Gas Chromatographic Verification of a at he ma tical Model Product Distribution Following Methanolysis Reactions R. B. Lam,' F. A. Palocsay, T. N. Gallaher, and J. J. Leary2 James Madison University, Harrisonburg. VA 22807 Kinetic and thermodvnamic narameters of alcoholvsis reactions have heen carefully studied by gas chromatography. T h e mt.thanolvsis of diethvl acetnl wairtudied hv Juvet and Chiu ( I ) and later modifiei as a gas chromatogra~hyexperiment for determination of the rate constants for the t w o - s t e ~ reaction (2).Juvet and Wachi (3)reported equilibrium constants for the alcoholysis of propionate and butyrate esters, and estimated the precision of the gas chromatographic method to he about 1-2%. Rowland et al. ( 4 , 5 ) extensively studied malonate transesterifications and provided kinetic, equilibrium thermodynamic, and mechanistic information; Johnson et al. (6) carefully studied the kinetics of the methanolysis of diethyl malonates. The application of binomial statistics to the chemical systems herein described is in many respects analagous to the recently developed theory of electron transfer with reactants having multiple electroactive centers (7). In a general acid-catalyzed methanolysis reaction the overall sequence may he written as ~~~~~~
~
~
where (PFR)EN is a polyfunctional molecule with N ethyl suhstituents. As the transesterification begins, one of the ethyl groups (E) is lost and replaced by a methyl group (MI. This process continues until the reaction mixture contains N 1 different esters, each with varying numbers of ethyl and methyl substituents. This scheme may he viewed as a continuous, though discrete, process with one methyl group being suhstituted a t a time. Furthermore, the relative amounts of the methyl substituted esters should be a function of the number of methyl groups available, and therefore controlled by the amount of methanol and ethyl ester present in the reaction mixture. This assumption that the concentration of a given ester is dependent only upon the relative amounts of the components (ethyl and methyl suhstituents), has led to the adoption of the binomial distribution as a mathematical model describing the reaction process. The model is verified experimentally by gas chromatographic analysis of oxalate, malonate, and succinate systems. The binomial equation (8)is given by eqn. (1)
+
N!
,
F ~ " ( n ' = ( N - n).n.,P"+"
(1)
where, in this application N is the total number of possible ester sites on a given molecule; n is the number of methyl substituents on the ester; p is the probability that a given site will be methyl substituted; q is the probability that agiven site ) probability of will be ethyl substituted; and F N , ~ (is~the exactly n sites on a given molecule being methyl suhstituted.
Since the binomial distribution is only concerned with dichotomous events, p q = 1.
+
Development of the Model Considering only statistical limitations and excluding differences in activity, bonding, or steric effects between ethyl and methyl substituents, p can be defined as the ratio of methyl to methyl plus ethyl groups present in the reaction mixture. methyl groups present ethylgroups present
= methyl groups present
+
(2)
Since q = 1- p, the working form of the model for the general ester system described above is obtained.
Using eqn. (3), it is possible to calculate the probability, or fraction (F), of any given ester present a t equilibrium knowing only the total number of ester sites ( N ) and the relative amounts of ethyl and methyl suhstituents available in the reaction mixture. For hifunctional ester systems ( N = 2), three equations can be derived from eqn. (3); each of these describing the fraction of one ester present a t equilibrium. Thus, for the dimethyl ester, n = 2, both sides are now methyl-substituted and eqn. (4) is obtained.
Similarly, for the ethylmethyl ester (n = 1) only one site is methyl-substituted and eqn. (5) is obtained.
If n = 0, the equation for the diethyl ester eqn. ( 6 ) is obtained.
These three equations describe the amount of each ester relative to the total amount of ester in the reaction mixture. Therefore, the sum of the fractions of all esters present equal one [F(MM) F(EM) F(EE) = 11. These quantities may he calculated by knowing the initial amounts of diethyl ester and methanol. I t should he noted that the working equation of the hinomial model (eqn. (3)) is applicable to the general sequence of methanolysis reactions on a symmetrical polyfunctional ester
+
+
Part of the work herein described was presented at the thirty-first Southeast Regional Meeting of the American Chemical Society. Roanoke. VA 1979. Present address: The Foxboro Company. 140 Water Street. Norwalk. CT 06856 Author to whom all correspondence should be addressed.
