = y(u,v,z) 14.63 First consider the derivative xu, which is taken holding v and z constant. The value of xu is simply the value of the Jacobian that governs the transformation from the variables (x,v,z) to the variables (u,v,z):

xu = ∆(x,v,z)∆(u,v,z) =

∆(x,v,z)

∆(x,y,z)

∆(u,v,z)∆(x,y,z)

-1 = vy

∆(u,v,z)

∆(x,y,z)-1

14.64

The Jacobian of the transformation from (x,y,z) to (u,v,z) is

∆(u,v,z)∆(x,y,z) = uxvy - uyvx 14.65

Hence all of the terms on the right hand side of 14.64 are known. The partial derivative xz can be found in the same way:

xz = ∆(x,u,v)∆(z,u,v) =

∆(x,u,v)

∆(x,y,z)

∆(u,v,z)∆(x,y,z)

-1 =

(uyvz - uzvy)(uxvy - uyvx) 14.66


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where we have used the fact that the sequence (u,v,z) is an even permutation of the sequence (z,u,v) to write the second form on the right hand side. The other partial derivatives follow easily by the same procedure. 14.5 PROPERTIES OF A ONE-COMPONENT FLUID The most familiar thermodynamic properties are those that govern the behavior of one-component fluids. The coordinates that characterize the states of a fixed quantity of a one-component fluid are the volume, V, and entropy, S. The forces conjugate to these are the negative of the pressure, P, and the temperature, T. Since there are two independent variables there are n(n+1)/2 = 3 independent thermodynamic properties of a mole of a one-component fluid. A convenient set of thermodynamic properties is the isobaric specific heat, CP, the isothermal compressibility, ˚T, and the coefficient of thermal expansion, å, where

CP =

∆H

∆T P = T

∆S

∆T P 14.67

˚T = - 1V

∆V

∆P T 14.68

å = 1V

∆V

∆T P 14.69

The three properties are independent of one another, and are relatively easy to measure since they are based on the operational coordinates {P,T}, which are easily controlled in simple experiments. Since {P,T} are the operational variables for the properties CP, ˚T and å, these properties should be the ones determined by the Gibbs free energy, ¡G(P,T). To show that this is the case we evaluate the second derivatives of ¡G. The results are

∆2¡G∆T2 = -

∆S

∆T P = -

CPT 14.70

∆2¡G∆P2 =

∆V

∆P T = - V˚T 14.71

∆2¡G

∆P∆T = ∆2¡G

∆T∆P = -

∆S

∆P T =

∆V

∆T P = Vå 14.72


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It is also common to use the volume and temperature as experimental variables, particularly when the "fluid" is actually a solid at atmospheric pressure. The variables {V,T} are the natural variables for the Helmholtz free energy, which determines the three thermodynamic properties

∆2¡F∆V2 = -

∆P

∆V T =

1V˚T

14.73

∆2¡F∆T2 = -

∆S

∆T V = -

CVT 14.74

∆2¡F

∆V∆T = ∆2¡F

∆T∆V = -

∆S

∆V T = -

∆P

∆T V 14.75

where CV is the isometric specific heat. To evaluate these properties we require expres-sions for CV and (∆P/∆T)V in terms of CP, å and ˚T. We obtain these by using the Jaco-bian function to effect a coordinate transformation from the variables (T,V) to the variables (T,P)

CV = T

∆S

∆T V = T

∆(S,V)

∆(T,V)

= T

∆(S,V)

∆(T,P)

∆(T,P)∆(T,V)

= T

∆V

∆P T

∆S

∆T P

∆V∆P T

-

∆S

∆P T

∆V∆T P

= CP - T

∆S

∆P T

∆V∆T P

∆V

∆P T

= CP - TVå2

˚T 14.76

and


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∆P

∆T V =

∆(P,V)∆(T,V) =

∆(P,V)

∆(P,T)

∆(P,T)∆(T,V) = -

∆V

∆T P

∆P∆V T

= å˚T

14.77

The set of coordinates {S,P} is often controlled during fluid flow, since there are many situations in fluid flow where mechanical relaxations must be considered but heat conduction can be ignored. The governing potential for this case is the enthalpy, H. It de-termines the three thermodynamic properties

