Relaxation methods were introduced bv Eieen and co- . - workers (I) for studying the rates of fast reactions, with re- laxation times in the 1 sec to 10-9 sec range. The basic prin- ciple of relaxation methods is the small perturbation of an equilibrium system, followed by the measurement of the rate and amplitude of the subsequent reequilibration. To date, relaxation methods have been used mainly in the liquid phase. The instrumental details of the popular step-perturbation (i.e, temperature-, solvent-, electric field-, and pressure-jump) techniques have been reviewed also in this Journal (2).

I t is seldom recognized that relaxation methods owe their conception to prior developments in the field of the thermo- dynamics of irreversible processes, notahly by Onsager (3), Meixner (4 ) and Prieoeine (5). Here. the word "thermodv- namics" & k e d in t h i Gue sense of the word, with emphasis on the dynamic nature of time-dependent thermophysical processes with which it deals. This usage is in obvious contrast to the traditional use of the word thermodynamics for the description of static equilibrium states, whichcorrectly should be called thermostatics.

Chemical relaxation (and chemical reactions in general as well) is an irreversible process by the criterion that the equation describing the time dependent process is variant with regard to the algebraic sign of the variable time t. For re- versible processes the equations are invariant with respect to time.

In discussine the thermodvnamics of relaxation methods. the major aspects of consideration are usually (1) the mode of perturbation of chemical equilibria (forcing parameters), (2) the enforced change of the concentrations (relaxation amplitudes) and (3) the chemical contributions to equations of state. In the present paper, after reviewing the pertinent basic definitions and forcing parameters, the driving force of relaxation and the thermodynamic meaning of the relaxation time T are investipated, using the framework of irreversible . thermodynamics.

Definnlons and the Perlurbatlon of Equilibria Consider the reaction

UIAI + mA2 +. . . e &Ai + Y ~ + ~ A ~ + ~ +. . . . (1)

where, by definition, the stoichiometric coefficients vi are positive for products and negative for reactants. Due to con- servation of matter during chemical reactions, eqn. (1) can also be written as 2iu;Ai = 0.

The extent of reaction E at any time is defined according to De Donder by eqn. (2)

ni = .ni + v i t (2)

where ,n; are the reference values of the mole numbers n; present f i r each species Ai. The equilibrium constant K and the rate of reaction (1) are defined asK = :a? and the rate = dtldt , where a; are the equilibrium values of the activities of A;.

If energy (in the form of heat or work) is exchanged re- versibly between the system and the surrounding, the change of internal energy is given by the first law of thermody- namics

dU = EFjdBj j

(3)

where Fj are intensive variables (generalized forces) and do j

are differentials of extensive variables (generalized dis- placements). Examples of these conjugate variables are the following:

T dS -PdV

E . dD* and E YIP* (4) pi dni A dt H dM

In addition to the familiar thermod)namic symbols (T, S, P, V. n.. €1. E stands for elertric field. P' for nolari~ation. D' is relatedto "electric displacement"'(2), pi is the chemical po- tential, A stands for affinity (defined by De Donder: see eqn. (20)), H for magnetic field and M for magnetization.

Due to the sum in em. 3, indicating that there are several ways in which work can be done on orby the system, the dif- ferential of the Gibbs free enthalpy G becomes ( 1 ) dG = VdP - S d T - D* . d E + Zpidni or in general2

Since the equilibrium constantK and AGO of reaction (1) are related by AG" = -RT In K , the (usually sudden) change of any of the conjugate variables Fj or Bj may affect the equi- librium activities, i.e. may shift the equilibrium. The relaxa- tion time T of the shift is shorter if during eauilibration the extensive variables 0; instead of the intens&e~~ariat,les F, are kept constant tfij. Nevertheless, ex~erimentallv it is easier to vaiy the intensive variables Fj (seilist in eqn.(4)) which led to the development of the T-jump, P-jump, E-iump, solvent (pi)-jump, as-well as the sound absorption or dispeision (os- cillating P and T variation), and dielectric relaxation (oscil- lating variation of E ) methods. Typically, one chooses such a forcing parameter Fj for perturbation to which the equilib- rium is sensitive.

Driving Force of Relaxatlon Once the value of a forcing parameter has been suddenly

changed, reaction (1) starts to relax irreversibly to new equi- librium concentrations. Let us compare the relaxation with other irreversible processes, for the case where the.systems are "not too far" removed from the final equilibrium (linear reeion). " .

