Analysis of Pressure Changes for Simultaneous First-order Decomposition Reactions in a Gas-kinetic System with Dead-space BY PETER J. ROBINSON* Department of Chemistry, Manchester Polytechnic, Manchester M 1 5GD GRAHAM G. SKELHORNE AND Lankro Chemicals, Manchester M30 OBH Received 2iid March, 1918 Methods are described for treating the problem of unreactive dead-space when extracting kinetic parameters from total pressure measurements on simultaneous first-order decomposition reactions in a constant-volume gas kinetic system (A - B --f C'+D'). The proposed methods are verified by applying them to computer-gene~~eCd+'Pexperimental" pressure against time data for systems with a wide range of parameter combinations. In the course of kinetic studies on the thermal decomposition of a mixture of cis- and trans-4-chloropent-2-ene, it was found necessary to investigate the mathe- matical treatment of a pair of first-order reactions in a constant-volume gas apparatus with unheated dead-space.The results are of general interest and are reported here. The case of a single first-order process A + B + C has already been discussed.2 The final pressure p m is equal to 2"p0, where p o is the initial pressure and a is the " dead- space factor," eqn (1) where V,, T, = volume and temperature of dead-space, Vh, Th = volume and temperature of reaction zone. The pressure of reactant at any time (pA) is approxi- mated only poorly by (2p"-p), where p is the total pressure at time t , and plots of In (2p"-p) against time are seriously curved. However, plots of In (p" - p ) or In (2"~"-p) against time are close to linearity, and the first-order rate constant k can be obtained from the initial slope, which is (2"- 1) k/a.In the present work we consider the decomposition of a mixture of two reactants, A and B, by parallel first-order reactions (2) and (3) a = (1 + VcTh/ V,T,)-' (1) kA kB A -+ C+D B --P C'+D'. (3) In the absence of dead-space, the combined reactant pressure is given by eqn (4), in which the symbols have the obvious meaning. Provided that kA and kB are not too similar in value, and that the shorter lived reactant is not present in great excess, the rate constant for the longer-lived reactant can be obtained from a PA +PB = p" - p = 2p" - p = p i eXp ( - kAt) +pi eXp (- kBt) (4) 27552756 GAS-KINETIC SYSTEM WITH DEAD-SPACE plot of In ( p a - p ) or In (2p0 -p) against time, which at sufficiently long times becomes linear.For example, fig. 1 (upper curve) illustrates the case with p s = 4 p i and kA z 10kB ; the slope of the linear portion is equal to -kB, and the extrapolated intercept at t = 0 is In p i . For times in the initial curved section, p B can be calculated as p i exp (- kBt), and p A can then be calculated from eqn (5) A plot of In p A against time then gives - kA from the slope and p i from the intercept (fig. 1, lower curve). The precise combinations of rate constants and concentrations for which this treatment is practicable depend strongly on the precision of the experi- mental results and the time resolution available. Useful data can typically be abstracted for cases where kA/kB > 2 when p i / p g i2l 0.1, or where kA/kB > 10 when 2p" - p - p i exp (- kBt).( 5 ) PA = 2p"-p-p, = n f 4 I a ' I 1 40 80 time/s FIG. 1 .-Method of analysis of pressure-time data (no dead-space). If we now consider the same reactions in a system with dead-space, derivations similar to those given previously show that the rate of pressure change beconies eqn (6) and the rates of change of partiaI pressures are given by eqn (7) and (8). In eqn (7), and similarly in eqn (8), the second term on the r.h.s. accounts for the movement of reaction mixture out of the reaction vessel resulting from the pressure increasc dp/dt = a(kA PA + kB P B ) (6) (7) (8) dpA/dt = -kAPA-(PA/P)(l -a)(kAPA+ ICBPB) dpB/dt = -kBpB- (PB/P)(l -a)(kAPA+ kBPB)* In the absence of dead space, a = 1 and the previous equations are recovered.In the presence of dead-space, it will be expected by analogy with previous work that p" # 2p", and that plots of In (2p"-p) against time will be curved and give erroneous rate constants.P . J . ROBINSON A N D G . G . SKELHORNE 2757 SEMI-EMP IRICAL TREATMENT in our previous work,' it was thought that eqn (6)-(8) could not be solved analytically. We therefore proposed, evaluated and used a semi-empirical treatment, which, by analogy with the case of a single reactant,2 was expected to provide a reasonably accurate correction for dead-space effects : (a) plot In (2"~" -p) against t ; (b) from the slope SB of the linear part, determine kB by eqn (9); (c) from the intercept I B of the linear part extrapolated back, determine p i by eqn (10); ( d ) plot In (2"pO-p-exp ( I B + S B t ) ) against time; and from the slope SA determine k A by eqn (1 1); (e) from the intercept IA, determine p i by eqn (12).(9) k B = - SB(2"- l)/a p i = exp (IA)/(2"- 1). A N A LY TI C A L TREAT MEN T During the preparation of the present paper, it was discovered that eqn (6)-(8) can be integrated by suitable substitutions to give eqn (13) for the total pressure, eqn (14) for pA and similarly for pB. For a system with dead-space, the 1.h.s. of eqn (13) can be used in place of the 1.h.s. of eqn (4), and the various parameters thus determined directly without any further correction for dead-space (pi +pi) - Por(PiPo)i/~ - 11 = p i exp (- kAt) + pg exp (- kBt) (13) eXp (- kAt).