J . Phys. Chem. 1992, 96,4938-4944
4938
Measurement of Quantum Yields for Photochemical Systems Interconverted by Both Light and Heat James R. Sheah*.+and James J. Diamond* Hewlett- Packard Laboratories, Palo Alto, California 94304, and Linfeld College, McMinnville, Oregon 971 28 (Received: February 5, 1992)
We present an analysis of the kinetics of photochemical reactions in which two molecules (or groups of molecules) are interconverted by light and one is thermally unstable, in the presence of arbitrary amounts of absorbance, for both stirred and transport-free media. We show that all photokinetic parameters can be determined in a single bleaching experiment that stops well short of achieving a steady state, regardless of the time scale of the reaction (provided it is a one-photon process). Explicit analytical formulas for the fitting of data to calculation, with error estimates, are provided, along with a discussion of implementation using common commercial software.
I. Introduction Photoreactions of the type described by eq 1 are very widely studied systems in photochemistry.’ This equation describes, for hv
example, the reversible cycloaddition reactions of polycyclic aromatic compounds2-8 and photochromic isomerization^.^-^^ Such isomerizations are of interest for their relevance to the study of molecular energeti~s,~q~*~*’*~ as probes of the interaction of molecules with their environment,” and for various applications in imaging,I4Js permeabilityI6 and solubility” control, actinometry,I8 and data storage.19 In many of these experiments, a substantial absorbance cannot be avoided and may indeed be desired. This results in the coupling of the reaction kinetics and optical extinction (Beer’s law), and the solution of this pair of coupled nonlinear equations presents some difficulties. In a paper referred to hereafter as SDS,20we have shown that in the absence of the thermal term in eq 1 the equations admit an analytical solution for arbitrary reaction rate laws and extents of absorbance by reactants, products, and medium, for both fully stirred and unstirred (e.g., transport-free) systems. The rate constants may be determined from optical absorbance data at the irradiation wavelength. (To separately determine primary quantum yields 4~and extinction coefficients a requires that a for either the reactant or the product be measured separately, since and a are multiplicative factors in the rate coefficient .) In our previous paper we did not consider the presence of a thermal (i.e. light independent) reaction, which is nevertheless exhibited by many photochromic systems. Recently Rau, et al. reported2’ a method for determining quantum yields in such a system by irradiating at a specified wavelength to a photostationary state (PSS) at various intensities and recording the absorbance at that and some different wavelengths. An extrapolation to infinite intensity gives the desired parameters, including a for the thermally unstable form. Although this is clearly a useful approach, and is quite simple to apply, it has some limitations. In order to ensure a good extrapolation, many separate runs are required, each irradiated to stationarity; and the measurement of absorbance at a second wavelength necessitates either multiple irradiation sources and detectors or the transporting of the sample from the irradiation system to a spectrometer (which puts an upper limit on the allowable rate of the thermal reaction). Errors may also be introduced if the system experiences any side reactions, since the irradiation must fully achieve a stationary state. The method is not applicable to the solid state. Finally, it is necessary @J
Hewlett-Packard Laboratories. College.
1Linfield
that the quantum yield be identical at the two wavelengths. In this paper we show that under a wide range of important circumstances our previous analysis can be extended to systems in which the thermal reaction is present. The general case of eq 1 cannot be solved analytically. However, the unidirectional reaction R P both thermally and photochemically can be solved exactly for stirred systems with little more effort than without the thermal term; this is an experimentally significant case. Similar equations are obtained for a transport-free system with a first-order rate law and absorbance only by the reactant; more generally one is led to a nonlinear integral equation whose convergence appears satisfactory (although this is still under investigation). In the bidirectional case, the thermal component must be neglected in order to use our previously derived methods. We will see, however, that the range of applicability is not thereby diminished, since the analysis uses “real-time” absorbance data (recorded during photolysis) rather than measurements taken after reaching stationary state. Thus, in principle one can always analyze the system successfully by using a sufficiently high light intensity, such that an adequate amount of photoreaction for the desired accuracy is induced before a significant amount of thermal reaction occurs. Satisfactory accuracy is obtainable from an irradiation that stops far short of the PSS,and a thermal reaction with a relaxation time several times longer than this irradiation time can be tolerated. Only a single sample, with one irradiation, is required (for the kinetic parameters at a given wavelength), which is of considerable importance for systematic studies of temperature or solvent dependence, etc. We give explicit equations for the analysis of systems obeying eq 1 with first-order reactions in each direction. We then illustrate their utility with the data from the paper of Rau et al.,z’ on the photoisomerization of p(dimethy1amino)azobenzene. Analytical equations describing the precision with which the parameters can be determined are derived and plotted. Finally, we discuss briefly the relevant experimental and computational methods.