∆2¡H∆S2 =

∆T

∆S P =

TCP

14.78

∆2¡H∆P2 =

∆V

∆P S = - V˚S 14.79

∆2¡H∆P∆S =

∆2¡H∆S∆P =

∆T

∆P S =

∆V

∆S P 14.80

where ˚S is the isentropic compressibility. To evaluate these in terms of CP, ˚T and å we require expressions for ˚S and (∆V/∆S)P. To accomplish this we again use the Jacobian determinant to find the effect of a transformation to the independent variables (T,P). Since

∆V

∆P S =

∆(V,S)∆(P,S) =

∆(V,S)

∆(P,T)

∆(P,S)

∆(P,T)

=

∆(V,S)

∆(V,T)

∆(V,T)∆(P,T)

∆(P,S)

∆(P,T)

=

CV

CP

∆V

∆P T 14.81

the isentropic compressibility is

˚S =

CV

CP ˚T 14.82

It is also easy to show that

∆V

∆S P =

TVåCP

14.83


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Note that we do not need to know the precise form of the fundamental equation, or even the set of constitutive equations to determine the thermodynamic properties. We only require a complete set of thermodynamic properties in one representation (choice of coordinates). The thermodynamic properties in any other representation can then be found with the help of the Jacobian determinant. 14.7 THE LOCAL CONDITION OF STABILITY We now consider the local stability of a homogeneous phase. If the system is homogeneous the volume densities of the entropy, s, the mechanical coordinates, {Ë}, and the chemical coordinates, {n}, are constant. It follows, as discussed at the end of chapter 6, that the forces, T, {p}, and {µ}, are also constant. The stability of the system (ignoring its interaction with its environment) requires that the internal energy of the system be a minimum, at least with respect to small changes. Mathematically, (∂E){u} ≥ 0 14.83 The variation considered in equation 14.83 is to be taken at constant values of the total entropy, S, mechanical coordinates, {q}, and chemical coordinates, {N}. Given that the system is homogeneous these constraints can be incorporated by the use of Lagrange multipliers. The condition 14.83 then reads ∂(E - TS - ∑

k pkqk - ∑

k µkNk) ≥ 0 14.84

where T, {p}, and {µ} are Lagrange multipliers. By the energy function the variation of E is, to second order,

∂E = ∑k

∆¡E

∆uk ∂uk +

12 ∑

jk

∆2¡E

∆uj∆uk ∂uj∂uk 14.85

Inserting 14.85 into 14.84 regenerates the known relations

∆¡E

∆S = T 14.86

∆¡E

∆qk = pk 14.87

∆¡E

∆Nk = µk 14.88


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and leaves the residual condition for local stability

∑jk

∆2¡E

∆uj∆uk ∂uj∂uk = ∑

jk Ejk∂uj∂uk ≥ 0 14.89

The inequality 14.89 shows that a homogeneous phase is stable with respect to in-finitesimal changes in its state only if the quadratic form on the right hand side is not negative. This requirement constrains the (nxn) matrix of thermodynamic properties, Eij. The applicable constraints can be most easily written in terms of the eigenvalues and eigenvectors of the matrix Eij. 14.8 EIGENVALUES OF THE PROPERTY MATRIX 14.8.1 Coordinate transformations in state space While eigenvectors and eigenvalues are general mathematical properties of symmetric matrices, it may help in understanding if we define and discuss the eigenvalues and eigenvectors of the matrix Eij in terms of their physical meaning. This particular matrix, the property matrix, is central to much of the thermodynamic theory of materials. An arbitrary change of state is specified by a set of variations {∂u} in the indepen-dent variables of the energy function. This moves the state of the system from the state {u} to the state {u+∂u} in the state space of the system. We can represent the change graphically in an n-dimensional space that is spanned by the orthogonal coordinates, {u}, as shown in the two-dimensional example given in Fig. 14.1. The change of state is the n-dimensional vector, ∂u, that connects the points {u} and {u+∂u}, as illustrated in the figure.

u

˙

˙ 1

2

2

u 1

∂u

{u}

{u+∂u}

Fig. 14.1: A change of state specified by the vector ∂u in a two

dimensional state space. The states may be referred to coordinates {u} or {˙}.


