LeChatelier Dynamics

Transcription

1 LeChatelier Dynamics Robert Gilmore Physics Department, Drexel University, Philaelphia, Pennsylvania 1914, USA (Date: June 12, 28, Levine Birthay Party: To be submitte.) Dynamics of the relaxation of a perturbe thermoynamic system to equilibrium is etermine by the generalize eigenvalue equation ( K αβ + λs αβ )y β =. Here S αβ is a matrix of linear response coefficients an K αβ is a matrix of kinetic coefficients. We escribe LeChatelier s Principle an both its static an ynamical symmetries, an treat cases in which there are multiple relaxation channels. The treatment simplifies when the relaxation channels have wiely separate time scales. I. INTRODUCTION LeChatelier s Principle is a statement about how a system in thermoynamic equilibrium respons to a perturbation. In brief, after the perturbation the system relaxes back to an equilibrium. This equilibrium is the highest possible entropy state, or the lowest possible energy state, that is available to the system uner the constraints that have been impose. If one or more relaxation channels are close, the new equilibrium will be ifferent from the original equilibrium. LeChatelier s Principle is usually state in qualitative terms. When it is escribe quantitatively it is almost always in terms of responses to a perturbation from equilibrium when a single reaction channel opens up [1, 2]. The case of intermeiate responses when there is a succession of relaxation channels with wiely separate time scales has been iscusse quantitatively in [3]. In this work we escribe the ynamical response of a system with any number of relaxation channels to an arbitrary perturbation. The relaxation time constants are etermine from a generalize eigenvalue equation. This equation involves the static linear response functions an their ynamic counterparts, the kinetic coefficients. The eigenvalues are well-efine functions of both sets of coefficients. The generalize eigenvalue equation is first expresse in the entropy representation. In this representation the entropy, S(t), continuously increases after the perturbation until the new equilibrium is reache. We then transform to the energy representation using what is for all practical purposes a similarity transformation. In this representation the internal energy, U(t), continuously ecreases after the perturbation until the new equilibrium is reache. Constraints that prevent the system from returning to its original equilibrium configuration are expresse naturally in this eigen-representation. Such constraints are represente by vanishing eigenvalues. Natural extensive variables, an their conjugate intensive variables, are summarize in Table I for the two ther- TABLE I: Natural conjugate variable pairs (Extensive, intensive) in the entropy an energy representations. Representation Entropy Energy X α y α E α i α U 1/T S T V P/T V P N µ/t N µ moynamic representations. Throughout we use contra- ( α ) an co- ( α ) variant notation for extensive an intensive thermoynamic variables. This notation is base on the geometric formulation of classical thermoynamics [4, 5, 6, 7]. In Sec. II we briefly review the properties of the matrices of static an ynamic coefficients, an then write own an justify the generalize eigenvalue equation that escribes LeChatelier ynamics. In Sec. III we transform to the more familiar energy representation. We illustrate how constraints are hanle in Sec. IV in terms of a simple example with a single relaxation channel. In Sec. V we review the symmetries that exist in single channel processes. They are of two types. One involves the ratios of asymptotic responses uner ual perturbations. The other involves a single relaxation time constant, also uner ual perturbations. In Sec. VI we treat the case in which there are multiple relaxation channels with wiely separate time scales. The results are summarize in Sec. VII. II. ASSUMPTIONS AND EQUATIONS - ENTROPY REPRESENTATION Two funamental equations escribe the ynamics of LeChatelier s Principle. In the entropy representation these are