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11. Bidirectional Reaction, Fully Stirred, No Thermal Component
Equation 1 defines the basic system under consideration, with rate constants k l , k-l, and kAfor the forward photochemical (R P),reverse (P R), and thermal reactions, respectively. We will use the notation S for the more thermally stable species and U for the relatively unstable species, in diferent cases; depending on the transformation under consideration, each of these may be designated “R” (for reactant) or “P” (for product) to correspond with the notation of SDS. We now consider eq 1, with S = R, U = P, and kA = 0; the concentrations are denoted C, for R and C, for P, with superscript 0 for initial values. This problem has already been fully solved in SDS, although explicit expressions for the requisite intermediate functions were not worked out from the general formulas. We do so here, assuming first-order re-
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0022-3654/92/2096-4938%03.00/0 @ 1992 American Chemical Society
The Journal of Physical Chemistry, Vol. 96, No. 12, I992 4939
Light and Heat Interconverted Photochemical Systems
TABLE I: Extent of Renctioo Needed To Achieve 10%Uncertainty in Calculated Parameterso
Y
rr
r,
e
4Arf)
v(Ak-d
0.2128 0.2128
3.523
17.552 12.255
1.70 1.85
3.05 3.25
0.40
1.90
3.523
3.523
2
2
"The last two columns give the value of u at which the uncertainty in the parameter (rfor kJ has dropped to 10% when determined by least squares with the corresponding number of points (Au = 0.05 per point).
-
- 1 408
0
1
2
3
4
5
6
scaled dose
7
a
9
1
0
v
Figure 1. Transmittance b vs scaled dose u for = 3.523, Pr = 0.2128, and 8 = 17.552. Also shown is the relative uncertainty in the determined parameters, as calculated from eq 36, assuming that the parameters were determined by fitting the data by two-parameter least squares and that the data have an uncertainty ( a ) of 1%.
actions in both directions (other rate laws can be treated in the same way). The complete solution for the concentration i5 (a = C J e , the bar indicates spatially averaged quantities) and transmittance b as functions of the scaled dose v is provided by the following formulas:
b = dr,
r = rf+ p a ( v )
- 0 880
0
1
2
3
4 5 6 7 scaled dose v
8
9
Figure 2. Relative derivatives d(ln b)/d(ln p), for p = (labeled on plot) for the parameters used in Figure 1.
1
0
rr,k,,and kl
with respect to each of the parameters to be determined (in each case the other parameters are held constant).
(3)
where ys = rr+ pg(s) db
-- -
where p = ri - I",; ri is the initial absorbance and rrthe absorbance when all R has been converted to P (note that throughout we use natural logarithms; e.g., the transmittance b = Z/Zo = e-r). In the remainder of this section we set = 0 for simplicity, since that is expected to hold for systems with a thermally unstable P. There are three unknowns: k, ( = L Y ~ C P ~ + ~ ) ,k-, (=auCPu+s), and rr. The measurement of as is presumably not a difficulty, and so kl determines @s+u. The determination of rf from the photolysis experiment gives au and hence CP from k-,. The nonlinear algebraic equation for 4, denoted a-fthe tilde distinguishes it from the closely related but distinct function G used in transport-free systems), is readily solved with the NewtonRaphson algorithm (cf. Appendix I); the required derivative is
where
e,
(7) Because the equations are nonlinear, a least-squares fit to experimental data may appear to present some difficulties. Provided a reasonable rough guess of the parameter values can be made, however, a nonlinear least-squares problem may be reduced to a linear one. As will be shown below, such is the case for the present problem, and in fact one parameter (k,)can often be determined alone, leaving only a tweparameter fit. The details of the least-squares analysis, including the estimation of error in the values of the parameters that are determined, are presented in Appendix I, along with a discussion of useful software. Here we merely list the required derivative of the measured quantity
