2076
NOTES
two selected inert gas experiments.21m22Although a pressure dependence of TDMC/CDNIC has been found by Ring and Rabinovitch,22only one value of the ratio has been included in Table 111, at 3 atm, which corresponds most closely with the pressure of Frey's experiment2' and is also reasonably close to the maximum value of 0.5 found at 2 atm. The TDMC/CDMC ratios in inert gas experiments under all conditions show a much larger degree of stereospecificity than both the present experiments and the Hg photosensitization work. The data in Table I11 show excellent agreement between the TDMC/CDMC values form the last two types of experiment and clearly show that they are different for the two different substrates, cis- and trans2-butene. The discrepancies with inert gas experiments are probably due to CH2X'2 complications noted above.
\a/
-
CH~NJ
3.7 3 . 5 ( 3 .l ) h 3.8 9.3
3.2 3 . 1 (2.4)'" 3.3 2.4
CHzCOa CHzC06 CHzCO" CHzCOd CHzN," CH2Nz' CHzN,B
1.8 1.9 1 . 6 (2.2)& 1.3 0.97 0.30 0.40
0.84 1.2 0 . 7 4 (0. 97)h 0.64 0.97 0.75 0.75
Table I11 : Comparison of Triplet Yields
Substrate
Precursor
L.\
CHzCO" CH2CO' CHzCOd
L/
L A
a From Table 11,3130 A. * From Table 11, 3660 8. Reference 20. Reference 12. Ref 21; 1600-fold excess Ar, totalpressure 2100-3000 mm. Reference 22; 1500-fold excess Nz, total pressure 3 atm, unfiltered AH-6 Hg arc. Reference 22; 1500-fold excess Nz, total pressure 3 atm, 4358-A radiation. Corrected for a 12% 'CHI component.
'
in Table I11 are different from the equilibrium values of 2.2 reported. in ref 23. (23) C. McKnight, P.8. T. Lee, and F. S. Rowland, J . Amer. Chem. SOC.,89, 6802 (1967).
Calculation of AH and IC Values from Thermometric Titration Curves
by A. C. 11.Wanders and Th. N. Zmietering Central Laboratory, Staatsmijnen D S M , Geleen, The Netherlands (Received November 1% 1968)
The calculation of the reaction enthalpy ( A H ) and the equilibrium constant ( K ) from thermometric titration curves has received much attention.1-3 Very recently, Izatt and coworkers2 compared several methods for evaluation of thermometric titration data. Their conclusion is that for simple equilibria involving only one reaction (e.g., A B = AB), the "simultaneous solution of equations" (see below) is adequate but that for more complex systems iterative methods have to be used. We independently did some unpublished work on the iterative evaluation of thermometric titration data and found theoretical and experimental evidence indicating that even in the simplest cases iterative techniques are a necessity. This will be elucidated by showing that the method of simult,aneous solution Hansen4 applied to his experiments on the equilibrium reaction Sod2- H + @ HS04- yields biased AH and K values. Much better agreement between observed and calculated heat of reaction values can be obtained with AH and K values derived by means of the iterative method; in addition these AH and K values agree more closely with literature values.
+
+
Theory The simple equilibrium A
+ B = AB gives a set of
N generalized equations of the form (see eq 6 of ref 1) McKnight, et al., reported TDlVIC/CDRIC = 2.9 when 3CH2from CH&O at 3660 adds to either cisor trans-2-butene in the presence of CH31, where 95% of the methylenes were estimated to be triplet.23 From this they concluded that the rate constant for ring closing of the biradical adduct, kl, was considerably smaller than that of internal rotation, k2. The results of ref 12 and 20 of this work are more consistent with a biradical adduct of 3CH2and 2-butene in which cyclization to form dimethylcyclopropanes occurs a t a rate competitive with rotation about the C-C single bond in the position of the former double bond than the rapid rotation which has been advanced by McKnight, et The presence of CH31 could be a complicating factor in their experiments. In addition, the T2P/C2P shown T h e Journal of Physical Chemistry
AH K
-=
+ E,AH + F i
Di(AH)2
(i
=
1 . .. N ) (1)
where N is the number of data points taken from one experimental curve. These N equations in two unknowns constitute an overdetermined system, which may be solved for the proper values of AH and K by means of the least-squares technique. If we assume (1) J. J. Christensen, R. M. Izatt, L. D. Hansen, and J. A. Partridge, J. Phys. Chem., 70, 2003 (1966). (2) R. M. Izatt, D. Eatough, R. L. Snow, and J. J. Christensen,
ibid., 72, 1208 (1968). (3) S.Cabani and P. Gianni, J . Chem. floc., A , 647 (1968). (4) L. D. Hansen, Ph.D. Dissertation, Brigham Young University, Provo, Utah, 1965 (University Microfilms, No. 65-14656, Ann Arbor, Mich.), aee Appendix E.