-'
Volume GO
Number 9
September 1983
769
with N ester sites. While t h e experimental aspect of this work covers a series of bifunctional esters, t h e approach is applicable t o either mono-esters o r svmmetrical oolvfunctional e s t e n . It is possible to qualitativeiy evaluate t h e aiplicahility of t h e model bv, oerformine a series of exoeriments at different values of p a n d cumpartng t h e expertmental results with t h e rorrrsoondine oredicted valueobtatned from a nlot of FIMM). . .. F(EM), a n d F ~ E E )versus p. For t h e case of the diethvl ester. a auantitative measure of t h e applicability of the m&lel can be ihtained by considering eouilihrium constants. Combination of t h e individual chemical equations allows t h e elimination of t h e ethanol a n d methanol concentrations a s shown in ean. (7). An overall equilihrium constant (eqn. (8))can t h e n bk calculated which is dependent only upon t h e three ester concentrations.
.
A theoretical value of the equilibrium constant based on the statistical model may be calculated by substituting eqns. (41, (5). a n d (6)into (8);givingeqn. (9).
Comparison of meoretical curves and experimental data fw melhanotysis of diethyl oxalate.
-
T h e fieure shows a olot of the oredicted mole fractions of dimethyi(eqn. 4)), etl&lmethyl (eqn. ( 5 ) ) ,a n d diethyl ester (eon. v D. . . (6)) . . . versus the a u a n t i t. I t is immediatelv- atmarent .. t h a t at either e n d of t h e p axis ( p 0, o r p z 1). t h e concentrations of two esters are verv small comoared to t h e concentration of t h e third ester a n d i h e slopes of the curves are large. T h u s , t h e uncertainty in determining t h e value of t h e equilibrium constant at these extreme values of p is larger t h a n t h e uncertainty near t h e midpoint.Vherefore, t h e individual values of K used t o calculate t h e average value (ff) should he appropriately weighted.
Efperlmental Diethyl axalate, diethyl malonate, diethyl succinate, dimethyl oxalate, and dimethyl succinate were obtained from Aldrich Chemical Company, Inc. Dimethyl malonate was purchased from Sigma Chemical Company. Reagent grade methanrrl was obtained from J. T. Baker Chemical Company and 100%ethanol was purchased from IMC Chemical Group, Inc. All of the previous chemicals were chromatographically analyzed for purity. However, p-taluenesulfonic acid monohydrate, ohtained from Aldrieh Chemical Company. Inc.. was dried at 90° for 2 hr in a vacuum oven and stored in a desiccator. All gas chromatographic work was performed on a Perkin-Elmer 3920 gas chromatograph equipped with a flame ionization detector. Column temperatures were monitored with a Fenwal Electronics. Inc. 15 kohm iso-curve thermistor. The resistance was measured on a Sabtronies Model 2000 digital multimeter. All reaction solutions were maintained at 50 1°C wlth a Haake Model R2C Constant Temperature Water Bath. Chromatographic cnnditions for the separation of the oxalate esters and the malonate esters were the same: 3 m hy 1' s in. column; 109o OV-101 on 1001120 mesh chromasorb W-HP; column temperature 138%; detector temperature 212'C. The succinate esters were separated using: 3 m by 1'" in. column: 10% 0'4-17 on 100/120 mesh chromosorb W-HP; column temperature 163'C: detector temperature OlODP &.A u.