These equations have been used to calculatethe timedependent transmittance for the data published by Rau et al.21on p-(dimethylamino)azobenzene, using the parameters obtained by them at 405 nm: p = 3.31,B = 17.552, k, = 24000 L/(mol cm) (base 10 absorbance), k-, = 1450, F,= 0.2128. The result is displayed in Figure 1 along with the uncertainty (1u) that would be attached to the determination of these parameters if a fit were made to data having a relative l a uncertainty of 1%. The uncertainty is shown as a function of the number of points used for fitting (there are 200 in the bleaching curve). From this it can be seen that error limits of less than 10% are reached for exposure doses that are only about 20-30% of that required to reach steady state (cf. also Table I). At the highest intensity used by Rau et al. (1.76 mW/cm2), steady state is reached at about v = 10, corresponding to 30.4 s; this can be compared with the 240-s half-life of the thermal reaction. Clearly the present methods could be satis-
4940 The Journal of Physical Chemistry, Vol. 96, No. 12, 1992 TABLE II: Rate of Convergence of Iterated Liaerrized Lenst-Sauares h l v s i s O rr= 0.2128; e = 12.255 rr= 2; e = 2
rr
e
rr
e
0 0.183 0.215 0.2125
11 11.44 12.25 12.24
0' 0.200
ll* 12.29
3 1.93 2.14 2.07 2.01 2.00
5 1.59 2.98 2.21 2.02 2.00
100 13.18
3* 1.92 2.05 2.00
5*
0.3 0.190
1.59 2.27 2.02
"he first line in each group of numbers gives the starting parameters ("guess"); the next line is the result from the analysis ("estimate"). Subsequent lines give the estimate resulting from using the previous line as a guess. rr = 3.523 for both cases. Sets marked with an asterisk use points only up to Y = 3.2.
factorily used for this system even at a rather low light intensity. There are two questions that must be answered before this analysis can be adopted generally: (1) is the fit robust (are the effects of the parameters well separated, so that an error in one is not masked by a compensating error in the other); and (2) can a satisfactory initial guess be readily made? There are two ways to examine the fmt question. In Figure 2, we show the logarithmic derivatives (i.e., the fractional change in b per unit fractional change in the parameter) computed for the parameters of Figure 1. The effects for this system are indeed well separated. The kl curve is opposite in sign to the others and quite different in magnitude; the rrand curves are also of significantly different form and are especially well separated in the early stages of reaction, with r, stronger. This is because there is "instantly" a change in absorbance when a molecule of R is converted to P, while the reverse reaction rate is not significant until enough P has built up to make it so. It is also possible to gain useful analytical insight into this question, by looking at the solution for small v (early times), which is derived in Appendix I1 (where 'small Y" is defined quantitatively). The result is
The transmittance near the beginning of irradiation depends only on kl and rr,and not on k1; this is in accord with Figure 2. In addition, we can see that the dependence of r on k, is explicitly different from the dependence on rp With respect to the second question, eq 12 shows that when rf is small (as in the present case), it has very little effect for sufficiently small Y, and kl can be determined simply from the initial slope of b(t). Because of the large difference in effect of rfand kl in the relatively early stages of reaction, as illustrated both by Figure 2 and eq 12, it is feasible to estimate rrby fitting the data in that regime as well as possible by adjusting rfalone, inserting an arbitrary small value for Ll. An approximate kl is then easily found, and the resulting values of the unknowns can then be used in the least-squares routine. However, we have found that such a procedure is more complex than typically needed. Table I1 shows the results of linear least-squares fitting for data using the parameters of Figure 1 except with B = 12.255, using two sets of initial guesses as listed. It is seen that, even starting from values that are quite far from correct, only two iterations arc actually needed to obtain results correct to 1% (a third iteration confirms the convergence and provides essentially exact values). It is also seen that if the calculations are restricted to the points below Y = 3.25, the convergence is even faster; this is presumably due to the fact that the derivatives are most distinct from each other in that region. (The noise in real data will of course limit the ultimate precision, but this will not substantially affect the rate of convergence if the noise is not unreasonably large.)