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While the coordinates {u} may be the natural choices, they are not the only coordinates that might be used for the state space and are not always the most convenient. The state space can equally well be referred to alternate coordinates {x}. Let the coordinates {x} be simple rotations of the coordinates {u}, as illustrated by the coordinates {˙} shown in the figure. The new coordinates {x} are linear combinations of the {u}, xk = ∑

i lkiui 14.90

where the lki are coefficients that give the component of the coordinate xk along the coordinate ui. Since the volume element of the state space is preserved in the coordinate transformation its Jacobian determinant is

dV({x})dV({u}) =

∆(x1,...,xn)∆(u1,...,un) = det

∆xi

∆uj = det[lij] = 1 14.91

which shows that the determinant of the matrix, lij, is unity. If the transformation from {u} to {x} exists then the reverse transformation {x} “ {u} also exists, and equations 14.90 can be solved for the {u}: ui = ∑

k l-1ik xk 14.92

where the matrix of coefficients, l-1ki , is inverse to the matrix lki. If equation 14.90 is substituted into 14.92 the result is ∑

k l-1ik lkj = ∂ij 14.93

where ∂ij is the Kronecker ∂. It can be shown that ∑

k lkilkj = ∂ij 14.94

and, hence, that l-1ki = lik 14.95 Since the coordinates {x} are linear combinations of the coordinates {u} the gov-erning thermodynamic potential for a system that is specified by the coordinates {x} is the internal energy. The transformation rule for the property matrix, Eij, can be easily found from the condition that the variation of the energy in a change of state must be numerically the same whether the coordinates {x} are {u} are used to describe it. Hence


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∑

ij Eij∂ui∂uj = ∑

ij ∑km

Eijlkilmj∂xk∂xm = ∑km

E'km∂xk∂xm 14.96

and the property matrix, E'km, with respect to the coordinates {x} is E'km = ∑

ij Eijlkilmj = = ∑

ij

lki Eijl-1jm 14.97

14.8.2 The diagonal coordinates: eigenvectors Equation 14.97 shows that the elements of the property matrix, Eij, have different values for different choices of the coordinates {x}. The simplest possible form of the matrix is a diagonal one, in which all off-diagonal elements vanish. It is a general result of matrix algebra that it is always possible to find a set of coordinates {x} that diagonalize a symmetric matrix. Since the property matrix, Eij, is symmetric, there is always a choice of coordinates {x} that are linear combinations of the coordinates {u} for which E'km = ¬k∂km 14.98 where ∂km is the Kronecker ∂. The diagonal coordinates are the normal coordinates or eigenvectors of the matrix, and have the very useful property that they decouple the second derivatives of the energy function. Let the normal coordinates be designated by the set {˙}. Then the energy function can be written E = ¡E({˙}) 14.99 where

∆2¡E

∆˙i∆˙j =

∆2¡E

∆˙i2 ∂ij = Eii∂ij 14.100

and the second variation of the energy function is ∑

ij Eij∂˙i∂˙j = ∑

i Eii(∂˙i)2 14.101

The normal coordinates are the eigenvectors of the matrix Eij. The diagonal elements of the matrix, ¬k, are its eigenvalues.


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To find the eigenvectors consider a particular variation of state that is specified by the set {∂u}. The variation can alternately be described by the changes in the diagonal coordinates, {∂˙}, in the directions of the eigenvectors. It follows from equations 14.98 that the {∂˙} satisfy the equations ∑

m E'km∂˙m = ¬k∂˙k 14.102

The general mathematical condition for the existence of a solution of a set of linear simultaneous equations of this form is that the determinant det[E'km - ¬∂km] = 0 14.103 which is obviously satisfied when E'km is diagonal and ¬ has any one of the n values, ¬k. Equation 14.103 can be converted back to an equation in the matrix Eij by using the transformation 14.97. The result is det