The second law of thermodynamics ascertains that the entrow of an isolated svstem increases monotonicallv until . - i t reaches its maximum at equilibrium,

dS/dt > 0 (6 )

This formulation can be extended easily to open systems

Presented in part at the YATO Advancrd Study 1nut:tute un Sew Applicarion~ of Chemical Reloratron Sperrromrtry and other Vast Renetion \lethodc i l l Sulutiun. Abcryztwyth, \Val~r, 10-20 Septrmht'r !"-e L",".

Support by the Robert A. Welch Foundation and the Organized Research Fund of UTA is gratefully acknowledged.

In principle, a magnetic term should also be part of the sum in eqn. (5) . However, since chemical equilihria in solution are insensitive to perturbation hy magnetic field of practical strengths, it is not in- cluded. The equivalence of Zpjdni and -.Ad€ follows from eqns. (19) and (20).

Volume 57, Number 4, April 1980 1 247

which exchange energy and matter with the surrounding. Here d S is composed of two, and only two, terms: dS,, the transfer of entropy across the boundaries, and dSi, the entropy pro- duced within the system (7). The second law assumes that dSi > n - ".

If the assumption of "local equilihrium" (5) is accepted, one can obtain for the entropy production per unit time ( 8 )

where J , are the rates of various processes p involved (chem- ical reactions, heat flow, diffusion) and the X, are the corre- sponding generalized forces (affinities, gradients of temper- ature, gradients of chemical potential, etc.).

Near equilibrium, linear homogeneous relations exist he- tween the fluxes Jo and the forces X, (9). If several processes occur simultaneously, the associated fluxes J , are given by

with Lo,, being the phenomenological coefficients. One im- portant result of linear thermodynamics of irreversihle pro- cesses is the Onsager reciprocity principle (3) which states that the coefficients for coupled processes are symmetrical (if the fluxes J , are rates of changes of state functions):

L , , = L,,, (9)

In other words, if the flux J , corresponding to the irreversihle process p is influenced by the force X,, of another irreversihle process p', then the flux J,, is also influenced by the force X, through the same coefficient.

If only one irreversihle process occurs in a system, J = LX describes the flow. Well-known examples in the linear region are the laws of heat conduction

J = - nT grad T Fourier's law (10)

where KT is the coefficient of thermal conduction; electric current J

J = - x grad 4 Ohm's law (11)

with the electric conductivity K , and electric potential @; fluid flow

J = -y grad P Poiseuille's law (12)

where y is the friction coefficient; and diffusion

J = -D grad C Fick's law (13)

with the diffusion coefficient D and concentration C. In analogy with the physical processes described in eqns.

(10) through (13), next we shall consider chemical reactions, where the rate of the reaction can he considered as the flux J I dtldt , that in the linear region equals to LX. We shall identify the phenomenological coefficient L and the driving force X. For simplicity, instead of (l) , let us use the elemen- tary reaction

A + B & A-l

(14)

for the derivation (7). The rate of reaction (i.e., the rate of equilibration or relaxation) is

dCcldt = ~ICACB - ~ L I C C (15)

Since Ci = nilV, it follows from eqn. (2) that dCi = vjdtllr. Hence eqn. (15) can he written as

dtlVdt = klC~Ce( l - ~ - I C C I ~ ~ C A C B ) (16)

if the forward rate is factored out on the right-hand side of eqn. (15). Since near equilihrium the forward rate will ap- proach the forward equilibrium rate vl

klCnCe - ~ , C A C B -vl (17)

near equilihrium and at low concentrations where Cj - ai eqn. (16) becomes

because k- l lk l =K- ' , and the non-equilibrium ratio of the concentrations CclCnCe can he called Q. Now, QlK can he related to the affmity A , if one considers the total differential of G (at constant T, P and E):

dC = x@Glan;)dni = 1p;dn; = Euirid( (19)

where the last equality follows from eqn. (2). From eqn. (19), the differential quotient dGldE is

dGIdt = Xvip; = AG = -ff (20)

which is the definition of A. But AG = AGO + RT In Q and AGO = -RT in K, hence

ff = -RTln(Q/K) or QIK = exp(-AIRT) (21)

Near equilihrium AIRT << 1, therefore exp(-AlRT) can be expanded into a series and terminated after the second term, resulting in(1-AIRT). Substituting this for QIK in eqn.(l8), we have

dEldt - (VVIIRTIA (22)

Thus, L can he identified as the constant VvIIRT and the driving force of relaxation as the affinity, X = A .