2 2 δx α = S αβ δy β (Statics) (1) t δxα = K αβ δy β (Dynamics) (2) These equations escribe very ifferent physical processes. The first equation escribes static or aiabatic (meaning very slow) equilibrium processes. At equilibrium the entropy is a function of its natural extensive variables X (c.f., Table I). We assume there is an equilibrium for values X α of the extensive thermoynamic variables, an y α (X ) = S X (X α ) are the values of the conjugate intensive thermoynamic variables at this equilibrium. If the extensive variables are change to new values X α + δxα slowly, so that the system moves on the equilibrium surface S = S(X), the intensive variables will slowly change to new values y α (X ) + δy α. For small isplacements the linear relation between the isplacements of extensive an intensive thermoynamic variables is given by Eq. (1). The n n real symmetric matrix S αβ of( static susceptibilities is the inverse of the matrix S αβ = 2 S(X). This matrix is negative efinite at equilibrium, since the entropy is a maximum at X α X )X β equilibrium. The susceptibilities S αβ an S αβ are intrinsic to the thermoynamic system: they are inepenent of the container holing the material. The secon equation above relates forces to fluxes. The generalize forces are the ifferences in the intensive variables, δy β, across some sort of barrier separating the system of interest (A) from the outsie worl (B). The generalize fluxes, t δxβ, are the time rates of change of isplacements of the extensive variables as the system tries to relax to equilibrium: the highest available entropy state. Eq. (2) is a linearization of the general equation δx α = F α (δy) near the equilibrium manifol [8]. The matrix K αβ of kinetic coefficients obeys the Onsager symmetry K αβ (t) = K βα ( t) [9] an is positive semiefinite. The kinetic coefficients K αβ are extrinsic: they epen on the container holing the material an can be change from one experiment to another. The positivity property of the matrix of kinetic coefficients can be seen by computing the time rate of change of the combine entropy of the two systems A + B: t (S A + S B ) = S A X α A + S B X α B t XA α t XB α = ((y A ) α (y B ) α )XA α /t = δy Kαβ α δy β (3) We have exploite conservation of extensive quantities: X α A + Xα B = Xα Tot = const., so that Ẋα A = Ẋα B. We point out here that by simple imensional consierations, the imensions ([ ]) of the kinetic coefficients are closely relate to the imensions of the corresponing equilibrium linear response coefficients: Kαβ] = [ [ ] S αβ (time) 1. We now assume that uner a suen perturbation from equilibrium X α Xα + Xα, the subsystem remains homogeneous (no soun waves) an the relation between the extensive an intensive variables given in Eq. (1) remains vali. In this case we can write Eq. (2) as t Sαγ δy γ = K αβ δy β (4) Since this is a linear equation we can assume an exponential time epenence of the form δy β (t) = δy β ( + )e λt. This leas irectly to a generalize eigenvalue equation ( Kαβ + λs αβ) δy β ( + ) = (5) The eigenvalues λ are nonnegative because S αβ is negative efinite an K αβ is positive semi-efinite. These are thermoynamic stability conitions. The number of inepenent ecay channels is the number of nonzero eigenvalues of this generalize eigenvalue equation. Equivalently, it is the number of nonzero eigenvalues of the matrix K αβ. The number of vanishing eigenvalues is the number of inepenent constraints preventing ecay. The eigenvectors v i with eigenvalues λ i an components v αi of this generalize eigenvalue equation are mutually orthogonal with respect to the metric S αβ an can be normalize as follows: v αi ( S αβ )v βj = δ ij v αi (+ K αβ (6) )v βj = λ i δ ij They evolve in time like v αi e λit. The time evolution of the intensive isplacements is given by δy α (t) = n a j v αj e λjt (7) j=1 The coefficients a j are etermine in the usual way - by matching initial conitions: S αβ δx β ( + ) = δy α ( + ) = v αj a j (8) an inverting this matrix relation. Equation (5) shows how the relaxation time scales λ 1 i are etermine as functions of both the static an kinetic coefficients. Equation (8) escribes how the initial conitions enter into the ynamics.

