SOLIDSTATEREACTION KINETICS coupling constants in diazirine and we have tried to fit the 14N field gradients with sets of localized orbitals. Walsh orbitals having sp orbitals in the plane of the ring (one pointing away from the N and containing a lone pair of electrons; the other entering into a C-N
2253 u bond) have been used with unhybridized pa and p,, orbitals to describe the I4Nfield gradients.
Acknowledgment. The support of the National Science Foundation is gratefully acknowledged.
Solid State Reaction Kinetics. 111. The Calculation of Rate Constants of Decomposition for a Melting System Undergoing Volume and Surface Changes by Ernest A. Dorko* and Robert W. Crossleg Department of Aero-Mechanical Engineering, Air Force Institute of Technology, Wright-Patterson Air Force Base, Ohio 46455 (Received June 20, 1971) Publication costs assisted by the Air Force Institute of Technology
A new physical and mathematical model for autocatalytic decomposition of a solid which undergoes melting is described. The object of the new model is to allow the entire reaction to be described by one set of kinetic parameters. The model includes a time variant parameter, p , which incorporates spatial distribution of
reactants and products in the sensible region, reaction surface distribution, and phase, as well as total material loss. The mathematical manipulations and numerical operations performed to obtain values of p and the kinetic parameters are presented. A computer program employing a grid search technique to determine
Arrhenius activation parameters over short temperature ranges was also developed for use with the decomposition data. From the results of this program it is possible to distinguish regions of the reaction which are controlled by a single mechanism from those which are influenced significantly by competing mechanisms.
Introduction The utilization of the differential scanning calorimeter operated in the isothermal mode to determine the decomposition history of organic materials which are initially in the solid state has been reported previously. la The computer reduction of the data and the calculation of refined autocatalytic rate constants by a computer has also been described.Ib I n previous work,l a sigmoid decomposition history was demonstrated for the cupferron-tosylate system. It was concluded that the decomposition was autocatalytic. The factor contributing most significantly to the autocatalytic or sigmoid decomposition behavior was felt to be liquefaction. However, the kinetic analysis of the mechanism was based on the nondimensionalized parameters, reactant fraction, and product fraction. It was found that a correlation based on the Bernoulli differential equation could be made for this system that was analogous to the Prout-Tompkins2~3analysis for a svstem decomposing completely in the solid phase. Also analOgOUS to solid phase decomposition, a change in kinetic parameters was noted at ca. 50% reaction.4 The present report describes analytical work related to a more comprehensive physical model than con-
sidered previously. The object of using the new model is to allow the entire reaction to be described by one set of kinetic parameters. The physical model is discussed in the next section, and the resulting differential equation is presented. Then, the derivation and numerical processing of the equations which make up the mathematical model is described. Finally, a method for finding Arrhenius parameters for the rate constants is presented. The detailed application of the mathematical techniques described herein to specific systems will be reported subsequently.
Development of Physical Model The physical model with which data are to be com(1) (a) E. A. Dorko, R. S. Hughes, and C. R. Downs, Anal. Chem., (b) R. w. CrossleY, E. A. Dorko, and R. Dims, “Analytical Calorimetry,” Vol. 2, R. S. Porter and J. F. Johnson, Ed., Plenum Publishing CO.. New York. N. Y.. 1970. DD 429-440. _(2) E. G. Prout and F. C. Tompkins, Tram. Faraday S O ~ .40, , 488 42,253 (1970);
I
(1944). (3) D . A. Young, “The International Encyclopedia of Physical Chemistry and Chemical Physics,” vel. 1, Pergamon Press, New York, N. Y., 1966,p 49. (4) L. G. Harrison, “Comprehensive Chemical Kinetics,” Val. 2, C. H. Ramford and C. F. H . Tipper, Ed., Elsevier, Amsterdam, 1969, 395. The Journal of Physical Chemistry, Vol. 76, No. 16, lB7d
2254
ERNEST A. DORKO AND ROBERTW. CROSSLEY
pared was extended from the work of Dubovitskii, Manelis, and Merzhanov.6 These authors do not state explicitly their assumptions, but they appear to consider the disappearance of an extensive property of a specie to be proportional to some function of that property and functions of intensive properties of other species involved in the reaction. As applied to the present analysis, the model is (rate of disappearance of moles of reactant) (moles of reactant) (mole fraction of condensed product X moles oi reactant) or
-
--dA dt
klA
+
+ kzCBA
Equation 1 is appropriate t o systems in which mass may disappear from the experimental region of measurement by escape of gaseous products and in which the spatial distribution of the mass remaining in the experimental region may vary throughout the experimental time, e.g., by such occurrences as melting, macroscopic particle shattering, and coalescence. To bring eq 1 into a nondimensional form that is applicable t o the experimental data, it is normalized with respect to the initial amount of reactant, A o ; the density of reactants plus condensed product is assumed constant; and CBis recognized as
where B represents the moles of condensed product.
