P. R. NoIi and 0. K. Selinger Australian National University Canberra, Australia
II
A * h $ e Rule Underlying Kinetia
The use of analog machines in chemical kinetics is described in the literature,' but the approach in some instances is rather ad hoc2 We have examined the actual basis of the analog appnoach in an attempt to formulate a philosophy in programming which is economic and generally applicable. This has lead us to a greater understanding of the structure of chemical stoichiometric schemes which is of particular relevance to kinetics.
librium constants are added there arc four indepcndent equations for the four unknown solution concentrations.
Theory
= 2; C = S - R = 1. Thus, there is only one independent conscrvation equation. There are three differential equations describing the time rate of change of each species' concentration. Any three of this set of four equations can be used to solve for the three unknown time dependent concentrations. We need a way of integrating our time dependent cquations. This can be done by an electronic differential analyzer (analog computer) in which concentrations are rcprcsented by time depcndcnt voltages. Using such a computer to calculatc rate constants is a four step process.
A Dynamic Sysfem
Liudemann Mechanism for "unimolecular" reactions
R
Any stoichiometric chemical schemc has associatcd with it certain conservation conditions. A consrrvation equation can be written for each cntity in the schemc. Any atom, group of atoms or othm proprrty such as charge which remains unchanged throughout thc scheme is defined as an cntity. Each scheme has a fixed number of independent conservation equations. By this we mean that any additional conscrvation cquation could have been derived from the others. If S is thc number of species and C the numbrxr of independent conservation equations, then c = s - R
where R is the number of degrees of freedom lost through stoichiometric relationships. A stoichiometric relationship is definrd as a species transformation i
7
(e.g., A-B-C or A e B + C or A-B-C all represent cases where R = 2). If there are no stoichiometric relationships each species is an entity, and there is an independent conservation equation for pach species. Each relationship combines two previously indeprndcnt conservation conditions into one. What we have said holds true for both static (e.g., equilibrium or a continuous steady state process as represented in a flow diagram) and dynamic (e.g., kinetic) schemes. There have to be as many independent equations as there arc species whose concentrations are to be dctermined. A Static System
(a) The experimental data are plotted as concentrations versus time ( b ) The reaction scheme postulated is simulated (c) The mechine parameters are altered until the output matches the experimental plots (d) The rate oonstants are evaluated from the soding factors
We can thus propose a logical programming system with the following rules (1) The set of equations chosen must be linearly independent, i.e., none of them may he derivable from the others (2) Use C = S - R conservation equations, with the ones used chosen t o have the fewest inputs (3) The remaining S - C = R equations must he differential ones
The conservation equations arc related to the differential ones in the following way: any linear combination of differcntial equations cqual to zero yields a conservation equation upon integration between the limits 0 and t. This can be done on inspection. Thus a set of equations which includes a conservation equation and differcntial cquations that may be linearly combined to yield it, breaks rule (1). Experimental
R
2;C = S - R = 2. Thus, there are two independent conservation cquations, and when the two cqui=
'TARBUTT,F. E., J. CHEM.EDUC,,44, 64 (1967). CROSSLEY, T. R., AND SLIFKIN, M. A,, Progr. React. Kinet., 5, 409 (1970). G n o s s w l i ~ ~ ~L.n ,I., et al., "The Applioation of Analogue Computers in Chemical Research," Ill. