J. Phys. Chem. 1988, 92, 6264-6269
6264
ground state of RhH2. The ground state of RhH2 was found to be of 2Al symmetry with r,(Rh-H) = 1.51 A and B,(H-Rh-H) = 84'. The ground state of RhH2 was found to be more stable than Rh(4F) H2 or Rh(2F) H2. The Mulliken population analyses of the electronic states of RhH2 reveal considerable ionic character for the Rh-H bond (Rh'H- polarity). The linear electronic states exhibit considerable dsp hybridization of the Rh
+
+
atom while the bent electronic states exhibit mixing of the d*sl and d9 configurations of the rhodium atom. Acknowledgment. This research was supported by the Chemical Sciences, Basic Energy Sciences Division of the U.S. Department of under Grant No. DE-K32-86-ER-13558. Registry No. RhH,, 80800-00-6.
Activation Parameters and Multistage Reaction Schemes Michael J. Blandamer Department of Chemistry, The University, Leicester, LE1 7RH, England
and John W. M. Scott* Department of Chemistry, Memorial University, St. Johns, Newfoundland, Canada (Received: October 2, 1987; In Final Form: March 8, 1988)
-
-
-
Dependences of rate constants k(obsd) on temperature are examined for chemical reactions described by three reaction schemes. Scheme I is A B, Scheme I1 is A == B C, and Scheme 111 is A F= B == C D. Activation parameters are related for Schemes I1 and 111 to the dependence of k(obsd) on temperature and rate constants characterizing the component reactions. Comparisons are drawn between activation parameters calculated by using complete descriptions of the kinetics of reaction and those calculated by using the steady-state hypotheses. Situations are identified in which such approximations can have major and minor impacts, thereby influencing conclusions drawn about the mechanisms of reactions from the dependence of k(obsd) on temperature.
At constant temperature and pressure, rate constants for a given chemical reaction measure standard Gibbs functions of activation, A*Go.A hierarchy of activation parameters exists in which first derivatives with respect to temperature and pressure yield enthalpies A*."" and volumes of activation. A second differentiation with respect to temperature yields isobaric heat capacities of activation A*Cpm.In principle, the dependence on temperature of a rate constant for a given chemical reaction in solution offers an insight into the energetics of reaction. One might reasonably expect that A*Cpmprobes quite subtle features of reaction processes. This is certainly one view,' but often the outcome is not clear-cut.2 In view of our interest in the interpretation of activation parameters for reactions in solution3 we have been concerned with the interplay between kinetic and thermodynamic complexities in contributing toward an understanding of the energetics and mechanisms of reaction^.^^^ The problem is quite general, being encountered in analysis of kinetic data involving inorganic and organic substrates. A striking example that has attracted our attention concerns the solvolysis of tert-butyl chloride in aqueous solution. Good kinetic data from three laboratories are available for this reaction,>' eq 1. Conventionally this reaction (CH,)$Cl(aq)
+ H20
-
( C H W O H ( a q ) + H+(aq) + W a q ) (1) is followed through the increase with time in electrical conduc(1) Robertson, R. E. Prog. Phys. Org. Chem. 1967, 4, 213. (2) (a) Blandamer, M. J.; Robertson, R. E.; Scott, J. M. W.; Vrielink, A. J . Am. Chem. SOC.1980, 102, 2585. (b) Blandamer, M. J.; Robertson, R. E.; Scott, J. M. W. Prog. Phys. Org. Chem. 1985, 15, 149. ( 3 ) Blandamer, M. J.; Burgess, J.; Engberts, J. B. F. N. Chem. SOC.Rev. 1985, 14, 237. (4) Blandamer, M. J.; Burgess, J.; Robertson, R. E.; Scott, J. M. W. Chem. Rev. 1982, 82, 259. (5) Moelwyn-Hughes, E. A.; Robertson, R. E.; Sugamori, S. E. J . Chem. SOC.1965, 1965. (6) Albery, W. J.; Robinson, B. H. Trans. Faraday SOC.1969, 65, 980. (7) Adams, P. A,; Sheppard, J. G. J . Chem. SOC., Faraday Trans. I 1980, 76. 21 14.