Sheats and Diamond 0.0 -0.2 -0.4 1;' -0.6
k-1,
0
2
4
3
5
6
0='2.255
7
8
Figure 3. Relative derivatives as in Figure 2 for p = = 3.523, rr= 2, and two values of 0 as noted.
0.160 0.144 0128-
I
I
'
1
,
c
I
,
9 ' 0
v
scaled dose
rrand k-,,for r,
1
-
rf=0.212 8 ; 0 = 2
+
4
0.032 0 0'6 0000 0
r,=3523 "
1
"
"
2
3
"
4
'
'
5
'
'
'
6
scaled dose
'
7
I
"
'
8
9
1
0
v
Figure 4. Transmittance b vs scaled dose Y for values of rrand e as noted.
rr= 3.523 and various
2 09
-3.9ot'
0
'
"
"
1
2
3
"
4
"
5
"
6
scaled dose
"
7
"
8
"
9
1
' 1
0
v
Figure 5. Relative derivatives as in Figure 3, for ri = 3.523, rr= 0.2128, and 0 = 2.
If rfis not small, then eq 12 cannot be used in the same way to determine kl. However, because of the stronger effect of rr compared to early in the reaction (cf. Figure 3), the low dose data (including the region somewhat beyond the applicability of eq 12) can still be approximatelyfitted to the two parameters kl and rrby inspection, using eq 12 and the initial slope to fix one for a given choice of the other. Final values of all three parameters are obtained by least squares. As can be seen from the sample curves in Figure 4, there is no qualitative difference between the effect of large product absorbance and large backward reaction rate. Figure 5 shows that when kl is large while rris small, the effects of the two parameters are comparable at low doses, and kldominates subsequently. This makes it somewhat harder to determine good initial guesses. However, the rate of convergence does not appear to be seriously impaired. In Table I1 we give the linear least-squaresresults for = kl = 24000 (a a data set with ri = 3.523, rr= 2, and parameter set chosen because it results in a very small total change in transmittance; cf. Figure 4), starting with rfand k-,off by 50 and 150%, respectively. It now takes four iterations to converge,
The Journal of Physical Chemistry, Vol. 96, No. 12, I992 4941
Light and Heat Interconverted Photochemical Systems with fairly large oscillations initially. This is reduced to three iterations if the smaller subset of points (up to Y = 3.2)is used. Table I shows that very good accuracy is reached at such relatively early reaction times. (The uncertainty curves are similar in shape to those in Figure 1, except that they drop to a slightly lower value at large Y.)
III. Bidirectional Reaction, Unstirred, No Thermal Component
Equations 4 and 5 in section I1 are applicable;in addition one needs the following:20 4 P , 4 = d#) (13)
w(x) = &‘&) ds = ~ ~- (e-U) 1
+ -X0
(15)
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IV. Unidirectional Reaction, FuUy Stirred
In this section we consider a system U S by both thermal and photochemical reactions. Although this may seem at first glance to be relatively uninteresting, we note that if the conversion S U occurs at some wavelength (with or without U S),and in a different region of the spectrum the conversion U S (alone) occurs, it is in fact quite useful. A sample with a known, stable concentration of U can then be prepared by irradiation at a low enough temperature that the thermal reaction is negligible (the methods of section I1 or I11 are used to determine the concentration). A rapid temperature jump then brings the system to the desired reaction temperature, and its bleaching behavior is recorded. The following analysis is then sufficient to determine the kinetic parameters. The widely-studied merocyanine-spiropyran isomerization is such a ~ y s t e m . ~ ” ~ ~ The development follows that of SDS;equation numbers from that paper will be designated here by an asterisk. We identify R = U and P = S. The scaled kinetic equation aa oa - 1 - = -f(a)b, flu) = (20,42*) 0- 1 aY
-
--
is first transformed into an integral equation:
which are used in place of eqs 2,3,6,and 7;{ = Ff/p. Although the equations differ in detail from those in section 11, the form of the resulting curve is similar,20and their behavior with respect to the effect of the parameters and irradiation time required can be expected to be qualitatively similar as well. The three derivatives required are given below. It should be noted that in solids, where the front face of the sample is typically exposed to air or other gas, the sample reflectivity changes with bleaching. This effect is discussed in SDS and can be incorporated here if necessary. It introduces a multiplicative factor in front of b to determine the measured sample transmittance T. In principle one would take derivatives with respect to T rather than b for the fitting analysis. However, since the effect is a small one (changing the transmittance by a few percent for even a strongly absorbing sample), the extra effort is probably not worthwhile; the equations given here will not be altered by including it.
