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3379
kcal mol-', AG = -0.66 kcal mol-', A S = -26.6 cal mol-' deg-l, and TAX = - 7.94 kcal mol-'. Table I: * Equilibrium Constants a t Different Temperatures for the Reaction of Acetaldehyde with tert-Butyl Hydroperoxide (Eq 1) Temp,
K ,M-1
O C
12.2 i 0 . 8 9.3 f 0.5 6 . 7 =k 0 . 5 5.6 0.5 4 . 8 i0 . 4 3 . 1 i0 . 4
-1.0 5.5 10.0 12.0 17.5 25.5
The value of K for the reaction of tert-BuOOH with acetaldehyde is slightly less (a factor of 4 a t 0"; a factor of 2 after statistical correction) than for hydrogen peroxide and acetaldehyde. Since conditions could not be made identical, the difference is only an estimate; nevertheless, it is clearly small. Also there are small differences in the values of A H and AS (more negative for HzOz addition). It seems clear that the addition process is essentially the same for HzOaand tert-BuOOH; the replacement of H by a tert-butyl group has little significance on the reaction. Acknowledgments. We are grateful to the U. S. Air Force Office of Scientific Research (Grant No. 70-1839) for continued support.
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Dependence of the Glass Transition Temperature on Heating Rate and Thermal History Publication costs assisted by Catholic University of America
Sir: In a recent paper Rasmussen and MacKenziel have reported glass transition temperatures, T,, measured by differential thermal analysis (dta) as a function of heating rate for water-alcohol solutions. Using a treatment proposed by McMillan,2 they calculated activation entropies and enthalpies for the glass transition relaxation from their data. We wish to report here some preliminary results of an analysis of heat capacity measurements in the glass transformation region which indicate that no fundamental significance can be attached to the kinetic parameters derived by the method of the above authors. l s 2 McMillan's treatment2 is in error in that he considers the heat capacity, C , rather than the enthalpy, H , to be the relaxing quantity in the glass transition region. (Mchlillan's treatment cannot account, for instance, for the maxima commonly observed3 in heat capacitytemperature plots near T,.) The correct approach is to consider the total enthalpy of a glass-forming liquid to be the sum of a nonrelaxing part, Ho, and a relaxing part, Hr
H(T,t) = Ho(T)
+ Hr(T,t)
The heat capacity changes and the breaks in the dta curves observed near T , are then associated with the
time-temperature dependence of the relaxing part of the enthalpy, for which a corresponding relaxational heat capacity may be written
C,(T,t)
=
bH,(T,t)/bT
The simplest assumption for the time dependence of the relaxational enthalpy is the first-order kinetic expression
b[Hr(T,t) - Hr"(T(t))l -= bt
-
[Hr(T,t) - Hrm(T(t)) 1 r(T(t)) (1)
is the relaxation time and H," is the equilibrium relaxational enthalpy, such that at constant temperature
Hr,~= m lim H,,T t+
m
In eq 1 it is presumed that the temperature-time schedule of the system is known, so that H," depends ultimately only on T . This, along with the assumption of linearity, also makes r a function only of T . Previous relaxation experiments have shown that eq 1 is generally inadequate both in that more than one time constant r is needed to account for the observed data and in that first-order kinetic expressions fail at fairly (1) D. H. Rasmussen and A. P. MacKensie, J . Phys. Chem., 75, 967 (1971). (2) J. -4.McMillan, J . Chem. Phys., 42,3497 (1965). (3) U. E.Sohnaus, C. T. Moynihan, R. W. Gammon, and P.B. Macedo, Phvs. Chem. Glasses, 11,213 (1970).
The Journal of Physical Chemistrv, Vol. 76, N o . 91, 2071
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3380 1.5
E (kcal/mole)
\ \
;.0
0.5 Cr
0.c
11 210
I
8
8
2.18
2.14
2.22
io3/ T - C.E
-IC
Figure 1. Calculated heat capacity curves for various heating rates of glasses quenched a t a rate of -40 deg/min.