∑

ij lkilmj(Eij - ¬∂ij) = det[ ]Eij - ¬∂ij = 0 14.104

where we have used the fact that the determinant of a matrix is an invariant whose value is unchanged by a coordinate transformation. Equation 14.104 is an nth order polynomial in ¬ whose solutions are the n eigenvalues, ¬k. Given equation 14.103, there are independent solutions to the linear equations ∑

j Eij∂uj = ¬k∂ui 14.105

for each of the n values of ¬k. The kth solution is a set of values {∂uk} that define a vector ∂uk that is in the direction of the kth eigenvector, ˙k. To complete the proof that the eigenvectors {˙} that are the solutions of equation 14.102 are the normal coordinates we need to show that they are orthogonal. The proof is simple when the eigenvectors correspond to different eigenvalues. Let ∂uk be a variation of state in the direction of ˙k, whose eigenvalue is ¬k and let ∂um be a variation in the di-rection ˙m with a different eigenvalue ¬m. Given equation 14.102 and the fact that Eij is symmetric, ∑

ij Eij∂uk

i ∂umj = ¬k∑

ij ∂uk

i ∂umj = ¬m∑

ij ∂uk

i ∂umj 14.106

which can only be true if


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∑ij

∂uki ∂um

j = ∂km 14.107

This result fails when the eigenvectors ˙k and ˙m have the same eigenvalue, in which case we say that they are degenerate. However, two eigenvectors that have the same eigenvalue lie in a plane in state space that has the same eigenvalue for every direction. The eigenvectors can always be made orthogonal by choosing perpendicular vectors in the plane. The orthogonality of the eigenvectors has the physical consequence that it is possible to change the value of one of the normal coordinates without affecting the value of any other. The variables {˙} are independent. An arbitrary variation of state can be described as the sum of independent variations from the original state in the directions of the eigenvectors. The normal coordinates, or eigenvectors, are very useful because they simplify the expression for the variation of the energy. However, they often have the disadvantage that they depend on the precise state that is under consideration. Since the properties, Eij, are functions of the state, {u}, the directions of the eigenvectors often are as well. When the normal coordinates can be written in a form that automatically accounts for this variation it is useful to make the transition {u} “ {˙} and solve problems in the normal coordinate frame. As we shall see, this is what is done in most modern theories of the properties of real solids. When the normal coordinates cannot be expressed in a simple form they still have value, since several important results that concern the stability of thermodynamic systems can be easily established from the fact that they exist. The following section is an example. 14.9 STABILITY CONSTRAINTS ON THE MATERIAL PROPERTIES Equation 14.89 must hold for all variations of the state of the homogeneous system if it is to be stable with respect to local changes. The results of the previous section show that this condition can be rewritten ∑

ij Eij∂ui∂uj = ∑

k ¬k(∂˙k)2 ≥ 0 14.108

where the {˙} are the eigenvectors of the matrix Eij. It follows immediately that

it is a necessary condition of stability that ¬k ≥ 0 14.109 for all eigenvalues of the property matrix, Eij. If all ¬k are positive, the system is stable.


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If any one of the ¬k were negative it would be possible to lower the energy of a homoge-neous phase with given T, {p}, and {µ} by changing its state in the direction specified by the eigenvector ˙k associated with ¬k. On the other hand, if all of the ¬k are positive then, by equation 14.108, the energy must increase for any change whatever, and must, therefore, have a minimum value in the initial state. To apply this condition of stability we need only know the value of the least eigenvalue, ¬min. If ¬min < 0 14.110 then there is at least one variation of state that leads to a decrease in the energy and the system is unstable. The condition of stability can also be phrased in terms of the determinants of the matrix Eij and its various minors, which is the way Gibbs first wrote it, and is particularly useful in practice because it sets constraints on the values of the properties that can be found without computing the eigenvectors. The general condition is found from the iden-tity det[Eij] = det[E'jk] = ∏

k ¬k 14.111

which shows that the determinant of the matrix Eij is just the product of its eigenvalues. It follows that a homogeneous phase is stable with respect to small changes of state only if the determinant of its property matrix, Eij, is positive. Now consider small changes of state that involve only some of the coordinates {u}. For specificity, let the hypothetical transition involve variations of the first r of the n coordinates in the set {u}. For the system to be stable with respect to these restricted changes it is only necessary that the eigenvalues of the (rxr) matrix Er