As before, a and b are the normalized reactant concentration and transmittance, respectively; ~1 and v are scaled distance (along the optic axis z ) and time, respectively, andfla) is the scaled rate law exclusive of intensity m a ) = F ( C , ) / F ( c ) ;F(C,)is the rate function). The essence of the remainder of the derivation is the solution of the right hand side. This procedure therefore depends on the ability to separate the concentration dependence and intensity dependence into separate multiplicative factors as in eq 20,which, as noted earlier, is not possible for the general rate law of eq 1. The situation is simplified, however, if reaction in only one direction is considered. The rate equation is
Normalizing and scaling variables as in SDS and making the assumption that F(C,) = FA(C,),we obtain
\-. , 4(P,V)
+ f v (24)
The thermal reaction results in a constant term that is easily integrated. The steps described in Appendix I of SDS lead immediately to
This equation is quite similar to eq 6 in section I1 and reduces to it for f = 0. (Note that although rfmay in general be nonzero, it would typically be zero for the type of reaction described in the introduction to this section.) For the case of a simple first-order reaction, F(C,)= C,, and f l u ) = a; then g(s) = P. The desired quantities ii and b are computed with the same equations as those in section 11; only a single parameter is to be determined, and preliminary fitting is simply a matter of scaling the calculation. The derivative required for least-squares analysis is The + or - sign in the f expressions corresponds to the derivative with respect to k, or k-,,respectively.
4942 The Journal of Physical Chemistry, Vol. 96, No. 12, 1992
The elementary quantum yield = k/au. In order to obtain a value for 9,au must be independently known (regardless of the method of data analysis; the procedure of Rau et al. is not applicable here). If the system is prepared as suggested at the beginning of this section, this information will be known.
V. Unidirectional Reaction, Unstirred Using the same starting equations as in section IV and analysis along the same lines as in SDS, one can derive a solution in terms of a nonlinear Volterra integral equation that appears to be well behaved for a wide range of parameters, although no general proof of convergence has been obtained. Since the equations are considerably more complicated, we will defer this discussion to a separate publication. Quite important, however, is the special case in which the reaction is first order and { = 0 (no product absorbance); this corresponds to the conditions cited in section IV. Then the differential equations are - -1 aa - b + 5, a(p,O) = 1 a %t ab - -1 a, b(0,v) =1 b aP This pair of equations is exactly the same as the pair describing an ordinary photobleaching reactionm (without any thermal term) b, p Y, and 5 t. Thus the with the interchange of a solutions previously obtained can be immediately carried over to the present case, using these substitutions in eqs 13, 14, and 16, with As) = P , and w(s) = 1 - e-s. If the sample is prepared as described in section IV, there will in general be a gradient of initial concentration which prevents solution for t # 0. When f = 0, however, the reduced variable p may be definedZoas
--
- - -
VI. Discussion The primary application of this analysis is expected to be the determination of kinetic parameters of photoreactions and mechanistic investigations based on photokinetics; this utility has been demonstrated in previous publication^.^^,^^ The drawback of nonlinear fitting is largely compensated by the robustness of the linearization process and its susceptibility to manipulation by commerciallyavailable software. The experimental setup is simple: a single monochromatic light beam and a photodetector with digitized output having good time resolution compared with the reaction time. A single experiment provides all necessary data. The analysis can be used in transport-free media such as polymers, where the spatially averaged light flux formula used by Rau et al. is not applicable. Although the thermal component of a bidirectional reaction is not directly included, we have shown under what conditions it can be rendered insignifcant; the maximum tolerable thermal rate scales linearly with light intensity. There are three limits to the minimum reaction time. The reaction must remain a one-photon process; the usual steady-state approximation for the excited state must be applicable (so that an integrable rate equation pertains); and it must be p i b l e to record the time dependent transmittance during the reaction (which implies either following the time dependence of a single pulse or a high enough repetition rate to give many pulses during the reaction). The third condition can be satisfied down to the nanosecond regime. The first and second conditions depend very much on the specifics of the system. For a reaction proceeding via singlet states, the excited-state lifetime is short enough for the steady-state approximation if the reaction lasts of the order of many tens of nanoseconds. Deviation from singlephoton behavior may arise from nonlinear absorption (multiphoton absorption, saturation, or other nonlinear effects that give rise to an apparent absorption) or from absorbing intermediates. The regime of onset of nonlinear effects varies substantially but rarely cxcurs below a b o ~ t 0.1-1 ~ ~ -MW/cm2 ~ ~ and is often greater. Intensity dependent photochemistry based on the presence of absorbing intermediates can occur at much