small departures from eq~ilibriurn.~-~ Since the calculations t o be described here are intended only to be illustrative of the effect of thermal history and heating or cooling rate on T,, however, it is sufficient for our present purposes to assume the simple relaxation kinetics given in eq 1. Integration of eq 1, followed by differentiation with respect to T , allows calculation of C, as a function of temperature. The value of H , at the initial time and temperature is a required boundary condition for the integration and is determined by the previous thermal history of the system. In Figure 1 we show some C, us. temperature curves calculated in this fashion. (Details of the calculation will be presented in a future publication.) These are sample calculations using a value of 7
=
A exp(E/RT)
with A = 1 x 10-22 sec and E = 50,000 cal/mol, which gives a glass transition around 200". The C, values are normalized to zero and unity at the respective low- and high-temperature limits. The C, curves shown in Figure 1 were calculated for various heating rates of glasses which were first quenched at a rate of -40 deg/min from a temperature well above the glass transition region to a temperature well below it. T , has been taken as the midpoint of the line AB, t o correspond with the T , definition of Rasmussen and 1lacKenzie.' In Figure 2 Arrhenius plots of heating rate us. T , are shown for the curves of Figure 1, along with similar plots for glasses of other thermal histories and for the relaxation time 7. The actiThe Journal of Physical Chemistry, Vol. 76,hro. 21, 1971
Figure 2. Arrhenius plots and activation energies for heating or cooling rate us. T, and relaxation time us. temperature: 0, Tgmeasured for a cooling schedule starting above the glass transition region; 0 , T Bmeasured for a heating schedule of glasses quenched a t - 2.5 deg/min; A, Tgmeasured for a heating schedule of glasses quenched at -40 deg/min; - _ _ , Arrhenius plot for 1 / ~ .
vation energies, E , for the various plots are given in the figure. Two important points may be gleaned from Figure 2. The first is that the T , values measured for a given rate of change of temperature with respect to time depend both on the thermal history of the glass and on the direction of temperature change (heating or cooling). Hence it is not a fruitful exercise t o concern oneself overly with the assessment of an exact and "correct" T , value at a given heating rate for a given substance, as Rasmussen and MacKenziel have attempted to do for the case of water. That is, T , values at 5 deg/min outside their limits of -137 f 1" can be obtained for water samples subjected to different thermal histories, as is suggested by the scatter in their Figure 7.' (We do not mean here to denigrate the value of T , vs. composition studies for glasses of identical thermal history, as conducted by Rasmussen and 34acKenzie' and numerous others.) Second and more important, the activation energies (and other kinetic parameters) assessed from the dependence of T , on heating rate also 8epend on thermal history and type of heating or cooling schedule and do not necessarily correspond to the activation energy for the relaxation time controlling the glass transformation phenomena. For the sample calculations summarized in Figure 2, the apparent activation energy for T , was (4) M. Goldstein and M. Nakonecznyj, Phys. Chem. Glasses, 6 , 126 (1965). (5) P. B. Macedo and A. Napolitano, J . Res. Nat. Bur. Stand., 71A, 131 (1967). (6) L. Boesch, A. Napolitano, and P. B. Macedo, J . Amer. Ceram. Soc., 53, 148 (1970).
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3381
found to deviate by as much as 25% from the relaxation time activation energy. It should be evident from the foregoing that the extraction of fundamental kinetic parameters for the glass transition from the temperature dependence of the heat capacity will require analysis of the entire C vs. T curves in the transition region, taking into account the thermal history of the glass. Such an analysis will also require the use of kinetic expressions more realistic than eq 1. We are currently at work on this problem and hope t o be able to report its solution in a future p~blication.~ Acknowledgment. This research was supported by a grant from the Air Force Office of Scientific Research. (7) EDITOR’S NOTE. Drs. D. H. Rasmussen and A. P. MacKenzie (Cryobiologv Institute, R F D 5, Box 137, Madison, Wis.), the authors of ref 1, comment as follows. We are very pleased to learn of the work of Moynihan and Macedo. Lest we appear from their paper to have endorsed a particular theory of the nature of glass relaxation phenomena, we wish to take this opportunity to say we believe we made ourselves clear concerning our reservations regarding McMillan’s treatment. I n the absence of a treatment founded upon more generally accepted concepts of the glassy state, we restricted ourselves t o the derivation of empirically useful information.