B is expressed as B =
V -(A0
vo
- A) =
(1
- p)(Ao - A )
(3)
where 00
-v
/&E-
vo
the initial mass of reactant and final mass of condensible product. I n contrast, the p of eq 4 is based on the "effective volume" of reactants plus condensable products a t each instant during reaction. This p functions as a parameter incorporating spatial distribution of reactants and products in the sensible region, reaction surface distribution and phase, as well as total mass loss, There would appear to be no way to measure p of eq 4 experimentally; happily, its role is not to be measured. It is intended to serve as a parameter, variable throughout the reaction, that absorbs the nonchemical influences on the reaction rate. If successful, the rate constants, kl and kz, will truly represent only chemical effects as they are intended t o do. The method of determining p in conjunction with the determination of kl and kz in eq 5 is presented in the next section.
Mathematical Analysis of Physical Parameters The mathematical analysis of eq 5 to find kl, k,, and depends on experimental data giving the remaining reactant fraction, a, at each time throughout the reaction period. The method of processing the experimental data has been described previously1b and is incorporated in a computer programa dealing with the solution of eq 5 . The program is able to calculate rkte constants under the assumption of no mass loss ( p = 0), measured total mass loss ( p constant as determined by eq 6), and parametric p (eq 4). The analysis of the last case will be discussed next. Then, the specializations of that case to obtain the first two will be qentioned briefly. The solution for the case of parametric p at a fixed point in time consists of the successive solution of three simultaneous transcendental equations. These are derived as follows. First, eq 5, is rearranged algebraically to give p
(4)
vo is the initial volume of reactant, and v is a variable in time. Inserting eq 2 and 3 into eq 1 yields Next, eq 5 is integrated to give
Equation 5 is formally identical with eq 3 of ref 5a. However, the latter was derived assuming the amount of material departing the experimental region is proportional to the extent of reaction, and the p there is defined as
I n integrating eq 5 to obtain eq 7, p is held constant. The underlying condition is that p remain essentially constant relative to a over each time range of integration. This must be justified by the results in each ap-
where vi is the final volume of condensed products. The p of eq 6 is a constant related only to the total material loss. It would be determined by measuring
(5) (a) F. I. Dubovitskii, G. B. Manelis, and A. G. Merzhanov, Dokl. Akad. Nauk SSSR, 121, 549 (1958); (b) G. B. Manelis and F. I. Dubovitskii, ibid., 124, 475 (1959). (6) This program which is a modification of program PARACT (QCPE No. 168, Quantum Chemistry Program Exchange, Bloomington, Ind.) is available on request from the authors.
The Journal of Phyeical Chemistry, Vol. 76, No. 16, 1978
SOLIDSTATEREACTION KINETICS plication of the analysis. Various forms of eq 5, including eq 7 and 9 kz(1
- P> - pki
=
are inserted in eq 8 to eliminate kz. The result after extensive algebraic manipulation is
where
2255 have been chosen to obtain rapid convergence and to treat kl explicitly. This is necessary because lcl is expected to be relatively small and sensitive to empirical or numerical uncertainties.
Calculation and Interpretation of Arrhenius Parameters When the rate constants vs. time have been calculated, it is found that, at least within certain time periods during the reaction, they are essentially constant. When the rate constants in related regions have been calculated over a temperature range, it is desirable to express them in Arrhenius form. It has been the authors’ experience’ in working with compounds that decompose during melting that only small temperature ranges are possible and not insignificant scatter is probably unavoidable. Therefore, a grid search technique7 t o determine Arrhenius parameters for both the exponential, eq 12, and logarithmic (or linearized), eq 13, forms has been developed N
- p)(l - a) - :)I1 - p(1 - a)]
kl(1
z=1+
(-IC1
Sex, =
i=l
(kt
- Ae-Ea/RTi)2
In kt - In A i-1
Finally, eq 5 is differentiated with respect to t. When is equated to zero and the resulting expression is rearranged, the result is
u
where amaxis the value of time at which u = 0 (or b achieves its maximum). I n deriving eq 11, p was again held constant. The implication is that dp/dt is negligible compared to dmax. This also must be justified for specific applications. The three equations to be solved are now eq 7 , 10, and 11. The solution is started by numerically iterating eq 10 for kl with p = 0. Then, eq 7 is solved for an initial K , and eq 11 is numerically iterated for an initial nonzero p . Equation 10 is then reiterated for with the nonzero p , The cycle is repeated until the specified trial-to-trial agreements of IC1 and p are obtained. Finally, kz can be obtained from the definition of K if desired. When p is a constant determined by eq 6 or an assumed constant (zero or nonzero), the solution of eq 7, 10, and 11 is much easier. Equation 11 becomes superfluous. Equation 10 is iterated with the specified value of p until satisfactory trial-to-trial agreement of kl is achieved. Finally, kz is calculated once by eq 7. Two final comments are in order in closing- this section. First, the value of amaxused in the iteration is Obtained directly from the experimental data and is thus an independent quantity in the iteration for kl and pee. Second, neither the particular forms of equations presented nor the order of iteration is unique. They
