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J. Phys. Chem. 1980, 84, 2996-2998
Analysis of Relaxation Amplitudes for an Unrestricted Number of Coupled Chemical Equllibria, Using Laplace Transform Klyoshl Tamura' and 2. A. Schelly" Department of Chemistry, The University of Texas at Arllngton, Arlington, Texas 76019 (Received: August 7, 1960)
A new method is described for the analysis of the relaxation amplitudes of an unrestricted number of multiple coupled chemical equilibria, using Laplace transform. The method enables one to obtain a general expression for the relaxation amplitudes, and it offers significant reduction in mathematical labor compared to the usual normal mode analysis. A master equation is presented for immediate computation of the amplitudes.
Introduction In relaxation method studies2"of fast chemical reactions an equilibrium system is perturbed by a sudden change of an external or internal4thermodynamic parameter, while the relaxation af the system to equilibrium is monitored. Typically, the temperature, pressure, electric field strength, or composition4 is changed in a stepwise (jump methods) or repetitive fashion (stationary methods). In either case, the spectrum of relaxation times and relaxation amplitudes represents the primary experimental results. If the perturbation is small enough for the linearization of the relaxation rate equations, the relaxation times q are related to the equilibrium concentrations ci and rate constants ki in a relatively simple manner. The relaxation amplitudes are functions of the thermodynamic properties of the reacting system and, in the case of coupled equilibria, also of the relative relaxation rates of the normal modes of reactions. The relaxation amplitudes provide complementary information, and/or a means of checking on the reaction mechanism suggested by the analysis of relaxation times. Eigen and DeMaeyer have d e ~ e l o p e d the ~ ? ~general thermodynamic relations governing chemical relaxation, and presented also the general treatment for coupled equilibria. In principle, the relaxation amplitudes of multiple coupled equilibria can be calculated by using a normal mode analysis which involves tedious algebraic manipulations. In fact, expressions only for two-6 and specific three-'step systems have been derived. An equation developed for the overall mean relaxation amplitude of an unrestricted number of coupled equilibrias is applicable with the solvent and concentration-jump methods only. In the present paper we describe a new method for the analysis of the relaxation amplitudes of an unrestricted number of coupled equilibria, using the Laplace transform. The Laplace transform is especially suitable for the amplitude analysis, because it is particularly effective in solving initial value problems of linear differential equations with constant coefficient^.^ The major advantages of this method of analysis are the saving in computational labor and the possibility of simultaneously obtaining expressions for both the amplitudes and the relaxation times. Theory Let us consider R elementary equilibrium reactions simultaneously present in a system. If the number of different species is N , one can symbolize the total of the reactions by
where vi is the stoichiometric coefficient (by definition, positive for products and negative for reactants) of the ith species Ai. Simultaneous equilibria may be coupled energetically and/or chemically. We shall consider only pure chemical coupling, where each equilibrium has at least one species (the coupling species) common with other equilibria. We shall restrict our discussion to relaxations which are induced by a single jump of a thermodynamic variable, where the perturbation can be treated as a rectangular step function. In addition, we assume that the perturbation is sufficiently small for the linearization of the rate equations. For simplicity, first we discuss the case of two coupled equilibria, followed by the results of the general case. Two Coupled Equilibria. Due to the mass conservation relations, for two coupled equilibria only two rate equations with two independent concentration variables are needed for the kinetic description of the system. The rate equations can be transformed into the linearized relaxation equations2s3 -dxi/dt = ~11x1+ ~ 1 2 x 2 (2) -dx,/dt = ~ 2 1 x 1+ ~ 2 2 x 2 where the time t = 0 is set to the instant of the step perturbation. The deviations of the individual concentrations ci(t) from the final equilibrium values c i ( ~are ) given by xi = ci(t) - E i ( ~ ) .The coefficients aij are functions of the rate constants ki and concentrations ci at the thermodynamic conditions of the final equilibrium. Let L(f(t))be the Laplace transform of f(t) as defined by &(f(t))= lwexp(-st) 0 f(t) dt Applying the transform to the eq 2, one obtains (all + sly1 + a1zY2 = Xl(0) (4) U2lYl + (a22 + sly2 = x2(0) where yi and xi(0) are the Laplace transforms and the initial values of xi, respectively. The xi(0) are, of course, the deviations of ci at t = 0 from their final equilibrium values. The solutions of (4) are
