J . Phys. Chem. 1994,98, 5058-5064
5058
Oxidative Quenching of the Excited State of Tris(2,2'-bipyridine)ruthenium(2+) Ion by Methylviologen. Variation of Solution Medium and Temperature Hai Sun, Akio Yoshimura,' and Morton Z. Hoffman' Department of Chemistry, Boston University, Boston, Massachusetts 0221 5 Received: November 16, 1993; In Final Form: March 12, 1994'
-
The rate constant (k4)for the quenching of * R ~ ( b p y ) 3 ~by + methylviologen (MV2+) has been determined as a function of the mole fraction of CH3CN ( X A N )in aqueous mixtures; k, goes through a minimum at XAN 0.4, which is suggested to occur because of the dynamic solvent effect for the electron-transfer reaction in the mixed solvents, whereby the relaxation dynamics of the solvent is a strong function of its composition. The cage escape yield (q,J for the release of the redox products into bulk solution has been determined as a function of ionic strength, temperature, and XAN;the apparent activation energy for the back electron transfer (&t) between the geminate redox pair, R ~ ( b p y ) 3 ~and + MV'+, within the solvent cage produced upon quenching is obtained from the data through the application of the simple cage model. Ebt is independent of ionic strength in aqueous solutions, but is strongly dependent on solvent composition in mixed solvents. However, Ebt decreases smoothly from 9.6 kJ mol-' in aqueous solution to -5.5 kJ mol-1 in CH3CN-rich solvents as XANis increased, requiring that a modification of the simple model be made. The concept of a kinetically important reorientation of the geminate redox pair is introduced to account for the experimental observations; the simple solvent cage model can be viewed as a limiting case in the modified model.
Introduction The quenching of * R ~ ( b p y ) 3 ~(bpy + = 2,2'-bipyridine) by methylviologen (1,l -dimethyL4,4'-bipyridinedication; MV2+)still serves, after almost twenty years, as the archtypical model system for the study of bimolecular excited-state electron-transfer reactions in homogeneous solution against which other systems are compared.2 Because of the need to understand the behavior of this system in great detail in order for it to serve as a standard, we undertook several years ago to measure the quantum yield (9)of production of the one-electron reduced methylviologen radical cation (MV*+) in continuous and pulsed-laser flash photolysis under a wide range of solution medium condition^.^ Inasmuch as the experimental value of depends on the extent to which *Ru(bpy)32+ is quenched by MVZ+, which, in turn, depends on [MVZ+], the more fundamental quantity is vce,the cage escape yield of the redox products into bulk solution. Our work3 and that of others4 have demonstrated the general fact that, for this system, vw decreases with increasing ionic strength in aqueous solution;however, the dependence of qCcon temperature in neat and HzO-CH~CNmixed solvents has not been explored. Inasmuch as one goal in the development of artificial photosynthetic systems is the maximization of the yields of chargecarriers, the study of the experimental factors that impact on that goal is of great potential importance. The overall mechanism of the oxidative quenching of *Ru( b p ~ ) 3 ~by+ MV2+ is very well established and is expressed by reactions 1-4. In the absence of a sacrificial electron donor, the system is completely reversible; the redox pair generated from quenching reaction 3 is annihilated through reaction 4 in bulk solution with no net formation of products.
hu
Ru(bpy),'+
0
+ R ~ ( b p y ) ~+~MV" +
-
*Ru(bpy)32+ MV2+
*Ru(bpy)F
(1)
Abstract published in Aduance ACS Abstracts, April 15, 1994.
0022-3654/94/2098-5058$04.50/0
+
R ~ ( b p y ) ~ ~MV" +
+
Ru(bpy)32+ MV2+
k, k,,
(3) (4)
Theefficiencyofquenching, vS(= kq[MVZ+]/(k,+ k,[MV2+]) = (kd - k,)/kd, where kd is the rate constant of the decay of * R ~ ( b p y ) 3 in ~ +the presence of MVz+), measures the competition between reactions 2 and 3; the experimental value of 9 reflects the efficiencies of quenching and the release of the redox products in reaction 3 (9= qsqcc). In this study, we determined qce and k, as a function of temperature, the mole fraction of acetonitrile (XAN) in CH3CN-HzO mixed solvents,and the ionic strength ( p ) of the solution (at XAN= 0). The conventional models that have been used in the past to describe the general dependencies have been applied to the results; when unexpected correlations are observed as a result of the detailed examination, modifications to the models are suggested.