(12)
>’
+Ea RTi
(13)
Use of eq 12 implies that absolute errors are assumed equal, and use of eq 13 implies that relative errors are assumed equal. Keither case is demonstrably more realistic in the present situation, and the object of using both is to delineate cases of agreement or disagreement to gain insight into the underlying physical processes. The grid search technique was developed to calculate the parameters efficiently and accurately and to obtain quantitative estimates of the uncertainties involved. The program prints Sexpand 81, throughout a selected region of the (A,E,) plane. Final minimum values on the S surfaces are found by constricting the search region around the last calculated minimum values. When the minimum values have been determined, absolute and relative errors and confidence limits are calculated based on the minima of S,,, and XI, and the F statistic.8 The ranges of the Arrhenius parameters corresponding to the calculated confidence limits can then be found by examining the sums of squares printed during the iteration process. The more data and the less scatter in it, the smaller will be the region of the S surface surrounding the minimum for a specified confidence level. The lower the confidence levels associated with either S,,, or SI, which enclose the minimum of the other, the less sensitive are the results to details of the analysis. (7) R. W. Crossley and E. A. Dorko, program ACTEN (QCPE No. 179, available from the Quantum Chemistry Program Exchange, Indiana University, Bloomington, Ind.) (8) R. S. Burington and D. C. May, “Handbook of Probability and Statistics,” Handbook Publishers, Inc., Sandusky, Ohio, 1958, 158.
-
The Journal of Physical chemistry, VoE. 76, No. 16,1973
LAWRENCE DRESNER
2256
Finally, and of particular importance, if the analysis rate constant can be given special attention. This gives differences in the locations of Xexpminand Xl,,,,,,, should be directed two ways: to the details of deterfor a particular rate constant that are significantly mining the rate constant and to the implications of greater than for the other rate constants, the “bad” the poor result on the kinetic model itself.
Stability of the Extended Nernst-Planck Equations in the Description
of Hyperfiltration through Ion-Exchange Membranes1 by Lawrence Dresner Chemistry Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 87830
(Received January 26?1972)
Publication costs assisted by Oak Ridge National Laboratories
Some mathematical questions are discussed which arise when the extended Nernst-Planck equations are used to describe hyperfiltration (reverse osmosis) through ion-exchange membranes. These questions concern the stability of the extended Nernst-Planck equations. A simple means of calculating ionic rejections in the limits of infinite membrane thickness or infinite water flux has been suggested. Circumstances have been identified under which a particular one of the counterions may be completely rejected. Other circumstances have been identified under which some counterions may be negatively rejected. An experiment is quoted exhibiting both of these phenomena.
1. Introduction and Historical Review A number of authors have used the Nernst-Planck equations, extended to include convection, to describe ionic transport in ion exchangers.* I n this paper, these extended Nernst-Planck equations (ENPE) are used to describe hyperfiltration (reverse osmosis) through ion-exchange membranes. The ENPE are highly nonlinear equations, the nonlinearity being caused by the electrical coupling of the various ionic fluxesU3 Because of this nonlinearity, the solutions of the ENPE that describe hyperfiltration have some interesting properties. While it is difficult to discuss these properties without first laying a proper foundation, the brief summary of them given below may give the reader a foretaste of what this paper contains. There are two different kinds of solutions of the ENPE; which one we get depends on the composition of the feed solution. They are distinguishable by their behavior in the limit of infinite membrane thickness. The limiting behavior of the first kind of solution may be calculated easily from certain algebraic equations related to the ENPE, but the limiting behavior of the second kind cannot. I n the limit of infinite membrane thickness, there is complete rejection of a particular one of the counterions from any feed which leads t o a solution of the second kind. Finally, under certain circumstances solutions of the first kind exhibit negaThe Journal of Physical Chemistry, Vol. 76, No. 16, 1972
tive rejection of one or more of the counterions. (Negative rejection means that the effluent solution is more concentrated in the particular ion than the feed.) An example is discussed in which both of the latter phenomena have been observed. Schlog12”and Hoffer and Kedem4 have solved the (1) Research supported by the National Science Foundation-RANN under Union Carbide Corp. contract with the U. S. Atomic Energy Commission. (2) (a) R. Schlogl, “Stofftransport durch Membranen,” Dr. Dietrich Steinkopff Verlag, Darmstadt, 1964; (b) F. Helfferich, “Ion Exchange,” McGraw-Hill, New York, N. Y . , 1962. (3) The reader may be aided in seeing this by noting that e is a function of the ionic concentrations. If we multiply eq 1 by z & h ,
sum, and use eq 2 to show that d
Substituting (A) into (1) we have
(B)shows that the dc,/dz are uniquely determined by the ci. This means that for given j , and J , there is one and only one solution of the E N P E having given values of ca a t some given value of z (uniqueness theorem). (4) E. Hoffer and 0. Kedem, Desalination, 2, 25 (1967).