3 3 III. ENERGY REPRESENTATION The entropy an more familiar energy representation are relate by something like a similarity transformation. Specifically, the matrix transformation relating isplacements of the inepenent extensive variables in the two representations (c.f., Table I) is E α = R α β Xβ eg. S 1 P T T V = µ T 1 U V N 1 N (9) The intensive variables are relate by y = 1 T Rt i. In the energy representation the static an ynamic equations are δe α = U αβ δi β U = 1 T RSRt t δeα = K αβ δi β K = 1 T R KR t (1) The generalize eigenvalue equation in this representation is ( K αβ + λu αβ) δi β () = (11) The more familiar linear response coefficients in the ( ) energy representation are U αβ = 2 U(E) 1. E α E This matrix is positive efinite, since U is a minimum at equilib- β rium. Similarly, the kinetic coefficients K in this representation form a negative semiefinite matrix, by arguments similar to those surrouning Eq. (4). Since U an K are relate to S an K by ientical transformations, the eigenvalue spectrum is the same in both representations. This must be true on the basis of physical arguments. The eigenvectors iffer by the transformations in Eq. (1) involving the nonsingular change of basis matrix R α β. IV. CONSTRAINTS We illustrate what happens when constraints are place on the system by consiering a simple gas in a cyliner [2]. The generalize eigenvalue problem in the energy representation has the form {[ ] [ ]}( ) K 11 K 12 U K 21 K 22 + λ 11 U 12 T() U 21 U 22 = P() (12) The stanar linear response functions are [ ] [ ] U 11 U 12 CP /T V α U 21 U 22 = P V α P V β T (13) FIG. 1: The entropy is suenly increase insie a cyliner that is thoroughly insulate (K SS = ). The temperature, pressure, an volume evolve in time as shown in Fig. 2. [ ] U 11 U 12 1 [ ] [ ] U11 U U 21 U 22 = 12 T/CV 1/V α = S U 21 U 22 1/V α S 1/V β S (14) We have exhibite the negative signs explicitly for the negative semiefinite matrix of kinetic coefficients K, an we further take K to be iagonal. The matrix element K 11 = K SS is measure in entropy flux per unit temperature increase an K 22 = K V V efines the rate of volume change per unit ecrease in pressure. If the piston separating the gas in the cyliner from the reservoir is a goo insulator (Fig. 1), K 11 =, there is one constraint an one vanishing eigenvalue. ( The ) unnormalize eigenvector with λ 1 = λ S = is. The 1 secon eigenvalue is λ 2 = λ V = K 22 U( 22 an the ) corresponing unnormalize eigenvector is U 12 U 11. These two eigenvectrors are orthogonal uner the metric U. The intensive variables evolve in time like an

4 4 S Responses T V - P Time FIG. 2: Response of the perturbations to the experiment shown in Fig. 1.The single ecay time scale is etermine by λ 2 = K 22 U 22. [ ] [ ] [ ] T(t) = a P(t) 1 e t 1 + a 2 e λ2t U 12 U 11 (15) For this case S( + ) = S, V ( + ) = an Eq. (8) is explicitly [ ] [ ] [ ] [ ][ ] U11 U 12 S T( = + ) 1 U 12 a1 U 21 U 22 P( + = ) U 11 a 2 (16) The coefficients are a 1 = S/U 11, a 2 = S U 21 /U 11. The values of the two pairs of conjugate variables, uner the no heat flow constraint K 11 =, are given at t = + an t in Table II. The evolution of these perturbations is shown in Fig. 2. All variables except S have the same characteristic ecay time λ 1 V, since there is only one ecay channel. The ual constraint is illustrate in Fig. 3. In this case the piston is suenly isplace an maintaine fixe at its new position. This constraint is represente by setting K 22 = : the rate of volume change per negative unit pressure change is zero. In this case the zero eigenvalue is λ V = an the nonzero eigenvalue is λ S = K 11 U 11. The solution is compute as before, subject to initial conitions V ( + ) = V fixe an S( + ) =. The asymptotic behavior (t = +, t ) is summarize in Table II. As for the case shown in Fig. 1, all variables except V have the same characteristic ecay time λ 1 S, since there is only one ecay channel. The time evolution of these four variables is similar to that shown in Fig. 2, with the exchange ( S, T) ( V, P). If the ecay times are ientical (λ V in Fig. 1 an λ S in Fig. 3) the two processes exhibit ynamical symmetry as well as the asymptotic symmetry escribe in the next section. V. SYMMETRIES It is possible to compare proucts of extensive variables with their conjugate intensive variables, since all FIG. 3: The volume of a cyliner is suenly increase an the piston is glue at its new position (K V V = ). such proucts have the imensions of energy. Table II shows that the only nontrivial prouct of