Experimental Section [Ru(bpy)3]Cly6HzO (GFS Chemicals) was used as received. Methylviologen dichloride (Aldrich) was recrystallized three times from ethanol. Methylviologenhexafluorophosphate was obtained by converting methylviologen dichloride to the PFb- salt and crystallizing it three times from water. Acetonitrile (Aldrich Optima), Na2S04 (Baker), and NaC104 (Fisher) were used as received. Distilled water was further purified by passage through a Millipore purification train. Ionic strength was controlled by NazS04 and NaC104 in aqueous and CH~CN-HZOsolutions, respectively. All solutions contained 20 pM R ~ ( b p y ) 3 ~and + were freshly prepared prior to use. Values of the lifetimes of * R ~ ( b p y ) 3 ~in+the absence (7, = l/k,) and presence ( T ~= l/kd) of MVZ+ were determined by monitoring the excited-state emission from Ar-saturated solutions with the use of pulsed-laser flash photolysis (532-nm excitation) system that has been described b e f ~ r e .The ~ loss of ground-state R ~ ( b p y ) ~ 2and + the formation of MV'+ were obtained from the changes of absorbance at 450 and 605 nm, respectively. 0 1994 American Chemical Society
Quenching of *Ru(bpy)32+ by Methylviologen
The Journal of Physical Chemistry, Vol. 98, No. 19, 1994 5059
TABLE 1: Values of E, (kJ mol-'), and log A for the Quenching of *Ru(bpy$+ by MV*+ kq, lo9 M-' S-', fOrXAN=
'
0.00
0.19
0.34
0.58
0.80
1.00
1.5 1.5 1.7 1.9 2.1 2.3 2.3 2.6 2.7 2.9 3.2
0.86 0.97 1.0 1.1 1.2 1.2 1.4 1.4 1.4 1.5 1.6
0.75 0.78 0.85 0.86 0.88 0.95 0.99 1.1 1.2 1.2 1.2
0.94 0.98 1.1 1.1 1.1 1.3 1.3 1.3 1.2 1.5 1.5
1.5 1.6 1.7 1.7 1.8 2.0 2.1 2.1 2.3 2.3 2.5
1.9 2.0 2.2 2.3 2.4 2.5 2.6 2.9 3.1 3.1 3.4
11.6 11.31
8.8 10.58
7.2 10.20
6.7 10.21
8.0 10.64
8.6 10.86
IogAet"
7.8 11.08
6.5 10.51
5.7 10.43
5.4 10.58
7.0 11.17
8.5 11.65
log ketb
10.59
10.33
10.33
10.66
11.32
T, OC 10.0 15.0 20.0 25.0 30.0 35.0 40.0 45.0 50.0 55.0 65.0
E, log4 Eet
a
"
e
\
4 50 t 40
9.4
; -
9.2
9.0
I
I
I
t
I
I
T. 'C
i
+ 65
I
x*.+ A 0.00
-3 5 5 +- 4 5 -3 3 5 + 25 +2- 15 -A7
0.25-
0 0.19 0 0.34
- A
I
0.30-
For a = 10.7 A. For a = 7.9 A.
96
.
1
I
0 1.00
p
0.20-
0.15-
9.2
0.10-
I
0.0 9.01
8.6 2.9
c
I
I
I
I
I
1
I
3.0
3.1
3.2
3.3
3.4
3.5
3.6
0.5
1 .o
1.5
2.0
2.5
v, M
Figure 3. Plots of qOcvs p (NaZS04) for aqueous solution.
l O O O / T , K"
Figurel. Plotsoflogk,vs l/TatdifferentXAN(p =0.050M;NaC104).
Temperature regulation was achieved to fO.l "C with the use of a Brinkman Model RM6 controller.