rise to net observable chemical change, making them more amenable to study. In the present studies, the rate constants were evaluated, typically within a standard deviation of 5-10%, using the stopped-flow method with very low reactant concentrations, (1-10) X M , under reversible second-order condition^.^^^ In all, ferrocene and seven of its substituted derivatives of the type Fe(ChH5)(CsHdX) and Fe(C5HkXh were studied. The formal electrode potential for each compound was determined in the same medium by potentiometric titration. These values, summarized in Table I, permit the calculation of the equilibrium constant for each of the cross reactions as in eq 2.7 Rate measurements were made on 22 reactions. Figure 1 depicts the results of these determinations in the form of a plot of log kij us. log K i j (where i and j refer to the numbers 1-8 assigned the ferrocenes in Table I). The data appear t o be reasonably linear, and the slope of the line shown is 0.55. This graphical treatment is suggested by the Marcus relation* for adiabatic, outer-sphere electron transfer
CHEMICAL ENGINEERINQ AND CORNELIUS T. MOYNIHAN MATERIALS SCIENCE DEPARTMENT PEDROB. MACEDO Table I : Electrode Potentials and Calculated Electron AND VITREOUS STATELABORATORY Exchange R a t e Constants for Ferrocene and Substituted CATHOLIC UNIVERSITY OF AMERICA Ferrocenes, Fe(C5H4X)(C6H4Y)“ D. C. 20017 WASHINGTON, RECEIVED JUNE14, 1971 x,y CHaj CHs n-C4Hsl TL-CIH~ H, n-C4H9
Electron Transfer Reactions of Ferrocenes’
HJ
H, HgCl H, CHzOH H, C P ”
Publication costs assisted by the Ames Laboratory of the U.S . Atomic Energy Commission
Sir: The rate of electron exchange between ferrocene and ferricenium ions, as in reaction 1, has been of long-
Fe(CJQ2
+ *Fe(CJ%)2+
=
Fe(CsH5)2+
+ *Fe(C5Hs)2
(1)
standing interest. Successful kinetic studies using radiotracer methods have been limited to low temperaeven there the rate is so tures (-70”) in high as to be barely measurable. The use of nmr line broadening has likewise been frustrated owing to the high transverse relaxation time of ferricenium Our approach to this problem has been to evaluate the rates of net electron transfer reactions between substituted derivatives, rather than the exchange process itself. These data may then permit, under certain theoretical models, the calculation of a value for the exchange rate constants. For example, one reaction studied is reaction 2 .
+ Fe(C5H4CH3), Fe(CsH5)(csH4-n-B~)+ Fe(C;H4CH3)2+ (2)
Fe(C5H5)(C~H4-n-Bu) +
=
Although also quite rapid, these cross reactions give
HJ
I
+0.1899 f:0.0005 $0.2353 i 0.0004 +0.2556+0.0005 +0.2719 f 0.0005 +0.2797 i:0.0004 +0,2806 =k 0.0005 +0.3267 =k 0.0010 wf0.427
6.6 6.7 6.5 5.7 5.3 4.2 18 14
” Eo’and k refer to 25.0°, in 1:1 v/v n-PrOH/HaO with p = 0.050 M , Ba(C104)t electrolyte. * Electron exchange rate constant, as in reaction 1, computed fitting experimental k i j values t o e q 3. (1) Work performed in the Ames Laboratory of the U. S. Atomic Energy Commission. Contribution KO.3044. (2) D. R. Stranks, Discussions Faraday SOC.,29, 73 (1960). (3) G. Lang, M.S. Thesis, Washington University, St. Louis, Mo., 1956. (4) M. Dietrich, Ph.D. Thesis, Washington University, St. Louis, Mo., 1962. ( 6 ) A Durrum stopped-flow spectrophotometer having a Kel-F mixing chamber with a 2-cm optical path was used for these deter-
minations. The reactions were followed at wavelengths in the region 230-270 nm where the difference in molar absorptivity between the two ferrocenes was the greatest. The data were fit to the kinetic equation for reaction 2 using standard relations. (6) A. A. Frost and R. G. Pearson, “Kinetics and Mechanism,” 2nd ed, Wiley, New York, N. Y., 1961, pp 187, 188. (7) As a matter of convention the cross reactions in eq 2 are all written such that K > 1. The rate constants referred t o are those for the reactions proceeding in the forward direction, although in many cases kinetic determinations were carried out from both sides. (8) (a) R. A. Marcus, J. Phys. Chem., 67, 853 (1963); (b) W.L. Reynolds and R. W. Lumry, “Mechanisms of Electron Transfer,” Ronald Press, New York, N. Y . , 1966, Chapter 6.
The Journal of Physical Chemistry, Val. 76, N o . 81, 1971