the responing variables is E r ( ) i r ( + ). This can be compare with the cross prouct of the forcing variables E f ( + ) i f ( ). above is The result in the two cases E r ( ) i r ( + ) E f ( + ) i f ( ) = Ufr U fr (17) This is true in general. The ratios of all such proucts are given by the elements of the LeChatelier matrix L αβ = U αβ U αβ = L βα [1, 2, 3]. This matrix is real an symmetric. Its iagonal matrix elements are greater than one [1, 2, 3, 7]. These are thermoynamic stability conitions. They provie quantitative expressions for LeChatelier s Direct Principle. If an extensive forcing variable is suenly change an hel constant, i f ( + )/ i f ( ) 1. Dually, if an intensive variable is suenly change an hel constant, E f ( )/ E f ( + ) 1 [1, 2, 3]. Further, the sum of the matrix elements L αβ in each row an in each column is equal to 1. Finally, if the forcing an responing variables are interchange, the cross ratios as given in Eq. (17) are equal, as seen in Table II. Further symmetries are present for systems with two egrees of freeom. In such cases the constraint requires one of the two eigenvalues to vanish. The two systems have ientical relaxation time scales provie the nonzero

5 5 TABLE II: Values of the thermoynamic variables uner constraints for a gas in a cyliner. Fig.1 Forcing Responing Channel Channel S T V P λ S = λ V = K 22 U 22 t = + S S U 11 S U 21 t S S/U 11 S U 21 /U 11 Fig.3 Responing Forcing Channel Channel S T V P λ S = K 11 U 11 λ V = t = + V U 12 V V U 22 t V U 12 /U 22 V V/U 22 eigenvalues are equal. The conition is that the nonzero kinetic coefficient K jj multiplie by the corresponing covariant matrix element U jj is the same for both experiments. Finally, if the initial perturbations, S in Fig. 1 an V in Fig. 3, obey ( S) 2 /U 11 = ( V ) 2 /U 22, the time epenence of the internal energy, U(t), is exactly the same for both experiments. to an excellent approximation. The components v αj of the jth eigenvector v j can be constructe irectly from the j (j 1) submatrix in the upper left-han corner of the matrix U of susceptibilities. The component v αj is the minor of this submatrix obtaine by removing row α. The three eigenvectors of Eq. (19) are, to a very goo approximation 1 U21 U 11 +(U21 U 32 U 31 U 22 ) (U 11 U 32 U 31 U 12 ) +(U 11 U 22 U 21 U 12 ) (21) The eigenvalues are easily etermine from these eigenvectors an Eq. (21). Extensive an Intensive Variables i³ E¹ i¹ i² E³ log (t) E² VI. CASCADING CONSTRAINTS V = [v αj ] = The three eigenvalues satisfy K jj v 2 jj = λ jv t j U v j = λ j v 11 v 12 v 13 v 22 v 23 v 33 j α,β=1 (19) v αj U αβ v βj (2) FIG. 4: Response of all thermoynamic variables when three channels have wiely separate time scales an the slowest extensive variable is perturbe. When there are multiple relaxation channels with wiely iffering time scales, the general solution that In Fig. 4 we illustrate the ynamics when there are mixes the static an ynamic coefficients given in Eq. (5) three channels with wiely separate relaxation time simplifies. For specificity we consier a system with three scales T 3 1, T an T 1 1 8, with T j = 1/λ j. egrees of freeom for which the matrix of kinetic coefficients is iagonal with K 11 /K 22 1 an K 22 /K In this figure the extensive variable with the longest relaxation time, E 1, is initially perturbe. Immeiately The equation etermining the eigenvalues/eigenvectors following the perturbation all three intensive variables of this problem is assume nonzero values while the remaining two extensive variables remain zero. K11 K 22 λ U11 U 12 U 13 U 21 U 22 U 23 v As t passes through the shortest time scale, T 3 1, 1 all three intensive variables relax to new values; the force v 2 = K 33 U 31 U 32 U 33 v 3 (18) The matrix of eigenvectors is upper triangular: with the shortest time scale, i 3, rops to zero. Its conjugate extensive variable E 3, rises to a nonzero value. Although i 1 an i 2 relax to new values, E 1 an E 2 remain unchange at the initial value an zero, respectively. As t passes through the intermeiate time scale T 2, the force i 2 rops to zero an its conjugate extensive variable, E 2, becomes nonzero. The extensive variable E 3 with the shorter time scale relaxes to a new value while the extensive variable E 1 with the longer time scale continues to remain unchange. Finally, as (if) t excees the longest time scale T 1 the corresponing force i 1 also rops to zero. At this point, all perturbations have relaxe to zero when no eigenvalues of the generalize eigenvalue equation are zero.