Detector resoonse factors for the dimethvl relative to the diethvl esters were determined in the following manner. For each ester system, five solutions of varying weight percent (0-25%) diethyl and dimethyl esters in methanol were prepared and analyzed chromatographically. Defining the response factor of the diethyl ester in each system to be 1 area unitlmole, the response factorof thedimethylester was obtained by taking an average over the five standard solutions of the ratio (area unitslmole dimethyl ester)/(area unitslmole diethyl ester). Realizing that the flame ionization detector response is hased A detailed Inahmatical beatmenl of the random m popagation
is available from the authws. 770
Journal of Chemical Education
Table 1. Response lacton ( 1 ) f
Oxalate
fw
0.352 0.676
ku' -
--
Ester System MBlonate
Succinate
0.479 0.740
0.703 0.852
-
'Estimatsd lrom hu and k . on effective carbon number and that the homologous series of compounds dimethyl, ethylmethyl, and diethyl ester differ only by 1 carbon between adjacent members of the series, the ethylmethyl response factors were estimated to mid-way between the dimethyl and diethvl used bv Rowland . resnnnse . factors. This is the same anoroach .. et al. !5l when studying the malonate svstems. Ilrsyn~nr~iactors hr ea,weighted mean equilibrium constants have been calculated and are presented in Table 5. These results show that the equilibrium constants obtained experimentally agree with the value predicted by the model to within approximately 5%. A standard error analysis was used to obtain the estimated standard deviations also provided in Table 5. This work demonstrates the a ~ ~ l i c a b i l iof t vthe binomial distribution as a mathematical i b d e l of the chemical equilibrium of bifunctional esters. The model, as developed, enables a straightforward extension to symmetrical polyfunctional esters and presents a mathematical basis for estimating equilibrium constants for such systems. Acknowledgment This work was supported by a grant from the James Madison University Program of Grants for Faculty Research. The authors wish to thank G. W. Marrah of the James Madison University Mathematics Department for many helpful discussions.
Table 3.
Data for the Dlethylmalonate and Methanol and for the Dlmethvl Malonate and Ethanol Reactlons
Table 4.
Data for the Dlethyl Succlnate plus Methanol Reactlon EE
Table 5.
+M
Welghted Mean Equlllbrlurn Constants and Estlrnated Standard Devlatlons
-
Ester System
K
Oxalate Malonate
3.9 4.2 4.2
Succinata
0.03 0.02 0.05
Literature Clted i l l .luvet.R.
S..andChiu.J..J Am. Chrm. Soc.. 83.156011961!,
( 5 ) Pram...I 2.. Rodand.
11968).
S. P..Mack, C. H.. and Coll. E. E..
J. Go* rhrom.. 6, I73
16) dohnstun, D. 0.. Cottingham. A. 8..and Roland, W. P.. J. Am. Chrm. Sol.. PO. 244
,,"2n, ,."..",.
(7)Flenagsn.J.B..Msrgel,S.,Bad,A. J.,sndAnson,F.C.,J. Am. Chom. Sor., 103,4248
,."...,. ,,O"R,
1s) Young, H. D.,
"Statistical Treatment of Exwimontal Data: Company. Ine.. New York, 1962.
MeCraw~HillRook
ACS Polymer Division Topical Workshop The ACS Polymer Division will hold a workshop an High Resolution NMR of Polymers in the Solid State, Oct. 1 S 2 1 , at the Sheratan Patriot Inn in Williamsburg,Va. The focus of this workshop will be on discussion of high resolution NMR techniques (includingCPIMAS, multiple-pulse, and deuterium spin-echotechniques),their novel applications in polymer science, and a critical evaluation of their limitations in providing unambiguous information on polymer structure and dynamics. The workshop is scheduled to have five sessions, which will be of a morninglevening format.Lectures will he presented by recognized experts from academic, government,and industrial communities, and should provide a basis for active participation by all registrants, including those entering the field. The Thursday evening session will also include a selected poster session (8 to 10 posters) designed to illustrate novel applications of solid state NMR techniques in macromolecular science. The afternoonswill provide ample time for private discussions, as well as for extended discussion growing out of the individual sessions. For further information on the scientific portion of the meeting, contact James R. Lyerla, 1BM Research Laboratory, K42/282,5600 Cottle Rd., San Jose, Calif. 95193, (408)256-1164.
I
Volume 60
Number 9
September I983
771