Results Quenching Rate Constants. Values of k, were obtained as a function of temperature (10-65 "C)in neat and mixed CH3CN-Hz0 solutions from plots of T , / T ~vs [MVz+]for Ar-purged solutions containing Ru(bpy)3Z+and 0.47-6.3 mM MVZ+ at an ionic strength of 0.050 M (NaC104); the results are given in Table 1. Plots of log kq vs 1/T were linear (Figure l), from which the apparent activation energy (E,) and the preexponential factor (4)in the Arrhenius equation were calculated (Table 1). Figure 2 shows the variations of log k,, as a function of XANa t several temperatures. Quantum Yields. When excited, deaerated solutions containing only R ~ ( b p y ) ~ Zexhibited + the characteristic positive (A, 360 nm) and negative absorptions (A, 450 nm) that correspond to the formation of *Ru(bpy)32+ and the bleaching of the ground state (reaction 1). Decay of these features occurred via firstorder kinetics ( k o ) . Since the differential extinction coefficient at 450 nm, A6450 (=e* - €2, where e* and €2 are the extinction coefficients for the excited and ground states at 450 nm, respectively), is well established (-( 1.Ok 0.09) X lo4M-1 cm-1): the concentration of *Ru(bpy)32+can be determined from the evaluation of the differential absorbance a t 450 nm immediately after the flash ( u 4 5 0 ) . Thus, [*R~(bpy)3~+] = AA450/Atl, where 1 is the pathlength of the cell (2 cm); extrapolation of the recovery
of the 450-nm absorption back to the midpoint of the flash yields hA450. In the presence of MV2+, the decay of the excited state occurred more rapidly (kd); the absorption at 605 nm corresponds to the formation of MV*+ (6605 = 1.37 X lo4 M-1 cm-l).' Thus, [ M W ] = hAa05/taosl. Inasmuchas CP = [MV'+]/ [ * R ~ ( b p y ) ~ ~qce + ]can , be directly obtained. Values of k,, kd, AA605, and AA4%were determined for aqueous solutions ( p = 1.0 M; NazS04) as a function of [MVZ+](0.2-20 mM) and temperature (15-65 "C) and were converted to qq and CP. As expected, plots of 0 vs qq were linear; qccis obtained from the slopes. A similar treatment of the data for aqueous solutions containing 2.7-1 1.3 mM MVZ' as a function of ionic strength (0.012-2.6 M; Na2S04) and temperature 7-65 "C) yielded the data shown in Figure 3. For solutions with p = 0.05 M (NaC104) at 10-65 "C,the results were obtained as a function of [MV*+] (0.52-5.1 mM) and XAN(0-1). Again, plots of CP vs qq were linear, yielding the variation of qce as a function of solution medium. Generally, thevalue of q,increases as X A N is increased; the data a t several temperatures are shown in Figure 4.
Discussion The quenching reaction and the subsequent processes that lead to the formation of redox products in bulksolution can be described generically for both oxidative or reductive quenching by the following scheme, in which the initially formed precursor complex (reaction 5 ) can undergo either diffusional dissociation (reaction 6) or electron transfer (reaction 7); the successor complex, in turn, can engage in back electron transfer to form the groundstate reactants (reaction 8) or release the redox pair into bulk
The Journal of Physical Chemistry, Vol. 98, No. 19, 1994
5060
I
I
I
I
I
00
02
04
0 50 T
I
Sun et al. is commonly used to account for the value of qaand its dependence on solution medium and energy parameters.I0 From this model, eqs 15 and 16 are easily derived; k,, representing the diffusion of species into bulk solution out of the solvent cage, can be calculated from eqs 12-14. The only unknown that remains is kbt, the rate constant of intramolecular electron transfer within the geminate pair, which is considered to follow the Marcus theory. Thus, the value of kbt would be expected to be a function of the driving forceof the backelectron transfer reaction, A&', showing "normal" and "inverted" Marcus behavior.
I
'r.
8
F
t
' O r 06
1
I
08
10
XAN
Figure 4. Variation of qcc as a function of XANand temperature.
solution from the solvent cage (reaction 9). It is assumed that the reverse of reaction 7 is too slow to be important due to its large endoergicity.