ij be positive, which insures that the determinant of the (rxr) matrix of properties Er

ij is positive. This result is guaranteed if the more general condition stated above is satisfied. It is a well-known result of linear algebra that the least eigenvalue of an (rxr) submatrix of an (nxn) matrix must be greater than or equal to the least eigenvalue of the (nxn) matrix (the proof is simple: the eigenvector associated with the least eigenvalue of the (rxr) matrix is a possible eigenvector of the larger matrix). Hence if the matrix Eij has only positive eigenvalues all of the eigenvalues of the restricted matrix Er

ij are positive as well. If all of the eigenvalues of Er

ij are positive its determinant is positive. The considerations in the previous paragraphs lead to the condition of local stability that was presented by Gibbs:


Page 268

a homogeneous phase is stable only if the determinant of its property ma-trix, Eij, is positive, and if the determinant of every minor, Er

ij , of its property matrix is positive.

The "minors" of an (nxn) matrix are simply the (rxr) matrices derived by removing corre-sponding rows and columns from the parent matrix, i.e., they are the restricted matrices Eij discussed above. At this point we shall explore the local conditions of stability for two cases. First, one set of restricted matrices Eij is obtained by considering variations in the state that alter only one variable of the set {u}. Let this variable be the kth. Stability then requires that Ekk(∂uk)2 ≥ 0 14.112 or Ekk ≥ 0 14.113 It follows that if a homogeneous phase is stable all of the second derivatives of ¡E with re-spect to a single variable are positive. We anticipated this result in previous sections. Second, consider variations in the state that alter the values of only two members of the set {u}. Let these be the jth and kth. If the initial phase is stable the determinant of the matrix Eij that contains only j and k elements must be positive semi-definite. Hence EjjEkk - (Ejk)2 ≥ 0 14.114 14.10 LOCAL STABILITY OF INTERACTING SYSTEMS Alternate forms of the local conditions of stability can be obtained from the other fundamental functions. However, some care must be exercised in their derivation. The reason is that the other fundamental functions govern situations in which a subset of the thermodynamic forces is fixed by contact between the system and an appropriate reservoir. Since these forces are fixed, they are held constant during a change in the state of the system. Let the system be in contact with a reservoir that fixes the values of the forces p1,...,pr. Then the equilibria of the system are governed by the thermodynamic potential

Ïr = E - ∑k=1

r pkuk = ¡Ïr[p1,...,pr,ur+1,...,un] 14.115


Page 269

The local condition of stability for the system is that its energy cannot decrease during a transition that involves a small change in the quantities ur+1,...,un. It follows that det[Ïij] ≥ 0 (i,j = r+1,...,n) 14.116 and det[Min(Ïij)] ≥ 0 (i,j = p+1,...,n) 14.117 where Min denotes a minor of the (n-r)x(n-r) matrix Ïij. However, no similar condition holds for the second derivatives of ¡Ï with respect to the forces, p1,...,pr. These have fixed values that are determined by the reservoir. As an example, consider the Helmholtz Free Energy of a simple fluid. The condi-tion FVV = 0 14.118 ensures that the isothermal compressibility is positive definite, since

FVV = -

∆P

∆V T =

1V˚T ≥ 0 14.119

However,

FTT = -

∆S

∆T V = -

CVT < 0 14.120

The Gibbs Free Energy yields the condition

∆2¡G

∆Nk2 =

∆µk

∆Nk PT ≥ 0 14.121

However,

GPP =

∆V

∆P T = - V˚T ≤ 0 14.122

and

GTT = -

∆S

∆T P = -

CPT < 0 14.123

since CP is required to be positive semi-definite because it has the sign of the second derivative, HSS, of the enthalpy.