Relaxation Time

Next, let us focus on the time constant T of the relaxation close to equilihrium. As a starting point, we write eqn. (22) in the general form as

dE(T,P)ldt = Sff (T,P,E) (23)

indicating the functional dependencies of the variables. Also -. r a a function of T,P and 5, hut close to equilibrium it can he considered as constant. If A is expanded in a Taylor series about its equilihrium value at constant T, and only the first order terms are kept, we have

A = ~ ( i , P ) + ( a ~ / a t ) d t - F ) + (a f f IaP)~ ,g(P - P) (24)

where the bars_ indicate_eq+hrium values. But at equilibrium the affinity is zero, A((, P ) = 0. With eqn. (24) substituted into (23) yields

d ~ t = ? [ ( a n l a t ) ~ , ~ ( t - F) + ( a f f / ap ) , , ( p - P)1 (25)

or by factoring out -Z(&~/~[)T,P (25) becomes

(a f f laP)~ . t ( p - j5)] (26) = -?(aff~%)T,~[- (E-F) - (Dff Ia~)T,P

The negative quotient of the partial differential quotients in the brackets is equal to ( a t I b P ) ~ , ~ . This is easy to see from the total differential of the affinity A (T,P,[)

which at constant A and T can be rearranged to

( d U b P ) ~ , a = - ( ~ f f / ~ P ) ~ , g / ( ~ f f I d O ~ , ~ (28)

However, for the relaxing system A approaches zero as a function of time, reaching a constant value A = 0 only at final equilihrium. Thus, for a relaxing system, constant A (as in- dicated by the subscript on the left side of eqn. (28)) means fin-al equilihrium conditions, with the equilibrium value o f t = [ = constant, and d E = 0. Therefore,

( ~ £ / ~ P ) T , A ($/~P)T,A = 0 (29)

since is constant. Thus, eqn. (26) becomes (10)

~?:pTtp- ? ( ~ . A / ~ £ ) T . P = ( V V I / R T ) ( ~ A / ~ ~ ) T , P (32)

if F is identified through eqns. (22) and (23).

248 / Journal of Chemical Education

The final conclusion is reached in the last two equations: (21 s c h d y , Z. A. and ~yr ing , E. M., J. CHEM. EDUC., 18, ~ 6 3 9 and ~ 6 9 5 (19711.

(1) the rate of approach toequilibrium (relaxation) is directly i:j ~ ~ ~ , ~ : ; ~ ~ ; ~ ~ ; ; ~ ~ : ~ ~ ~ ~ ~ ~ ~ ~ ~ ; proportional to the distance ( t - 5) from equilibrium (eqn. 31); (51 prigogine, I., " ~ i u d e thermodynamique der p h ~ n u m h ~ e p irreuenib~es: ~ h e s i s . uni-

and (2) the rate coefficient T - ~ of the relaxation has a simple Se;~~~,",',"$;: ::A,, N,NF,9,259 119561. physical meaning, described by eqn. (32). (71 prigogino. I., " ~ h ~ ~ m o d y n a m i c s of kreversibtc ~rocesses: 3rd ~ d . , lnterscience

Literature Cited Publishers, New York, NY, 1967. (8) Prigwine, I., Science. 201,777 (19781.

(11 Eigen, M, and DeMseyer, L. in "Techniques of Organic Chemiluy." Vol. 8. Part 2. (91 DeGraot, S. R, and Mszur. P.."Non-Equilibrium Thermcdynemica? North-Holland, (Editors: Friess, S. L.. Lewis, E. S.. and Weisbergor. A), Interscience Publishers, Amsterdam, 1969. N~~ y&, NY, 1 9 1 , ~ . 895ff porthe deflnjtiun off ie for theequation fo rdo (10) For sn alternative derivation of eqn. (801 see Haase, H.. "Thermodynamics of lrre- see p. 945. versibleProc-s,"Addiaan-Wesley,NewYork, NY. 1969.p. 135.

Volume 57, Number 4, April 1980 / 249