M*
+Q
-
[M**-Q]
kdif
(5)
(l / q = ) - = kbt/ k,
(16)
Quenching. The effect of temperature on k, for the quenching of * R ~ ( b p y ) by ~ ~ MV2+ + in CH~CN-HIO mixed solvents has been described previously by Greiner et a1.l' They reported a dependence of k, on X A N not unlike that shown in Figure 2, where k, goes through a minimum at XAN 0.4. They evaluated the activation parameters of k,, although the temperature range of their studies and the details of the data and its management were not disclosed. Their calculated values of AH*as a function of XANwere very similar to those shown in Table 1;values of AG* ranged from 17.6 to 19.4 kJ mol-' with a maximum around XAN 0.4. However, calculatedvalues of kdif and kAifdid not exhibit the same variation with solvent; neither did k,,, when calculated by the treatment of Brunschwig et al.I2 By using the empirical AG*values and solvent dielectric relaxation times in the absolute rate theory equation for adiabatic electron transfer, calculated and experimental values of k, showed the same qualitative, but not quantitative, dependence on XAN. Instead, we have chosen to evaluate the various rate constants in terms of the Arrhenius equation, in order to explore the variations in the preexponential factors and activation energies as a function of solution medium. The numerical calculation of kdif and kAif,according to eqs 12 and 13 were performed with the following parameters: rA(Ru2+and Ru3+)= 7.0 A, rB(MV2+and M V + ) 3.3 A; r(C104-) = 2.4 r(S0d2-) = 2.6 UA = r A + r(C104-) = 9.4 8,and UB = rB + r(C104-) = 5.7 8,in H2OCH3CN solvents; UA = r A + r(S042-) = 9.6 8, and UB = rB + r(S042-) = 5.9 8, in aqueous solution; a = rA rB = 10.3 A. Values of e and q for aqueous solutionI4 and for CHSCN-H~O mixtures as a function of temperature15J6 were taken from the literature. From k,, kdif,and kdif,values of k,, as a function of temperature and XANcan be calculated from eq 10; plots of log k,, vs 1/ Twere linear, from which the activation energy and preexponential factor for the electron-transfer reaction, E,, and A,,, respectively, were obtained (Table 1). Comparison plots of Ap and &,, and E , and E,,, vs XANare given in Figure 5. It is curious to find that the activation energies show a minimum around XAN= 0.4, which, if the preexponential factor were independent of solvent composition, as assumed earlier," would result in a maximum in the rate constant. Clearly, the values of k, are determined to some extent by the values of the preexponential factor. Unfortunately, calculated values of k, and their temperature variation are critically dependent on the parameters used in the calculation of kdif and kAif, If a is taken as 7.9 A, instead of 10.3 A, representing, perhaps, a closer approach of the reactants mediated by the counterions, and the other parameters are kept unchanged, the calculated value of k,, is temperature independent in all the solvent mixtures, rendering the electron-transfer reaction activationless. These values of k,, are given in Table 1 and are plotted in Figure 6; a value of k,, in neat CH3CN could not be estimated inasmuch as k, = kdif under that condition. This analysis suggests that it is the preexponential factor, not the activation energy, that is responsible for the observed dependence of k, on XAN,whereby k, goes through a minimum
-
-
[M-/' ...Q' 1-1 M-1' + Q+/- k, (9) According to this scheme, the experimental value of k, can be expressed by eq 10; values of kdif and kAif can be calculated from the theoretical equations (1 1,12) of Smoluchowski, Debye, and Eigen.* Here, q is the solvent viscosity, r A and rB are the radii of the reactants, a = r A + rB, and r is the distance separating the two reactants. Values of r A and rB can be estimated from the dimensions measured along three molecular axes of CPK spacefilling models by r = [(d,d,~l~)~/~]/2. In addition, kr, is Boltzmann's constant, N is Avogadro's number, t is the static dielectric constant of the solvent, 2, and ZBare the ionic charges of the reactants, e is the electronic charge, and u is the radius of a reactant plus that of the dominant counterion in the ionic atmosphere. The terms w(r,p) and j3 are given by eq 13 and 14;8b,9w(a,p) equals w(r,p) when r = a. . )
kdif= IkBTN( 30007
+
2 5)
1
+
rB
A .'
aJOmr-'exp[w(r,p)/kBT] d r (1 1)
(
j3 = 1OOOtk,T 8*Ne2 ) ' I 2 Reactions 8 and 9 comprise the simple solvent cage model that
+
The Journal of Physical Chemistry, Vol. 98, No. 19, 1994 5061
Quenching of *Ru(bpy)32+ by Methylviologen
2.4
I
I
I
I
I
I
I
I
30
3 1
3 2
33
3 4
35
I
I
11
1
Oi,
I-
::]\ 10
(b) -
1
2 9
c
’-
9
E
3 6
tOOOTT, K”
Figure 7. Plots of ln(q=-I-l) vs 1 / T as a function of fi ( N a ~ S 0 4 )in aqueous solution. 25
6
I
I
I
I
XAN
000
5
t
1I 0.0
I
I
I
I
I
I
0.2
0.4
0.6
0.8
1.0
XAN
Figure 5. Plots of A, and E,and ktand Eet (for u = 10.7 A) as a function of XAN. 11.6
I
I
I
I
I
I
/ 00 2 9
I
I
I
I
I
I
30
3 1
3.2
33
34
3 5
It
3 6
100OiT K”
Figure 8. Plots of ln(q&I) HzO mixtures.