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14.11 BEHAVIOR AT THE STABILITY LIMIT We now consider the case when the second variation of the energy vanishes for at least one choice of the variation {∂u}: ∑

jk Ejk∂uj∂uk = 0 14.124

This situation is usually encountered at the stability limit. Let a stable system undergo a continuous transition. The eigenvectors of the property matrix change in value during the transition. The stability limit is reached when the least eigenvalue falls to zero: ¬º = 0 14.125 The second variation of the energy then vanishes for a variation in the state in the direction of the eigenvector, ˙º, that corresponds to the eigenvalue, ¬º: ∂ui = ∂˙o

i 14.126 where ∂˙o

i is the value of ∂ui in an infinitesimal change in the direction of the eigenvector ˙º. The variation of the energy at the stability limit can be found by continuing the Taylor series to higher order terms. The condition of stability is, to the fourth order of the variation of the energy

∂E = 1

6 ∑ijk

Eijk∂˙oi ∂˙o

j ∂˙ok +

1

24 ∑ijkl

Eijkl∂˙oi ∂˙o

j ∂˙ok ∂˙o

l + ... > 0 14.127

where Eijk and Eijkl are, respectively, the (nxnxn) matrix of third derivatives of ¡E with re-spect to the {u} and the (nxnxnxn) matrix of fourth derivatives. If the system is to be stable in the limit the first term on the right hand side of 14.127 must be equal to zero: ∑

ijk Eijk∂˙o

i ∂˙oj ∂˙o

k = 0 14.128

The proof is straightforward. If the second variation of the energy vanishes for a positive variation along ˙º it also vanishes when the system is varied in the direction of ˙º in the negative sense. But the third variation, the left hand side of 14.128, changes sign when the signs of all the ∂˙o

i are reversed. Hence the third variation can be made negative unless it is zero, which violates the condition of stability. When equation 14.128 is


Page 271

satisfied the fourth variation of the energy determines stability, and must satisfy the condition ∑

ijkl Eijkl∂˙o

i ∂˙oj ∂˙o

k ∂˙ol ≥ 0 14.129

In the rare situation that the fourth-order variation vanishes, the fifth must be zero and the sixth must be positive definite to ensure stability. 14.12 THERMODYNAMIC PROPERTIES AT LOW TEMPERATURE The Third Law of Thermodynamics constrains the values of the thermodynamic properties that involve the temperature in any way. We discussed the most obvious and important of those in Chapter 7: the vanishing of the isometric specific heat. We are now in a position to establish the general rules that govern the limiting values of the thermodynamic properties. Let a system be characterized by the set of coordinates {T,{x}}, where the set {x} is some mixture of the molar geometric coordinates and their conjugate forces. The governing thermodynamic potential is Ï = ¡Ï(T,{x}) 14.130 The first-order thermodynamic properties are ÏTT and the sets {ÏTi} and {Ïij}, where the subscript (i) indicates partial differentiation with respect to xi. It can be shown that limT“0[TÏTT] = c{x} = 0 14.131 limT“0[ÏTi] = 0 14.132

limT“0

∆Ïij

∆T = 0 14.133

that is,

the first-order thermodynamic properties whose definitions include partial differentiation with respect to the temperature vanish in the low-temperature limit, while all other properties approach limiting values that are independent of temperature.

The proof is straightforward. Equation 14.131 follows from the identity

ÏTT = -

∆S

∆T = - c{x}

T 14.134


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which must remain finite as T “ 0. Equation 14.132 can be found from the identity

ÏTi = ÏiT = -

∆S(T,{x})

∆xi 14.135

which must vanish since the entropy is independent of the geometric coordinates in the low-temperature limit. The Third Law does not require that Ïij vanish, but its thermal derivative must, since

∆

∆T

∆2¡Ï

∆xi∆xj =

∆∆xi

∆2¡Ï

∆xi∆T 14.136

which vanishes by equation 14.132. It follows that Ïij approaches a limiting value that is independent of temperature as T “ 0 as required by equation 14.133. Specifying to the properties of the one-component fluid, these results show that at zero temperature, å = 0 14.137 CP = CV = 0 14.138

∆˚T

∆T P = 0 14.139

˚S = ˚T 14.140 14.13 LECHATELIER'S PRINCIPLE We finish this chapter by introducing an important general principle of thermody-namics that is known as LeChatelier's Principle. It can be stated in various ways, of which the most general is:

A perturbation that disturbs equilibrium induces a change of state that reduces the effects of the perturbation.