NOTES
2077
+ S02-
Table I: Heats of Reaction and pK Values for the Reaction HSOd- = H f from the Experimental Data of Ref 4 by Method I and Method I1 Method--I
r
7
as Calculated -Method I-
7-
Av Q" - QO,
Av -AH,
kcal/mol
Av Std dev
- Q",
Qo
PK
x
cal
101
-AH,
U.
oal
x
102
cal
PK
kcal/mol
x
102
U,
cal X l o 2
5.66 5.88 5.63 5.20 5.73 6.10 5.21 5.71 5.66 5.54
1.89 1.87 1.89 1.97 1.89 1.84 1.96 1.89 1.90 1.92
0.15 2.40 0.92 1.48 3.14 3.59 1.86 2.96 1.14 -0.02
0.31 1.69 0.90 0.81 1.80 2.16 0.88 1.63 0.77 0.32
5.65 5.28 5.25 5.57 5.33 5.42 5.19 5.39 5.43 5.43
1.89 1.95 1.94 1.98 1.94 1.92 1.96 1.93 1.93 1.94
0.07 0.14 0.08 0.04 0.08 0.20 0.01 0.07 0.11 0.04
0.26 0.59 0.33 0.26 0.32 0.66 0.28 0.28 0.39 0.24
5.63 0.28
1.902 0.041
1.76
1.13
5.34 0.16
1.938 0.024
0.08
0.36
that the experimental error in the heat of reaction Q is large compared with the other experimental errors, the problem reduces to the determination of those AH and K values, uix., AI? and 2, which mill minimize the error sum of squares N
U ( K ,AH) =
Cws(&ro i=l
(2)
does not use all combinations that can be made and, second, does not include the proper weighting of the solutions in computing the average. A further error is made when combinations of measurements giving abnormal values for K and AH (e.g., negative K ) are rejected before averaging.
Experimental Evidence
in which Qao and QtO are the observed and calculated heats of reaction a t point i, and w,is a weighting factor. Even for this simple equilibrium U ( K , A H ) is nonlinear in AH, and, therefore, U m i n must be calculated by iteration. The simultaneous solution of equations (SSE method) circumvents this complication because AH is directly calculated from the combination of data of two experimental points according tolJ
We intended to use the thermometric titration procedure of ref 1 in the study of complex formation in nonaqueous solvents,' but, in view of the above, we set out t o try the iterative approach. Considering the modest accuracy of our apparatus, we decided to try out the developed computer program on the very accurate experimental data of H a n ~ e n . ~ Equation 1can be rearranged as
+ Fg - F j = 0 (3) wherei = 1 , 2 , 3 , . . .,N - 1 , j = i + 2, i + 4, a n d j 6 N .
P,(AH)2
(Dl- D j ) A H 2 + (E, - Ej)AH
A corresponding value of K is then calculated from AH and ( l ) , and the final AH and K values are found by averaging the results of the various combinations. It is then assumed that these values correspond to A B and K , i.e., to the values giving Urnin. This assumption however, seems to be unjustified. Herschberge made an extensive study of overdetermined linear systems and proved that the classical least-squares solution for an overdetermined system of k linear equations in n unknowns, where k > n, has the property of representing the weighted average of the total number of solutions that can be obtained by choosing all k!/[(lc - n) !n!]distinct nonoverdetermined subsystems. If we assume that for AH and K values in the neighborhod of AI? and 2 this theorem holds also for the present nonlinear systems, we may conclude that, unlike the iterative technique, the SSE method, first,
+ W i A H1, + RtAH + Si = 0 (i = 1 . . . N ) (4)
where P i , W a ,R,, and S , are the appropriate constants obtained. As it is impossible to find values for AH and K which satisfy (4) for every observation i, we account for the experimental errors by introducing the residuals j t . Hence 1 Pa(AH)2 W i A H z R,AH L3t = jt
+
+
+
(i = 1 . . . N ) ( 5 ) Then, least-squares estimates of AH and K (viz., AI? and K ) may be obtained by minimizing the sum of the squared residuals
W H , 1/w
=
C f? i
(6)
( 5 ) See ref 4,p 110. (6) I. S. Herschberg, Thesis, Technological University, Eindhoven, 1966. (7) A. J. ,M.Teunissen and A. C. M. Wanders, unpublished results.