N
C Y , A=~0
i=l
(1)
0022-3654/80/2084-2996$01.OO/O
(3)
0 1980 American Chemical Society
Letters
The Journal of Physical Chemistry, Vol. 84, No. 23, 1980 2997
where
Such a case often arises for multiple equilibrium systems, when all but two equilibria equilibrate much more rapidly than the perturbation change. In such a case, the summation in eq 10 must be done over all the equilibria involved. General Case of Multiple Coupled Equilibria. The previous treatment can easily be generalized to multiple step systems. If the relaxation of such a system is described by n independent concentration variables xi, the relaxation equation is given by2p3J1
D is a function of the parameter s. By putting D(s) = 0, one finds the characteristic equation of the homogeneous system (2) which has only real negative eigenvalues;2let them be -AL and -A2. Thus, if D(s) is expanded, the polynomial can be written as a product of linear factors
D ( s ) = (s
+ A,)(s + A,)
Taking into account that the numerator determinants in eq 5 are polynomial functions of s by one order less than the denominator D,and assuming X1 z A,, eq 5 can be rewritten as Y1 Y2
= All/b = A21/b
(8)
where A, (for j = 1. or 2) are given by
-I:'
-
h.jI
From the known transform LC(exp(-At)] = l/(s + A), the inverse Laplace transformation of eq 8 results in x1 = All exp(-Alt) x, = Azl exp(-Alt)
+ A12exp(-X2t) + A22exp(-A,t)
i = 1, 2, ..., n
(12)
Applying the Laplace transform, we obtain n
C(aij + Sdij)Yj = X j ( 0 )
(13)
Using eq 15, the solution of (13) is yi = CAij/(s + X j )
(10)
2
C (dCi/d In Kj),,(A In Kj), j;.1
where 6, is the Kronecker delta. If we define the determinant (14) D = laij + ~ S i j l
n
As long as the eigenvalues are different, as we assumed previously (A, # A,), the physical content of eq 10 is clear. With the absolute eigenvalues being the reciprocal relaxation times, Le. Ai == l / ~eq~10, describe the time course of the independent concentration variables xi, with the preexponential factors Aij as the relaxation amplitudes.1° In typical jump experiments (T, P, E , solvent, and concentration-jump), where usually optical absorbance or electrical conductivity are used for detection, the experimental total signal amplitude is a linear crombination of the x i (Le,, alxl + a2xz,where the constants ai are determined by the experimental condition). For a comparison of the theoretical and experimental amplitudes, the latter ones are obtained directly from the analysis of the experimental relaxation trace, and the theoretical ones, Aij, are calculated using eq 9. The total theoretical amplitude is a linear combination of the A,. The initial values xi(0) needed in (9) can be calculated by using the equilibrium and mass conservation relations before and after perturbation. In the ordinary jump methods (except for the concentration-jump where the equilibrium constants are not a f f e ~ t e d )the , ~ ~initial ~ values xi(0) are given by xi(0) =
j=1
the characteristic equation D = 0 of the homogeneous system (12) has only real negative eigenvalues2of s. Let them be -A1, -A2, ..., -An, and assume that no two eigenvalues are equal, After factorization of the determinantal polynomial into linear factors, we have
a12 Q22
-dxj/dt = Cajjxj
j=l
+ A,) + A12/(s + A,) + A,) + A22/(s + A,)
A I , = ( k l ) ' ( A , - 1,)
n
(7)
(11)
where i refers to species, j to reactions, and the subscript to initial conditions. Equations 9 and 10 are essentially the same as those obtained by Thusius6 through the normal mode analysis of two coupled equilibria. Here one should emphasize that the treatment from eq 2 to 10 is applicable to more than two coupled equilibria, as long as the relaxation equations are expressed by eq 2.
(16)
j=l
which after inverse Laplace transformation becomes n
xi =
with
I
all
FAij exp(-Ajt) ]=1 - x/
0..
i,j = 1, 2,
,,-,
(17)
XI(0)
*'.
aln
I
...,n
where the product is calculated for 1 = 1 to n, except I = j ; IMi,,(Aj)l is the minor of the determinant la,, - hjijs,( with respect to the i,m element. Again, the experimental signal may be compared with the time course of the concentration variables xi (eq 17), and their indivildual preexponential factors Aij can be computed from the master equation (18). Thus, the Laplace transform offers a simpler alternative to the normal mode analysis of relaxation amplitudes. A similar treatment as presented in this paper can also be applied to nonrectangular single-step perturbations,12 which will be reported in due course. Acknowledgment. This work was partially supported by the R. A. Welch Foundation and the Organized Research Fund of TJTA. Acknowledgment is made to the donors of the Petroleum Research Fund, administered by the American Chemical Society, for partial support of this research.