6 6 The succession of steps is summarize in Table III. This table also inclues cases for which the extensive variables E 2 with intermeiate time scale, an E 3 with shortest time scale are initially perturbe. Throughout this relaxation process the linear relation Eq. (1) between extensive an intensive thermoynamic variables is maintaine. In the quiet regimes between time scales T j+1 t T j the three thermoynamic variables (labele with *) can be etermine from the three that are given explicitly in Table III by simple matrix methos previously introuce to simplify the computation of thermoynamic partial erivatives [5, 6]. When faster extensive thermoynamic variables are initially perturbe the relaxation takes place faster, as shown in Table 3. This comes about because the faster variables o not have sufficient time to fee into the slower extensive variables before they relax to zero. VII. CONCLUSIONS The ynamical aspects of LeChatelier s Principle are escribe by the generalize eigenvalue equation Eq. (5) in the linear response regime. The number of close an open relaxation channels is etermine by the number of zero an nonzero eigenvalues of the matrix of kinetic coefficients. The nonzero eigenvalues are complicate functions of the matrices of static an ynamic (kinetic) coefficients etermine through the generalize eigenvalue equation. These equations were constructe in both the entropy (Eq. (5)) an energy (Eq. (11)) representations. There is, as usual, a clean separation of the ynamics into the equations of motion an the initial conitions (Eq. 8). When there is a single relaxation channel there is only one ecay time scale. Uner such conitions the LeChatelier symmetries are exhibite [1, 2, 3]. These symmetries relate proucts of conjugate variables in the asymptotic limits t = + an t. If the nonzero kinetic coefficients are properly ajuste, so that the responing time scales in ual experiments are equal, there is also a ynamical symmetry. When two or more relaxation channels exist an are well-separate in time, the successive relaxations through each of the wiely separate time scales can be treate as if each was a single channel relaxation with a single time scale. In such cases the asymptotic an even the ynamical symmetries exist on each sie of the newlyopene relaxation channel, as shown in Fig. 4 an Table III. Acknowlegment: It is a pleasure to acknowlege the inspiration affore by Prof. R. D. Levine for this problem. TABLE III: Succession of steps as t increases when λ 1 λ 2 λ 3 or T 3 T 2 T 1 an the initial perturbation is in the extensive variable corresponing to the longest (top), intermeiate (mile) an shortest (bottom) time scale. Each matrix contains three pairs of extensive an conjugate intensive variables ( E α, i α ), orere by relaxation time scale from slowest (top) to fastest (bottom). t = + T 3 t T 2 t T 1 t E1, E 2, E 3, E1,, E 2,, E1,,, REFERENCES [1] R. Gilmore, LeChatelier Reciprocal Relations, J. Chem. Phys. 76, (1982). [2] R. Gilmore, LeChatelier Reciprocal Relations an the Mechanical Analog, Am. J. Phys. 51, (1983). [3] R. Gilmore an R. D. Levine, LeChatelier s Principle with Multiple Relaxation Channels, Phys. Rev. A33, (1986). [4] F. Weinhol, Metric Geometry of Equilibrium Thermoynamics, J. Chem. Phys. 63, (1975); Elementary Formal Structure of a Vector-algebraic Representation of Equilibrium Thermoynamics, J. Chem. Phys. 63, (1975). [5] R. Gilmore, Thermoynamic Partial Derivatives, J. Chem. Phys. 75, (1982). [6] R. Gilmore, Higher Thermoynamic Partial Derivatives, J. Chem. Phys. 77, (1982). [7] R. Gilmore, Catastrophe Theory for Scientists an Engineers, NY: Wiley, [8] C. Kittel, Elementary Statistical Physics, NY: Wiley, [9] L. Onsager, Reciprocal Relations in Irreversible Processes. I, Phys. Rev. 37, (1931); Reciprocal Relations in Irreversible Processes. II, Phys. Rev. 38, (1931).