(14) 0 (1 - l / o ) PA = P i ( P / P TESTS OF PROPOSED METHODS The semi-empirical method was used satisfactorily for the analysis of extensive experimental data in our previous work.' Such data do not, however, fulfil the requirements for a comprehensive test of the mathematical accuracy of the proposed treatments, since for this purpose the reactions must be 100 % clean, and very precise a priuri data are required for all the parameters, including mixture composition, dead-space and the rate-constants. Test data were therefore generated by computer " experiments " before the analytical solution was discovered. The coupled differ- ential eqn (6)-(8) were integrated by the Runge-Kutta-Merson technique embodied in an interactive software package known as CMS (Conversational Mode Simulation) on the PDP-10 computer in the U.M.I.S.T.Control Systems Centre. The correct operation of the integration routine was originally checked by test cases with a = 1.000, for which the analytical solution was known. By appropriate adjustment of the integration error control parameter, agreement could be obtained to better than 1 % of the initial partial pressure of each reactant, even at 80-90 % reaction for that component (e.g. table 1). It was later possible to show that the numerical integration agreed very well with the analytical solution for a # 1 (e.g. table 2). Computer data were generated for various cases with a # 1 and analysed as above by both the semi-empirical and the analytical methods, to extract the initial pressures and rate constants for the two components.The analyses were performed by least- mean-squares cafculation on a Hewlett-Packard 98 10A " programmable calculator ", which automatically plotted the required graphs and calculated slopes and intercepts from points selected after plotting by the operator. The plots were essentially linear2758 GAS-KINETIC SYSTEM WITH DEAD-SPACE n the limiting regions corresponding to those in fig. 1 and enabled the straight lines to be fitted in a reasonably objective manner. The results of the calculations are shown in table 3, which indicates the generally good agreement obtained between the results of the analyses and the original input data, over a wide range of parameter TABLE l.-COMPARISON OF ANALYTICAL SOLUTION AND NUMERICAL INTEGRATION FOR a = 1 (RATE CONSTANTS : kA = 3 .6 6 ~ S-', kB = 3 . 6 0 ~ S-') time/s 0 200 400 750 1000 analytical 10.4 4.99 2.41 0.67 0.27 numerical 10.4 4.99 2.41 0.68 0.28 timell02 s 0 60 180 310 440 analytical 89.6 72.2 47.0 29.4 18.4 pB/mmHg{ numerical 89.6 72.2 47.3 29.9 19.1 TABLE 2.-cOMPARISON OF ANALYTICAL SOLUTION AND NUMERICAL INTEGRATION FOR a = 0.90 (kA = 1o-l s-', k~ = low3 s-l, pX = 10 mmHg, pb = 90 mmHg) time/s 0 25 1000 2000 analytical 100.0 110.2 158.6 176.3 100.0 110.2 158.6 176.3 total TABLE 3 .-ACCURACY OF DATA EXTRACTED BY ANALYSIS OF COMPUTER-GENERATED PRESSURE AGAINST TIME CURVES pg/rnmHg pg/mmHg 10 kA/S-l lo3 kB1S-I input a 10.0 90.0 1 .oo 1 .oo semi-emp.b 10.5 89.5 0.94 0.99 analyt .b 10.0 90.0 0.99 1 .oo input analyt.input analyt. input analyt. a = 0.95 semi-emp. { 20.0 20.5 19.9 40.0 40.2 40.0 10.0 10.6 9.8 80.0 79.6 80.1 60.0 59.7 60.0 90.0 89.3 90.2 1 .oo 0.97 1 .oo 1 .oo 0.99 1 .oo 1 .oo 0.99 1.02 1 .oo 0.99 1 .oo 1 .oo 0.99 1 .oo 10.0 9.8 10.0 input 90.0 10.0 1 .oo 10.0 a = 0.95 semi-emp. 89.7 10.2 0.99 10.0 anal yt . 89.7 10.3 1 .oo 10.2 input 10.0 90.0 1.00 1 .oo 10.9 89.0 0.91 0.99 { r analyt. 10.1 90.0 1 .oo 1 .oo a = 0.90 semi-emp. a " input " = data fed into numerical integration SE " true values " ; " semi-emp " and " analyt " = results of analysis = " observed values ". combinations. Thus either method provides a fair treatment of the dead-space problem; without correction, the plots for kB are distinctly curved at all times (e.g. (fig. 2), and meaningful parameter extraction can be virtually impossible.The analytical method is more accurate than the semi-empirical method ; indeed the former should be exact if the simulated experimental results are correct and ifP. J . ROBINSON AND G . G. SKELHORNE 2759 sufficient precision is used in the data. Thus the residual errors (a few percent) can be attributed mainly to “ experimental scatter,” introduced by limiting the computer- generated pressure data to a physically realistic accuracy of four significant figures (e.g. to 0.1 mmHg in 100-200 mmHg).* Thus when data for two competing reactions are extracted from a single pressure against time curve, the scatter and errors will be significantly greater than for a single clean reaction, unless the original data are of unusually high precision. 5 n 4 a I (Je s - a 3 Lc n Q4 I O 2 3 M 1 f 0 0 0 2000 3000 4000 timeis FIG. Z.-Plots, with and without correction for dead-space, for a = 0.95, kB/kA = 0.10 and PSIPA = 5.0. SINGLE REACTANT The present analytical solution suggests a useful treatment for first-order decom- position of a single reactant into two molecules by puttingp; = 0 in eqn (13). The more general case in which one molecule of reactant decomposes to give q molecules of product can be obtained from the integral of eqn (7) of ref. (2), which gives eqn (15). This should provide a precisely linear plot of slope k and could be a useful alternative to the plot of In (qapo-p) against t, as previously recommended We thank the S.R.C. for a studentship (G. G. S.), and the referees for their constructive comments. P. J. Robinson, G. G. Skelhorne and M. J. Waller, J.C.S. Perkin 11, 1978, 349. P. J. Robinson, Trans. Furaday SOC., 1965, 61, 1655. * mmHg = 133.3 Pa. (PAPER 8/389)