0022-3654/88/2092-6264$01.50/0
tivities associated with the formation of H+(aq) and CI-(aq) ions. The kinetics of this reaction are accurately characterized by a first-order rate constant, k(obsd). The associated plot of In (k(obsd)/T] against Ti is curved, and so at each temperature T, k(obsd), A*H(app), and A*C,(app) are calculated; we use app here for reasons discussed below. In general terms eq 1 has the form given in eq 2:
SCHEME I A
kl +
B; k(0bsd) = kl
(2)
where -d[A]/dt = d[B]/dt = k(obsd)[A]
(3)
Isobaric heat capacities of activation at temperature T a r e given by eq 4, where TS = transition state. (We have ignored in eq A*Cp(app) = A*Cpm(aq)= C,"(TS;aq) - C,"(A;aq)
(4)
4 the small contribution arising from translation of the transition state along the reaction coordinate.') For the reaction represented by eq 1, A*Cpm(aq)< 0; this observation was interpretediv5by using Scheme I in terms of a negligible C,"(TS;aq) and a large positive C,"(A;aq) for the hydrophobic substrate (CH3)3CCI. For many reactions in solution, a satisfactory alternative to eq 2 is given by Scheme I1 (eq 5 ) in which the reaction proceeds via an interSCHEME I1 k3
A&B-+C kz
mediate. Scheme I1 has been discussed by many authors (e.g., ref 8-1 1) in terms of the relationship between k(obsd) and the (8) Pyun, C. W. J . Chem. Educ. 1971, 48, 194. (9) Szabo, Z. G. Comprehensive Chemical Kinetics; Bamford, C . H., Tipper, C. F. H., Eds.; Elsevier: London, 1969; Vol. 2, Chapter 1. (10) Ohkubo, K.; Sakamoto, H.; Tsuchihashi, K. Bull. Chem. SOC.Jpn. 1974, 47, 2141.
0 1988 American Chemical Society
Activation Parameters and Reaction Schemes rate constants for the individual steps ki,where i = 1, 2, and 3. This scheme forms the basis of mechanisms for reactions in solution involving some cited inorganic, organometallic, and organic substrates; eg., solvolysis2,6of tert-butyl chloride, substitution atI2 pentacyanoferrates(II), a n a l o g o ~ s ' ~cobalt(I1) ~'~ complexes, analogous15 pentacarbonylmolybdenum(O), and16 (cyclooctadiene)iron tricarbonyl. However, in relatively few cases2,4,6,17 have authors explored consequences of this scheme for the dependence of k(obsd) on temperature where k(obsd) is determined by the rate of formation of substance C. In an important paper, Albery and Robinson6 used a steady-state approximation in conjunction with an assumed dependence of In ki linear with 1/T for i = 1 , 2, and 3 . They showed that the predicted A'Cp(obsd) is negative; they also argued that a "thermodynamic" interpretation of this quantity in terms of Scheme I is consequently invalid. The ensuing debate2 was intense between those6 who favored interpretation of the kinetics of the reaction in eq 1 using Scheme I and those' who favored Scheme 11. Nevertheless, an important question concerned the impact of the steady-state hypothesis on activation parameters derived for Scheme 11. We attemptedI7 to answer this question by using the equations for k(obsd) based on Scheme I1 and by examining the first and second derivatives with respect to temperature. The approach was not subtle, had limited application, and, furthermore, was algebraically tedious. Therefore we have addressed the issue of describing a general approach to this problem. In fact the method described here forms the basis of an extension of the treatment to Scheme I11 (eq 6).