Volume 78, Number 6
June 1969
NOTES
2078 (This minimization demand is not equivalent to that imposed by eq 2; however, t o the extent that F , in the neighborhood of the minimum, approaches a quadratic form, the results will be identical.) The values of Al? and k cannot be found directly by standard leastsquares calculations, because eq 4 leads t o terms higher than quadratic in the arguments AH and K of the function F. Therefore, we search for the minimum by an iterative method, which consists of guessing a pair of values AH and K , approximating the function F ( A H , 1/K) in the neighborhood of this point by a quadratic expression G(AH, 1/K) having the same numerical values for its first and second derivatives as F ( A H , l/K), computing the minimum of G by solving two linear equations in two unknowns, and using the roots as a new guess for the next iteration. This procedure was programmed in FORTRAN IV and run on a UNIVAC I11 computer. The data analysis for the thermometric titration curves was based on formulas 32, 33, 36, and 38 of ref 1, and, as far as could be verified from Table I of ref 1, the calculated heats of reaction as a function of titrant added were identical. For each run, the final averaged AH and K values given by Hansen4 were used as starting values for the calculations. Indicating the present method as method I1 and that of ref 1as method I, we compiled the results of both procedures in Table I. For each experimental point, the value of Qio - Qro was calculated by substituting the AH and K values found. For these differences two statistics were determined, viz., the mean &" - &" value and the standard deviation g, both of which are included in Table I. It appears that the two methods of calculation give significantly differing results, although they originate from the same experimental data. For method I, the mean &" - Qa values are considerably larger than for method 11, as are their standard deviations. Therefore, it can be concluded that the iterative method is to be preferred for this type of calculations. The experiments of ref 4, performed with a titrant concentration of 0.991 M were not included in the comparison because the value for +T, the apparent molar enthalpy of the reacting species in the titrant, quoted in ref 4, seemed to be erroneous. We recalculated these experiments, using a corrected value of &. The final results obtained by means of method 11, including this correction, are AH = -5.41 f 0.04 kcal/mol and pK = 1.930 f 0.005 with 21 degrees of freedom. The values seem to be in better agreement with the literature values compiled in Table I11 of ref 2 than those obtained with method I.
Convergence The Of the calculations cited above 'Onverged within three Or four iteration steps when the AH and K values obtained by method I were used as startThe Journal of Physical Chemistry
ing values. However, the calculations turned out to be rather critical with respect to the initial values chosen for AH and K. Starting values deviating more than about 15% from the final values sometimes gave rise to strong fluctuations in the calculations and to large negative K values, which are physically meaningless. However, in view of the very strong correlation which turned out to exist between AH and K , this behavior was in agreement with expectation; the calculated correlation coefficients varied between 0.989 and 0.9997. This may, indeed, cause the successive values in the sequence of iterations to follow a rather erratic path. A cursory look at the surface P ( A H , 1/K) revealed that, in accordance with the strong correlation mentioned, the contour lines were very thin and possibly somewhat bended ellipses. This may give rise to wild excursions in the iterative procedure if AH and K are not close to Al? and k. The strong correlation between A€? and k may well be visualized also chemically; e.g., formula 4 allows the correction for too high values of Qo by lowering either AH or K . Summarizing, we may conclude that only the iterative procedure is capable of fully utilizing the high precision obtainable with the apparatuslts described by Izatt, Christensen, and coworkers. The two-point procedure may serve to compute the necessary starting values. (8)J. J. Christensen, R. M. Izatt, and L. D. Hansen, Rev. Sci. Instr., 779 (1965).
86,
Mass S p e c t r u m and Molecular Energetics of Krypton Difluoridel*
by P. A. Sessalb and H. A. RilcGee, Jr. School of Chemical Engineering, Georgia Institute of Technology, Atlanta, Georgia 30338 (Received November 14,1068)
The possible existence of fluorides or oxides of He, Ne, Ar, and Kr has been discussed by many investigators, and all suggested experiments have involved cryogenic considerations for an enhanced stability. A number of properties including the positive ion mass spectrum2 of gaseous krypton fluoride have been reported, and although the compound was once reported to be KrF4,a (1) (a) Research sponsored by the Air Force Office of Scientific Research, Office of Aerospace Research, U. 8. Air Force, under Grant AF-AFOSR-1308-67, (b) NDEA Predoctoral Fellow in chemical engineering. (2) E. N. Sloth and M. H. Studier, Science, 141,528 (1963). (3) A. v. Grosse, A, D, Kirshenbaum, A. G, Streng, and Le V. streng, ibid., 139, 1047 (1963).