Sheats and Diamond lower levels in certain systems with favorable conditions for the buildup of high concentration^,^^,^^ but these effects more commonly also occur only with ”laser” inten~ities,’~.~~ which typically means many kW/cm2 or more. Many photoreactions remain single photon at relatively high intensities; an example37is the isomerization of the fulgide Aberchrome-540. To obtain an order of magnitude estimate of how this affects our analysis, we may compute the relaxation time of a first-order reaction: C , / q = e-kft( k = a@),assuming that 9 varies between 0.1 and 1 and a between lo3 and lo5 L/(mol cm), with a rough “median” value of k = lo7 cm2/mol. For that value at 400 nm, 1 ~.lscorresponds to 30 kW/cm2. From these considerationsit appears that this analysis can be useful for some reactions on at least the microsecond time scale, if not shorter. One valuable application of this capability would be the determination of extinction coefficients for transient intermediates, an area in which conventional techniques involve considerable diffi~ulties.~’Although this will not be possible if the products undergo further photoreactions (which in general gives rise to rate laws that are not linear in intensity), there will remain a rather large range of reactions that are susceptible to treatment, and our analysis suggests that quite high accuracy is feasible. In general eq 1 may not be followed by such systems; in that case the specific functions g(x), w(x), etc., will have to be derived according to the basic equations in SDS. In the event that the rate law cannot be integrated, it may be integrated numerically and the result fitted to a suitable function. This requires numerical integration only of an ordinary differential equation (for which the potential instabilities are likely to be much less severe than for the nonlinearly coupled partial differential equations considered in SDS), and the subsequent analysis is made possible. The formulas in sections IV and V provide information that is not obtainable by other means (except of course by numerically solving the differential equations). One must have some means of separating 9 from a,as for example by the temperature dependent procedure suggested in section IV. Alternatively,if one makes the assumption that 9 is independent of h over the spectral range that includes a region where asand au are both nonzero, and an adjacent region where as = 0 but au # 0, then determination of k-, and au in the region of overlap (by the means of section I1 or 111) gives the (constant) quantum yield 9,and measurement of k-, in the as = 0 region gives au at that irradiation wavelength . If a stable PSS can be readily established (Le., no side reactions), then the absolute spectrum for a stirred system may be obtained in the overlapping region from the two equations rss= rf + (ri- rf)ass (30) ( k , + k - , ) c Z - k-lcZ + k a ( c =0 (31) in which we assume = 0; the sub- or superscript ss refers to PSS values. The first equation is just Beer’s law, and the second expresses the equality of forward and backward rates at PSS. These two equations together with a single measurement using photokinetics to specify values of au and the quantum yields at one wavelength determine the spectrum. This approach allows one to choose h to maximize the precision of determinationof k-, at short reaction times, which occurs where it is largest relative to k , . Clearly it is desirable to have a measurement of the spectrum that is independent of any of these assumptions, for example by a spectral technique that exhibits a clear separation of peaks for S and U, that can be applied simultaneously with the photochemical creation of U. However, there will be many cases where such a technique is inconvenient, unavailable, or nonexistent. The analysis we have presented provides an alternative solution for such problems. Computational Issues. Although the equations in this paper are not unusually complex compared to common usage in physical chemistry, the time required to write and debug a conventional program to solve them can be a significant deterrent to the utility of the analysis. The calculations in this paper were carried out using a combination of the spreadsheet Lotus 1-2-3 and the
e,
c)