SCHEME 111
This scheme has features in common with the mechanistic scheme proposed by Winstein et al.18*19 for solvolytic reactions. Further example^^*^^ include reactions involving Delelpine's salt and NCSions,20M 0 ( C 0 ) ~ ( p y (py ) ~ = pyridyl), and diimines in protic organic media.21 For both schemes I1 and I11 (eq 5 and 6), we probe the impact of kinetic complexity on signs and magnitudes of derived activation parameters. The approach follows the procedure described previously" in which we consider relationships between k(obsd) and derived activation energies of parameters characterizing rate constants for individual steps in the reaction schemes. The analysis shows the extent to which a measured dependence of k(obsd) on temperature can aid in identifying mechanistic features for a given reaction. Analysis. Chemical Kinetics. Schemes I and I1 describe the changes in composition of systems that are asymptotically stable.24925 Each path on the phase portrait for a system with given
(11) Ritchie, C. D. Physical Organic Chemistry; Dekker: New York, 1975. (12) Toma, H. E.; Malin, J. M. Inorg. Chem. 1974, 13, 1772. (13) Wilmarth, W. K.; Byrd, J. E.; Po, H. N . Coord. Chem. 1983,51,209. (14) Abou-El-Wafe, M. H. M.; Burnet, M. G. J . Chem. SOC.,Chem. Commun. 1983, 833. (15) Covey, W. D.; Brown, T. L. Inorg. Chem. 1973, 12, 2820. (16) Johnson, B. F. E.; Lewis, J.; Twigg, M. V. J. Chem. SOC.,Dalton Trans. 1974, 24 1 . (17) Blandamer, M. J.; Robertson, R. E.; Ralph, E.; Scott, J. M. W. J . Chem. SOC.,Faraday Trans. I 1983, 79, 1289. (18) Winstein, S . ; Clippinger, E.; Fainberg, A. H.; Heck, R.; Robinson, G. C. J. Am. Chem. SOC.1956, 78, 328. ( 1 9) Streitweiser, A. Soluolyric Reactions; McGraw-Hill: New York, 1962. (20) Hills, E. F.; Richens, D. T.; Sykes, A. G . Inorg. Chem. 1986,25,3144. (21) Elias, H.; Macholdt, H.-T.; Wannowius, K. J.; Blandamer, M. J.; Burgess, J.; Clark, B. Inorg. Chem. 1986, 25, 3048. (22) Elias, H.; Macholdt, H.-T.; Wannowius, K. J.; Blandamer, M. J.; Burgess, J.; Clark, B. Inorg. Chim. Acta 1986, 25, 3048. (23) Harmer, M. A.; Sykes, A. G. Inorg. Chem. 1980, 19, 2881. (24) Immirz, F.; Andronov, A . A,; Witt, A. A.; Khaikin, S . E. Theory of Oscillators; Fishwick, W., Transl. Ed.; Pergamon: Oxford, 1966. (25) Hayashi, C . Non-linear Oscillations in Physical Systems; McGrawHill: New York, 1964.
The Journal of Physical Chemistry, Vol. 92, No. 22, 1988 6265 [A] at time zero is unique to that system. Analysis. Scheme 11. The system of linear differential equations characterizing Scheme I1 is presented in matrix form in eq 7 , where Dois the operator d/dt. Solutions of eq 7 provide
expressions for [A], [B], and [C] as functions of time in terms of dynamic constants A, and h2, which are the nontrivial roots of the auxiliary cubic eq 8 with their signs reversed. Equations X[X2 + ( k , + k2
+ k3)X + k1k3] = 0
(8) describing the dependence of [A], [B], and [C] on time have the general form shown in eq 9, where [A] = X,(n=l), [B] = Xn(n=2), X , = a l nexp(-X,t) + a2, exp(-X2t) a3n (9)
+
etc. Calculation of the amplitude factors (a,,, r = 1-3, n = 1-3) is described elsewhere.26-28 The flow of reactant A to product C cannot be described rigorously by a single-exponential term incorporating one first-order rate constant. Under these circumstances, different approaches are adopted in characterizing the dependence of composition on time. We identify these approaches as subsets that are approximations to the overall scheme given by eq 7 . In Scheme IIa, a rigorous treatment is used in which compositions as a function of time are described by using the double-exponent expressions summarized in eq 9; the dependence of kinetic parameters on temperature is examined by using dynamic constants A, and X2. Granted that XI and X2 could be extracted from composition/time data, it is not obvious how the rate constants k,, k2, and k3 might be established. However, over a wide range of the component rate constants (k,, n = 1-3) the smaller exponent, arbitrarily X1, dominates the equations summarized by eq 9; Le., the kinetics of reaction are "pseudo-firstorder" with X, as the coefficient of time, t . Hence in Scheme IIb, a first-order rate constant is identified with X1 (eq 10). In most k(obsd) XI (10) applications (e.g., ref 8), the steady-state approximation is introduced with respect to intermediate B. We summarize this procedure under subheading Scheme IIc. A further assumption is discussed by some authors (e.g., ref 8) in which substances A and B are in equilibrium throughout the course of the reaction; Le., [A]/[B] = k 2 / k l at all stages in Scheme IId. We do not develop this point further because in most applications of Scheme I1 substance B is an intermediate having a very short lifetime. Scheme Ilc. Most attention has been directed toward simplifications introduced by a steady-state hypothesis, which sets d[B]/dt equal to zero. In these terms, k(obsd) is identified with the steady-state first-order rate constant k,. The latter is related to the component rate constants characterizing the individual steps by using eq 1 1 : k(0bsd) = k, = kik3/(k2 + k3) = k l / ( l a) (11)
+
where
= k2/k3 (12) Generally the further assumption is made that the component rate constants in eq 1 1 are given by Arrhenius equations in which each energy of activation is independent of temperature. These equations are written in either one of the following parametric (eq 13) or differential (eq 14) forms: k , = exp(a(n)/T + b(n)J; n = 1-3 (13) At fixed pressure CY
(26) Rodigiun, N . M.; Rodiguin, E. N. Consecutiue Chemical Reactions; Van Nostrand: New York, 1964. (27) Wylie, C . R. Differential Equations; McGraw-Hill: New York, 1979; p 147. (28) Hammett, L. P. Physical Organic Chemistry; 2nd ed.; McGraw-Hill: New York, 1970; p 73.