The utility of LeChatelier's Principle lies in its ability to predict the direction of thermodynamic changes whose mechanistic details are unknown or uninteresting. We shall develop LeChatelier's Principle in two cases. First, consider two sys-tems, K1 and K2, that are separated by a partition that only permits exchange of the geometric quantity, q, whose conjugate force is p. Let the initial equilibrium of the composite system be disturbed by instantaneously raising the value of the force, p2, in


3

K2. The change in p2 induces a change in system K1 which produces a further change in p2. According to LeChatelier's Principle the change in K1 should be in a direction that decreases the net change in p2. We prove that below. Let the force on system 2, p2, be increased to a value p2 > p1, and then let the sys-tem relax. The change in energy of the composite system during the relaxation is, to first order, ∂E = p1∂q1 + p2∂q2 14.141 which must be negative or zero since the relaxation is spontaneous. Assuming that q is a conserved quantity, ∂q1 = - ∂q2 14.142 Hence ∂E = (p1 - p2)∂q1 14.143 and, since p2 > p1 by hypothesis, the spontaneous change must be such that ∂q1 > 0. 14.144 But the stability of the initial state requires that

∆p∆q =

∆2¡E

∆q2 > 0 14.145

and the change in the imposed force, p2, during the relaxation of the system is

∂p2 =

∆p

∆q ∂q2 = -

∆p

∆q ∂q1 < 0 14.146

It follows that the response of a system to a change in a single imposed force is in a direc-tion that relaxes that force. To illustrate this principle in a non-trivial example consider a piece of metal that is subject to a tensile stress and contains a sharp crack perpendicular to the loading axis. It is well known that the elastic stress in the body is intensified at the crack tip to a degree that depends on the "bluntness" of the crack, as measured by the radius of curvature of the crack tip. Now assume that the metal is capable of plastic deformation. It follows immediately from LeChatelier's Principle that if plastic deformation occurs at all it will have the effect of blunting the crack tip, which decreases the applied stress.


Page 274

A slightly more complex form of LeChatelier's Principle occurs when the system has two or more degrees of freedom in its interaction with the environment. For speci-ficity, let the system be surrounded by a wall that permits exchange of the geometric quantities x and y, whose conjugate forces are X and Y. Let the system be perturbed by adding a small quantity of x from the reservoir, and let the system initially relax so rapidly that the quantity y is not changed. The instantaneous change in the force X is

(∂X)y =

∆X

∆x y ∂x 14.147

where

∆X

∆x y = Xx = Exx > 0 14.148

and we have used the simplified notation for the derivative of X(x,y). Eventually the variable y will change to restore equilibrium with respect to the force, Y, which is not changed. The ultimate change in X is, therefore,

(∂X)Y =

∆X

∆x Y ∂x 14.149

But

∆X

∆x Y =

∆(X,Y)∆(x,Y) =

∆(X,Y)

∆(x,y)

∆(x,Y)

∆(x,y)

= 1

Yy [XxYy - XyYx]

=

∆X

∆x y

1 - (Exy)2

ExxEyy 14.150

But from the conditions of stability, Eyy ≥ 0 and ExxEyy - (Exy)2 ≥ 0 14.151 Hence

∆X

∆x y ≥

∆X

∆x Y ≥ 0 14.152

and the ultimate change in X is less than it would be if a change in y were not permitted. If a system is in equilibrium with respect to interchange of both the quantities x and y, a


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change in x induces a smaller change in the conjugate force, X, than it would if the system were in equilibrium with respect to x alone. A simple application of this principle is to the equilibrium of a chemical reaction A + B = C + D 14.153 Let the volume of the system be decreased. According to LeChatelier's Principle, the chemical equilibrium (constant chemical potential) will shift so that the associated increase in pressure is as small as possible. Equilibrium is hence shifted toward the side of the equation that has the smaller total molar volume. One can increase the solubility of any gaseous species in a solid by increasing the total pressure on the vapor phase, and suppress the occurrence of a phase transformation that increases volume by increasing the applied hydrostatic pressure.

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