Light and Heat Interconverted Photochemical Systems Automated P r ~ g r a m m e r(AP), ~ ~ which is an application that converts graphics-based mathematical input into executable Fortran programs. Spreadsheets such as 1-2-3 can be easily programmed39to do all of the computations required here. Speed would become a vexing issue if large numbers of computations were desired, however: 200 b(v) values require about 90 s with the Fortran program (using an HP Vectra ES-12 PC; 12-MHz 286-based CPU, with 80287 math coprocessor), and while the corresponding time for 1-2-3 is unknown it would certainly be much longer (at least many minutes). In addition, the AP is far simpler to use, due to its graphical input procedure. There are also other suitable PC-based applications that avoid conventional programming difficulties, such as (and not limited to) Mathcad@ and M a t h e m a t i ~ a . ~The ~ Appendix refers to 1-2-3, but other spreadsheets have identical or similar functions. Experimental Issues. For systems in which the time window for each data point is of the order of 100 ms or more, common digital oscilloscopes,data loggers, and data acquisition boards will provide data directly to the PC at low cost. For faster reactions, high-speed transient digitizers are available, but simpler techniques using more commonly available equipment may suffice. If the time window is at least 1 ms, then a mechanical shutter is sufficient to gate the experiment, and the curve may be built up using a much slower recorder. Each point uses a new sample, which is irradiated up to the desired dose, at which time a track-and-hold circuit is triggered to hold the current value of the detector output while the slow digitizer records it. Sample usage can be kept low by an intermittent flow system with a small irradiation volume; this technique has been suggested in connection with high-peak-power laser spectroscopies.28 The corresponding setup for a solid film would be even simpler, involving stepping the sample across the beam. The same data collection system could still be used with reactions faster than 1 ms, along with a pulsed laser or electrical shutters (electro- or acoustooptic, liquid crystal light valve, etc.).
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MI. Conclusions We have extended our mathematical analysis of photobleaching kineticsz0to include the presence of a thermal reaction and have analyzed the conditions under which photokinetic parameters (including the absorption coefficient of a thermally unstable material) may be obtained. For unidirectional reactions (both thermal and photochemical in the same direction), the analysis is simple if the system is stirred or if it is transport free but first order and possesses transparent products; the general solid-state case requires more elaborate analysis. For bidirectional reactions, the thermal reaction cannot be explicitly included; however we show that the photochemical kinetics can be easily analyzed in the presence of a thermal component. The magnitude of the tolerable thermal rate scales linearly with available irradiation intensity; with a few mW/cm2, forward quantum yield -0.2, and (base 10) extinction coefficient t 2 X 104 L/(mol cm),a thermal half-life of -2 min is tolerable. In those regimes where other methods exist that can achieve the same results, this approach provides experimental simplicity at the expense of moderate computations (which can be executed by common commercial PC software) and avoids some limitations (the necessity of equality of CP at two wavelengths and the required absence of side reactions that might prevent the establishment of a true photostationary state). We find that even though there are three unknown parameters to be determined, they can be unambiguously determined to 10%precision or better by fitting of the order of 50 data points, and error estimates can be systematically calculated. Explicit formulas are given for these calculations. The analysis applies to both stirred and transport-free systems. Finally, we consider the lower limit to the permissible time scale and conclude that the analysis should be useful for many transient chemical intermediates, at least in the microsecond regime.
-
Appendix I The Automated Programmer can conveniently produce an ASCII file with values of v, b, and 4 in adjoining columns, which
The Journal of Physical Chemistry, Vol. 96, No. 12, 1992 4943 is the desired format for importing into 1-2-3 where it can be plotted and compared to the experimental data. When appropriate approximate values of the parameters have been chosen, the corresponding file with the values of 4 is read by the AP in calculating the derivatives (this eliminates the need for an iterated loop in that program). Once the derivatives are imported into 1-2-3, they serve as the coefficients in the least-squares analysis (1-2-3/DR command). We summarize here the method of linearization. We follow refs 42 and 43. The parameters are obtained as the solution to the matrix equation X = Ma (32) where X and a are vectors and M is a matrix, with elements defined by (33)
(34)
a i is the approximate value of the mth parameter a,,,, and the index i runs over the data points N, the uncertainty of which is described by the standard deviation ui;f i i is the computed value correspondin to the datum xi.The correct value of the parameter B where a: is the value determined in eq 32 and it om = a; a,,,, means evaluate is assumed that a;