Blandamer and Scott
6266 The Journal of Physical Chemistry, Vol. 92, No. 22, 1988
a In k,/aT
= E,,/RT?
E n = -Ra(n);
n = 1-3 (14)
Combination of eq 11-13 produces eq 14, which accommodates the temperature dependence of k(obsd) for the reaction in eq 1. Thus
+ b ) / [ l + exp(c/T + d)]
k(obsd) = exp(a/T
u = ~ ( 1 ) ; b = b ( l ) ; c = ~ ( 2 -) ~ ( 3 ) ; d = b(2) - b(3)
(16) derived. Hence only relative values of four of the six unknowns arise from the regression. Furthermore, two numerically distinct solutions are found when the data are fitted to eq 16. The second solution is obtained by dividing the numerator and denominator of eq 15 by exp(c/T + d ) . Hence
+ ( b - d ) ] / [ l + exp[-(c/T + d ) ] (17)
Equation 17 has the same form as eq 15. The fact that two numerical solutions satisfy the regression is an artifact of the assumptions represented by eq 11-13. The interpretative problem2 as to which solution is chemically significant disappears if more elaborate equations replace the assumptions represented by eq 11-13. However, this tactic increases the amount of information to be extracted from kinetic data. Expressions for the apparent energy of activation E(obsd) and the apparent heat capacity of activation A*C,(obsd) are derived by differentiating eq 3 twice with respect to temperature. One differentiation yields eq 18: E(0bsd) = El
+ E3
(k2E2
+ k3E3)/(k, + k3)
(18)
A slight simplification in the algebra emerges if we use energies rather than enthalpies of activation. This is equivalent to using the Arrhenius rather than the Eyring formalism to describe the temperature dependence of k(obsd). We have ignored the term -R in the equation A*Cp= (dE/dT), - R. In general terms, this approximation is acceptab1e.l A second differentiation yields eq 19: A*C,(obsd) = -kzk3(E2 - E3)2/[RP(k2
+ k3)2]
a In A,/aT = E * ( I ) / R P a In X , / ~ T = E * ( ~ ) / R ~ P
(24) (25)
Both E * ( l ) and E*(2) are functions of temperature. With reference to eq 22-25 we assume, as before, that the depencences of individual rate constants on temperature are described by eq 14. Hence
The quantities in the column vector on the right-hand side of eq 26 are given by eq 27 and 28: n= 1
P(1) = C k n E n
(27)
+ E,
(28)
n=3
P(2) = E ,
Equation 26 is solved for E * (1) and E*(2), yielding eq 29 and 30: = [(kl - XJEI
+ k,E, + (k3 - X,)E3I/(Xl
E*(2) = [(ki - XI)EI+ k2E2
- A,)
+ (k3 - XI)E~I/(X,- XI)
(29) (30)
Because rate constants k, (n = 1-3) are, in general, functions of temperature and both A, and X2 are functions of rate constants, the quantities E*(1) and E'(2) are functions of temperature and are characterized by isobaric heat capacities of activation A*C,(n) ( n = I , 2) defined by eq 31 and 32: aE*(i)/aT = ~*c,(i)
(31)
a~*(2)/= a ~~ * c , ( 2 )
(32)
Explicit expressions for these heat capacity terms are obtained by differentiating with respect to temperature the equations summarized by equation 26. Thus
The single finite element in the right-hand column vector is given by eq 34: n=3
Q(l)
n=2
= ICk,E,Z - CAn[E*(n)121/RT2 n= 1
n= 1
(34)
The solution of eq 33 yields eq 35 and 36:
These equations form the basis of a numerical analysis presented below. Analysis. Scheme III. Kinetic analysis of Scheme I11 is based on the equations given in matrix form in eq 37. As before, we
(21)
Equations 20 and 21 are differentiated with respect to temperature to yield eq 22 and 23: XI a In h , / d T + X2 a In X2/dT = k , d In k , / d T + k2 a In k 2 / d T k3 d In k3/dT (22)
+
(29) Dewar, M . J . S . The Molecular Orbital Theory of Organic Chemistry; McGraw-Hill: London, 1969; p 283.