vs 1 / T as a function of XAN in CH3CN-
TABLE 2: Difference in Activation Energies in Aqueous Solutions 1024,
00
,
I
02
0.4
I
0.6
u, M (Na2SOd
I
0.8
1 .o
&N
Figure 6. Plot of log ket (for a = 7.9
-
A) as a function of XAN.
at&N 0.4. This phenomenonmay beattributed tothedynamic solvent effect for the electron-transfer reaction 7 in the mixed solvents; it has been found that, for many electron-transfer reactions in polar solvent^,^^ k,, is a function of the relaxation dynamics of the solvent, which can be described by eq 17, where 7,is the lifetime of relaxation of the solvent coordinate. If eq 17 applies to the system discussed here, the results suggest that 7, is strongly dependent on solvent composition in mixed media. In any event, we believe that the values of the preexponential factor play a critical role in determining the values of k,. k,, = 7;’ exp(AG*/RT)
(17)
Cage Escape. The results clearly show that ,7 for the Ru(bpy)s2+/MV2+system is largest at high temperatures, high ionic strength, and highervalues Of-&. According toeq 12, k , (=k& should increase with increasing temperature and should be larger for CH3CN as the solvent than H 2 0 due to the differences in the viscosity and dielectric constant; on the other hand, k, should decrease with increasing ionic strength due to the positivecharges
0.012 0.026 0.030 0.10 0.16
E,
- EM,kJ mol-l 8.0 9.1 8.1 8.1 9.1
M,
M (Na2S0,) 0.33 0.61 1.o 1.8 2.0
EGO- Eb,, kJ mol-I
9.1 8.7 9.3 8.8
7.5 av 8.6 & 0.6
of the species within the solvent cage, although the exact analytical dependence is complex. If eq 15 expresses the relationship among q,, kbt, and k,, and the dependencies of kbt on the parameters are weaker than they are for k,, then the effect of temperature and solution medium on ,7 should follow those of k,. Indeed, that is exactly the case experimentally. However, up until this time, the interdependencies of temperature, ionic strength, and XANhave not been examined. If both kbt and k, are expressed in terms of the Arrhenius equation, then In kbt/k, (=ln(q,-l-l)) is given by A + ( E , Ebt)/RT, where A contains the preexponential factors and E, and Ebt are the activation energies for diffusional cage escape and back electron transfer, respectively. Indeed, plots of ln(7,-1-1) vs 1 / T as a function of p in aqueous solution (Figure 7) and XANin mixed solvent (Figure 8) are linear, from which values of E, - Ebt are calculated (Tables 2 and 3). It is clear from Table 2 that the values of E, - Ebt are independent of ionic strength, with an average value of 8.6 f 0.6 (std dev) kJ mol-’
5062 The Journal of Physical Chemistry, Vol. 98, No. 19, 1994
00
I
I
I
I
02
0 4
0 6
08
oxidized quenchers. On the other hand, only a very weak correlation between log(q,-l-l) and was obtained for the oxidative quenching of the excited states of a series of Ru(I1)bpy and phen (= 1,lo-phenanthroline) complexes, including *Ru(bpy)j2+,by MV2+.3j Similarly, Yonemoto et reported that a plot of log(q,-’-l) vs for the oxidative quenching of *Ru(II) by viologens, including MVZ+,was very scattered and did not show any evident correlation; on the other hand, when Ru(I1) and the viologens were covalently bound, the AGOdependence of the rate constants for forward and back electron transfer traversed the “normal” and ”inverted” Marcus regions. In addition, there is no evident dependency for the reductive quenching of a derivative of *Ru(bpy)j2+by aromatic amit1es.2~ These conflicting results raise the question of the general applicability of the simple mechanistic model described by reactions 8 and 9. The increasing body of evidence that values of q, in bimolecular quenching reactions involving Ru(I1) sensitizers can be weakly dependent, if at all, on AGbtoand the observation here of negative values of Ebt that result from the use of the simple model indicate that the events in the solvent cage cannot be described only by reactions 8 and 9 and that modifications must be introduced into the mechanistic representation of bimolecular electron-transfer quenching reactions. At the same time, any kinetic model will also have to account for the bell-shaped curves that have been observed in the relationship between log(q,-’-l) and AGbto. We propose the following modification to the simplecagemodel. It can be visualized that the geminate pair undergoes a reorientation within the solvent cage in order to facilitate back electron transfer. In the following scheme, [M-/+-Q+/-] I and [M-/+-.Q+/-] represent the solvent cage before and after the reorientation processes, respectively, k,, is the rate constant for reorientation, and kbt and kb( are the back electron transfer rate constants before and after the reorientation has occurred, respectively. It can be reasonably assumed that the values of k , of both geminate pair species are the same.