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References and Notes (1) R. A. Welch Postdoctoral Fellow. On leave of absence from the National Defense Academy, Yokosuka, Japan. (2) M. Elgen and L. DeMaeyer in “Technique of Organic Chemistry”, 2nd ed, Vol. 8, Part 2, S.L. Friess, E. S. Lewis, and A. Welssberger, Ed., Interscience, New York, 1963, Chapter 8. (3) G. G. Hammes, Ed., “Investigation of Rates and Mechanisms of Reactions”, 3rd ed, Part 11, Wiley, New York, 1974. (4) D.Y. Chao and 2. A. Schelly, J. phys. Chem., 79, 2734-2736 (1975). (5) M. Elgen and L. DeMaeyer, ref 3, Chapter 3. (6) D. Thusius, J. Am. Chem. SOC., 94, 356-363 (1972). (7) G. H. Czerlinskl, “Chemical Relaxation”, Marcel Dekker, New York,
1966. (8) 2. A. Schelly and D. Y. Chao, Adv. Mol. Relax. Roc., 14, 191-202 (1979). (9) W. E. Boyce and R. C. DiPrima, “Elementary Differential Equations and Boundary Value Problems”, 3rd ed, Wiley, New York, 1977. (IO) These relaxation concentrationamplitudes should not be confused with relaxation signalamplitudes, or with relaxatlon amplltudes expressed in terms of change of extent of reaction A t (ref 4 and 8). (1 1) G. W. Castellan, Ber. Bunsenges. phys. Chem., 67, 898-908 (1963). (12) The rectangular shape of the step perturbation treated In the present paper Is lmpllcltly expressed by the time independence of the equillbrium concentrations E,@), which are also chosen as reference values.
Effect of Pressure on the Lifetimes and Quenching of Transition Metal Complex Ion Phosphorescence A. D. Kirk and Gerald B. Porter’ Department of Chemistty, University of British Columbla, Vancouver, British Columbia, Canada VBT 1Y6 (Recelved: August 25, 1980)
A hydrostatic pressure of 2.3 kbar has almost no effect on the phosphorescence decay lifetime of Ru(bpy)32+, Wen):+, or Cr(bpy)?+,with AV = -1 mL as is expected from the unimolecular nature of the degrading processes. Quenching of the Ru(bpy)gB+phosphorescence by oxygen is also not affected by large hydrostatic pressures, however, for Co(aca& and methyl viologen, which quench by electron transfer rather than energy transfer, the quenching constants increase by about 40% at 2.3 kbar, with AV = -2.6 mL mol-l.
Introduction Little information is available about the effect of large hydrostatic pressures on spectroscopic and photophysical processes involving inorganic complex ions of the transition elements. Shifts have been reported in the emission spectra of ruby and sapphire, which have Cr(II1) in a nearly Oh environment, but no kinetic data are given.’ Drickamer et al.2-4have measured the small blue shifts in absorption spectra of transition metal compounds as solids, and Fyfe? the corresponding shifts in aqueous solution, that occur at high pressures. Large changes in emission quantum yields of several organic dyes have been studied? Recently, Angermann et al.7 have reported some measurements of the effect of pressure on photochemical quantum yields. The data are difficult to interpret because the quantum yield is a composite of rate constants, each of which may be differently affected by pressure. We have therefore made measurements of the pressure effect on luminescence lifetimes, and on the quenching of luminescence, which are more readily interpreted and which are necessary in the interpretation of the photochemical effects. Experimental Section Lifetimes were determined in aqueous solution at 16-18 “C in a two-aperture cell having sapphire windows.8 A nitrogen laser, 337 nm, 8 ns pulse width, was used for excitation, and a B & L monochromator and RCA 8645 photomultiplier were used as detectors, whose output was displayed on a Tektronix oscilloscope and photographed. The linear geometry of the high-pressure cell required that high optical density, nonfluorescent solution filters be used to exclude laser light from the detection system. Since the high-pressure cell severely vignetted the entrance and exit light paths, our studies were limited to 0022-3654/80/2084-2998$01.OO/O
those complexes with emission yields greater than and we placed emphasis on the well-characterized Ru(bpy),2+.l0J1However, we were able to obtain some useful data on two chromium complexes of photochemical importance. Results The phosphorescence lifetime of Ru(bpy),2+in aqueous deoxygenated solution decreases with increasing pressure, from 560 f 10 ns at atmospheric pressure to 515 f 10 ns at 2.3 kbar. The ratio of these lifetimes, 1.09 f 0.02, was established by repeated cycling between the high and low pressures. Well-degassed solutions of Cr(bpy)t+exhibited a similar small decrease in lifetime, from 71 f 2 to 64 f 2 ps, with a ratio of 1.11f 0.03. The measurements with Cr(ez~),~+ were less accurate because of a smaller quantum yield, and also because the exciting laser radiation is at about half the wavelength of the phosphorescence, hence scattered light in the grating monochromator is severe. The lifetimes are 1.75 f 0.15ps at atmospheric pressure and 1.45 f 0.15 ks at 2.3 kbar, and their ratio is 1.21 f 0.15. The apparent volumes of activation, calculated from eq 1, are -0.7 f 0.2, -0.9 f 0.2, and -1.6 f 1.0 mL mol-’ for
R ~ ( b p y ) , ~C+r, ( b ~ y ) , ~and + , Cr(en),,+, respectively. The quenching of Ru(bpy)gB+ phosphorescence was followed with three different quenchers: oxygen, Co(acac),, ion, and methyl viologen (N,iV-dimethyl-4,4’-dipyridinium MV2+). For a solution of R ~ ( b p y ) saturated ~~+ with O2gas at 1 bar, sealed in an optical cell without vapor head space and 0 1980 Amerlcan Chemical Soclety