(23)
We define two quantities, E * ( l ) and E*(2), which are apparent energies of activation related to XI and h2 given respectively by eq 24 and 25:
(19)
Equations 1 I , 18, and 19 summarize the energetic consequences of Scheme IIc, which accounts satisfactorily for the kinetic data characterizing the reaction in eq 1. Scheme IIa. A rigorous treatment is prompted by a concern that steady-state assumptions force a particular dependence on temperature for both energies and isobaric heat capacities of activation; e.g., a bell-shaped dependence for A*C,(obsd) as a function of t e m p e r a t ~ r e . ~If, ~XI~and X2 are the dynamic constants (eq 8), these quantities are related to the coefficients of the nontrivial part of eq 8 in the following way: A , + X2 = k , + k2 + k3 (20) Xihz = kIk3
+ d In X,/dT = a In k , / d T + d In k J a T
(15)
Equation 14 is based on the assumption that the constants in eq 12 and 13 are independent of temperature. The nonlinear dependence of In (k(obsd)}on T I arises solely from the complexities of Scheme I1 and not from any complexity in the dependence on temperature of component rate constants. Although three values of a(n) and b(n) are in principle required to describe the temperature dependence of k(obsd), only a( 1) and b( 1) are characterized absolutely by fitting kinetic data to eq 15. The relationship between eq 16 and the equations summarized by eq 13 are readily
k(obsd) = exp[(a - c ) / T
d In Xl/dT
L r ~ J1
Activation Parameters and Reaction Schemes
The Journal of Physical Chemistry, Vol. 92, No. 22, 1988 6267
concentrate on two aspects of Scheme 111. First we identify a steady-state approximation as Scheme IIIc and a rigorous treatment as Scheme IIIa. In Scheme IIc, d[B]/dt and d[C]/dt are set equal to zero and k(obsd) is identified as the steady-state rate constant k,. Hence k(0bsd) = k, = kIk,kS/(k2k4
+ k2k5 + k3k5)
The further assumption is made that the dependence of each contributing rate constant is given by eq 13 and 14 for kiwhere i = 1-5. The energy of activation E(obsd) is obtained by differentiation of eq 38 with respect to temperature. A second differentiation yields A*Cp(obsd). Hence E*(obsd) = El
+ E3 + E5 - X ( l ) / Y ( l )
I
I
(38)
The column vector quantities on the right-hand side of eq 52 are given by eq 53-55:
(39)
(53)
where X(1) = k2k4[E2 + E41
+ k ~ k [ E+ 2 EsI + k3ks[E3 + EsI (40)
Y(1) = k2k4
+ k2k5 + k3k5
(41)
The expression for A*C,(obsd) is given by eq 42: A*Cp(obsd) = -[Y(l)Z(l) - (X(1))2]/(Y(1)}2RT2 (42) where
+ k2k5[E2 + E5I2+ k 3 k 5 [ E 3+ E5I2
Z ( 1) = k2k4[E2 +
(43) It is useful at this stage to introduce contracted versions of expressions such as those in eq 40, 41, and 43 by adopting the following notation: X(1) = Cknk,[En + E,] [(2,4);(2,5);(3,5)1
(44)
Y(1) = Cknkr[(2,4);(2,5);(3,5)1
(45)
Z(1) = Cknkr[En + ErI2[(2,4);(2,5);(3,5)1
(46)
The trailing terms in square brackets signify two subscripts for each term over which the summation is taken. We turn our attention to Scheme IIIa in which no approximations are introduced. The auxiliary equations derived from the 4 X 4 matrix in eq 37 are given in eq 47: X[X3
+ (%")A21 n=
+
Cknk,[(2,4);(2,5);(3,5);(1,3);(1,4);(1,5)1h + klk3k51 = 0 (47) The roots are X = 0 and -Al, -A2, -A3. The corresponding dynamic constants are XI, X2, and X3. The changes in concentration with time of substances A, B, C, and D are given by the general eq 48:
+ a2,,exp(-X2t) + a3,,exp(-X3t) + a4,,
X,, = a l nexp(-X,t)