10
XAN
Figure 9. Calculated values of
Ebt
as a function of XAN,
TABLE 3 Activtaion Energies in Neat and Hz-CHJCN Mixed Solvents; ~.r= 0.05 M (NaC104) XaN
E,-Ebt. kJ mol-’ E,, kJ mol-’ Ebt, kJ mol-’
0.00
0.19
0.34
0.58
0.80
1.00
9.1 18.6 9.5
15.0 17.4 2.4
13.3
13.3 12.8 -0.5
15.0 9.48 -5.5
13.3 7.90 -5.4
16.7 3.4
in aqueous solution. Since the value of E, - Ebt (9.1 kJ mol-’) at XAN= 0 with NaC104 control of the ionic strength (Table 3) is within the error limit of that obtained with ionic strength controlled by Na2S04 (Table 2), E , - Ebt can be regarded as being independent of the ionic environment that surrounds the reactants. Inasmuch as k , can be evaluated from eq 12 with the use of the parameters given above, it is possible to obtain values of E,, as a function of XAN(Table 3) from linear plots of In k, vs 1/ T. Although the calculated values of k , depend on the parameters of the reactants (ZA,Zg, r, u, a ) and the ionic strength, the value of E, must be a property of the solvent, independent of the charges and dimensions of the reactants and the value of p. As a check on the validity of the calculated values of E,, it should be noted that the activation energy for the self-diffusion of H2O has been demonstrated experimentally tobe 19 kJmol-lJ*and theactivation energy for the oxidative quenching of Ru(I1) by quinones and nitrobenzenes at the diffusional plateau in CH3CN is -7 kJ mol-lJ9 all in very good agreement with the calculated values given in Table 3. Thus, values of Ebt can be calculated with confidence; the results are given in Table 3. Most interestingly, Ebtdecreases monotonically asXAN is increased from 9.5 kJ mol-’ in water to about -5.5 kJ mol-I in CH3CN-rich solvents; a plot of Ebt vs XANis given in Figure 9. Now, a negative activation energy for a single-step electrontransfer reaction is very unusual, although, in theory, it is possible if the entropy change for the reaction is very negative;20however, this is not expected to be the case for the reaction between Ru( b ~ y ) 3 and ~ + M V . We are led to theconclusion that the apparent negative values of Ebt arise because the reaction it is supposed to describe actually consists of more than one mechanistic step. Such an interpretation has been used to explain the negative activation energies of k, that have been observed for theoxidative quenching of *Ru(II) by quinones and nitrobenzenes.lg.21 Modification of the Solvent Cage Model. By the application of the simple solvent cage model (reactiofls 8 and 9), Ohno et ~ 1 . 2 obtained well-defined “bell-shaped” curves for plots of log kbt/ k,, vs AGbto,in accordance with Marcus theory, for the reductive quenching of *Ru(II) by aromatic amines and methoxybenzenes in CHXN-H,O mixtures: it was assumed in their treatment that k,; was aconstant value for the whole range of disparate
Sun et al.
M+O
k I’
[M-’f..,0+’-)2
By the application of the steady-state approximation to the geminate pair intermediates, equations for the rates of formation of the cage-escaped redox products (P)and ground-state starting materials (G) can be easily obtained. Inasmuch as 7 , can be expressed as eq 18, substitution results in eqs 19 and 20. d [PI/dt
’, = d[P]/dt + d[G]/dt
~
(18)
There are two limiting cases for which eq 19 and 20 can be simplified. If k,,