(48) Here [A] = X,, (n = l ) , [B] = X,,(n=2) .... In the following manipulations it is assumed that the nontrival roots of eq 47 are always real. Differentiation of explicit expressions for A,, X2, and X3 in terms of the cubic part of eq 47 is awkward. On the other hand, the differentiations are facilitated by the technique applied to eq 7 . The relationships between dynamic constants and the coefficients of eq 47 are given by eq 49-51: n=5
XI
+ X2 + X3 = C Ik n n-
XlX2
(49)
+ X1X3 + hzX3 = n=5
Cknk,[(2,4);(2,5);(3,5);(1,3);(1,4);(1,5)1 (50)
n=l
XlX2X3 = klk3k5
(51)
Equations 49-5 1 are differentiated with respect to temperature (see eq 31 and 32). Hence
The matrix [ A v j ] is identical with that on the left-hand side of eq 52. The column vector components on the right-hand side of eq 56 are given by the relationships in eq 57-59: n=5
n=3
Q(1) = ICknE,Z - CX,E*(n)')/RP n= I
Q(2) = [ C knk,(En
n= I
(57)
+ E,)'( ( 2 7 4 ) M ) ;(3,5);( 193) ;( 194) ;( 19-51}-
CXnh[E(n)+ E(r)12((l,2);(2,3);(l,3)~l / R F (58) Q(3) = 0
(59)
Scheme IZZb. This scheme is the analogue of Scheme IIb. In other words, with reference to solutions of eq 47 we assume that the dynamic constants are ordered in the sense XI Ih2 IA,. We also assume that for a wide range of rate constants kn (n = 1-5) the changes in composition as a function of time are dominated by XI. Hence k(0bsd)
XI
klk2k5/(k2k4
+ k2k5 + k3k5)
(60)
Results Normally the dependence of composition on time is analyzed to yield k(obsd) which, as commented above, is examined in terms of its dependence on temperature to yield enthalpies and energies of activation. These parameters are used in a discussion of possible mechanisms of reaction. Here we take the reverse route in the sense of starting with a predetermined set of rate constants and their dependences on temperature. In other words, we ask questions concerning the resulting k(obsd), E(obsd), and A*Cp(obsd). The equations derived in the previous section have been written to form a convenient basis for computer-based calculations. In this case we used a Hewlett-Packard 86B reporting 12 significant figures, a consideration that proved important in solving the cubic (eq 47) for dynamic constants XI, X2, and X3. Table I records four sets of a(n) and b(n) parameters that were used in conjunction with eq 13 to calculate rate constants kn ( n = 1-3) for set 1 and n = 1-5 for sets 2-4. The choice of a ( n ) and b(n) quantities in set 1 was dictated by previous1' considerations. In set 4 the a(n) and b(n) quantities were adjusted so that the ratios k 2 / k 3 and k , / k 5 are unity at 338 and 323 K, respectively. Scheme ZZ. In the analysis of Scheme I1 we compared the outcome of Scheme IIb (cf. eq 10) and Scheme IIc, the steadystate assumption. In particular we used set 1 (Table I) to compare calculated isobaric heat capacities of activation (Figure 1). The outcome confirms that steady-state treatments underestimate the
6268 The Journal of Physical Chemistry, Vol. 92, No. 22, 1988
Blandamer and Scott
TABLE I: Parameters Describing Dependences of Rate Constants on TemDerature"
n I
2
3
4
5
-10000
-2000 6.1889 -20 000 5.00 -2000 5 .OO -13 378 43.680
-7000 19.2358 -7000 20.47 -7000 20.00 -5155 20.42
-3000 7.0 -5000 10.00 -8684 29.48
-6000 22.00 -9000 25.00 -4224 15.69
28.3158
-10000 28.3158 -8000 25.00 -14.057 40.040
//;o';/;.
,
,
,
40.0
60.0
80.0
o /
Equations 13 and 14. 0
20.0
1 100.0
T / Celsius
Figure 3. Dependence on temperature of log k (Scheme 111): set 3. k = X I ( - - - ) and k = X, k = X3 (full line) and k = k, (0). (-e);
;
.~
j
2 200 0
4
0
C
,\d
150.0
C
0
0 0
0
20 0
LO 0
600
800
1000
T/ Celsius
Figure 1. Dependence on temperature of apparent heat capacities of activation (Scheme 11), calculated by using eq 33 (solid line) and using
eq 19 (0).
I
20.0
0
40.0
80.0
60.0
J
100.0
T/Celsius
Figure 4. Dependence on temperature of activation energies (Scheme 111) by using eq 39: set 2, E'(1) related to AI (full line), E'(obsd) (0); set 3, E'(1) related to XI (---), E'(obsd) ( 0 ) ;set 4, E'(1) related to AI (-*-), E'(0bsd) (m).
-8
$1500.0
.-u
-
m
\
0
0
20.0
40.0
60.0
80.0
100,o
T/Celsius
Figure 2. Dependence on temperature of log k (Scheme II); set 2, k = XI (full line) and k = k, ( 0 ) ;set 3, k = XI ( - - - ) and k = k , (0);set 4, k = XI (---) and k = k, (A).
magnitudes of apparent heat capacities of activation. Scheme III. For this reaction scheme, comparison between the steady-state treatment and the more elaborate calculations are presented in Figures 2-5. In Figure 2, the steady-state calculations (Scheme IIIc) are compared with the results of Scheme IIIb (cf. eq 60) for sets 2-4 between 273.15 and 373.15 K. For sets 2 and 4 the differences are not marked, but for set 3 the approximation that sets k, = XI is poor. A further examination of set 3 compares k, with h l , h2, and X3. Within the same temperature range as in Figure 2 the roots A, and A, provide an envelope for k,. However, it is not possible to draw a general conclusion from this pattern. Figures 4 and 5 respectively illustrate comparisons between (i) the apparent energies of activation and heat capacities and (ii) those quantities calculated on the basis of Scheme IIIb. For sets 2 and 3 differences between Schemes IIIa and IIIb for apparent energies of activation are relatively small, but for set 3 the differences are marked. Similar trends emerge for heat capacities of activation.
U
\ \
c
\
u I
0
\
20.0
LO.0
80.0
600
100.0
1 /Celsius Figure 5. Dependence on temperature of apparent isobaric heat capacities of activation (Scheme 111) by using eq 42: set 2, A*C,(l) related to X I (full line), A'Cp(obsd) (0); set 3, A*C,(l) related to XI (---), b'C,(obsd) ( 0 ) ;set 4, A*C,(l) related to X, A'C,(obsd) (m). (-e-),
Discussion In a recent review4 we commented that activation parameters derived from kinetic data for reactions in solution are frequently disappointing in the sense that such quantities seldom provide unequivocal mechanistic information. This disappointment perhaps explains Dewar's misgiving^^^^^^ concerning kinetic studies that report activation parameters. However, similar reservations (30) Winter, J. G.; Barron, J. P.; Scott, J. M. W. Can. J . Chem. 1975, 53, 1051.
J. Phys. Chem. 1988, 92, 6269-6212 can be advanced in the context of Gibbs functions calculated for the activation process from rate constants at a single temperature. A single rate constant at given temperature and pressure offers little information about a chemical reaction, particularly for reactions in solution. The debate referred to in the introduction is partly a matter of what one might term the direction of the argument. One direction moves from the dependence of k(obsd) on temperature to comment on the mechanism and thus to decide between schemes (e.g., I, 11, or 111). In the reverse direction a given scheme is used to predict the dependence of the dependence of k(obsd) on temperature, which is then matched against the observed dependence. Both are legitimate exercises, but the underlying difference between schemes is clearcut when trends in A'Cp(obsd) are considered. The calculations reported here are part of the argument based on the direction scheme k(obsd). As shown in the figures, the associated calculations confirm that simply on the grounds of the dependence of k(obsd) on temperature, different schemes cannot be necessarily distinguished. Moreover, to delineate possible dependences of activation parameters on temperature, good kinetic data covering a wide temperature range are required. The patterns formed by the dependences of A*C,(obsd) on temperature are very similar for Schemes I1 and 111 at all levels of approximation (Figure 4). Indeed, these patterns are common to stepwise processes for both kinetic and equilibrium processes. However, we are still unwilling to dismiss an important rdle played by the solvent in the activation processes. How we draw
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this factor into the analysis is a question that remains unresolved, although we have discussed various po~sibilities.**~' At this stage we suggest that, with respect to the reaction in eq 1, the case for Scheme I1 is supported by the self-consistency between the various data sets describing the kinetics of reaction in binary aqueous mixtures. The shift in the minimum in A'C,(obsd) when an organic cosolvent is added follows a pattern that is consistent with the impact of the various cosolvents on water-water interaction^.)^ Therefore we are doubtful that the dependence of k(obsd) on temperature can yield mechanistic information for reactions in solution. But we see considerable merit in using such data for the same reaction in a range of solvent systems, especially when the exercise is linked to the type of calculations reported here. In fact, the algorithms discussed above offer a convenient basis for these calculations using appropriate programs for a desk-top computer. Acknowledgment. We thank the Department of Chemistry at the University of Calgary (Alberta, Canada) and Professor Ross E. Robertson for their kind hospitality extended to both M.J.B. and J.M.W.S. over many years. J.M.W.S. thanks Dr. Danny Summers for advice and NSERC of Canada for financial support. (31) Blandamer, M. J.; Burgess, J.; Hakin, A. W.; Scott, J. M. W. J. Chem. Sot., Faraday Trans. 1 1986, 82, 2989. (32) Blandamer, M. J.; Burgess, J.; Duce, P. P.; Robertson, R. E.; Scott, J. M. W. J. Chem. Soc., Faraday Trans. 1 1981, 77, 1999.
Mechanism of Photochemical Conversion of 1,2-Naphthoquinonediazides in Solution Tsuyoshi Shibata, Ken'ichi Koseki, Tsuguo Yamaoka, Department of Image Science and Technology, Faculty of Engineering, Chiba University, Yayoi-cho 1-33. Chiba-shi, Chiba 260, Japan
Masayuki Yoshizawa, Hisao Uchiki, and Takayoshi Kobayashi* Department of Physics, Faculty of Science, The University of Tokyo, Hongo 7-3-1, Bunkyo-ku, Tokyo 113, Japan (Received: November IO, 1987; In Final Form: April 6, 1988)
The photochemical conversion of 1,2-naphthcquinonediazide-5-sulfonicacid (NQDS) in solution was studied by nanosecond time-resolved spectroscopy, and the observed transient absorption was assigned to the intermediate produced by the photodecompositionof NQDS. The intermediate is formed by the hydration of photochemicallygenerated ketene. The formation rate constant of ketene hydrate in the mixed solvent of l,Cdioxane/water increases with the water concentration. The conversion process from the ketene hydrate to 3-indenecarboxylic acid in aqueous solution is the acid-catalyzed reaction with a rate + (2.8 X lo7 M-I s-l )[H+], [H+] being hydrogen ion concentration (in M-' unit). constant of 5.2 X IO2
Introduction It is well-known that the photochemical conversion of 1,2naphthoquinonediazides (NQD) produces the corresponding indenecarboxylic Since this reaction is accompanied by the change of solubility in basic aqueous solutions, NQD derivatives can be used for the positive working photosensitive material that is dissolved in the exposed area. Novolak-type resins in which these compounds are blended or introduced as side chains are typically utilized as positive working photoresists5 in microlithographic industry. (1) Ershov, V. V.; Nikiforov, G. A,; De Jonge, C. Quinonediazides; Elsevier: New York; pp 261-8. (2) Schmidt, J.; Meier, W. Chem. Ber. 1931, 64, 767-77. (3) De Jonge, J.; Dijkstra, R. Rec. Traul. Chim. Pays-Bas 1948, 67, 328-42. (4) Huisgen, R. Angew. Chem. 1955, 67, 439-63. (5) De Forest, W. S. Photoresist: Materials and Processes; McCraw-Hill: New York, 1975; pp 132-62.
SUs6is one of the first who studied the conversion mechanism and described that the irradiation of 1,2-naphthoquinonediazide in an acidified aqueous solution produced 1-indenecarboxylic acid. On the other hand, it was demonstrated that the reaction product is not 1-indenecarboxylic acid but 3-indenecarboxylic acid by the subsequent studies of Melera et al.' and Pacansky et a1.,8 who used the model compound prepared by the carbonation of indenylsodium or indenyllithium. At present the latter assignment is generally accepted. The existence of carbene and ketene intermediate is now presumed in the conversion process, and the transformation of carbene to the ketene intermediate is known as the Wolff rearrangement.g*'O The high reactivity of these (6) Siis,0. Justus Liebigs Ann. Chem. 1944, 556, 65-85. (7) Melera, A.; Claesen, M.; Vanderhaeghe, H. J . Org. Chem. 1964, 29, 3705. ( 8 ) Pacansky, J.; Lyerla, J. R. IBM J . Res. Dev. 1979, 23, 42-55. (9) Wolff, L. Justus Liebigs Ann. Chem. 1912, 394, 23-59.
0022-3654/88/2092-6269$01.50/00 1988 American Chemical Society
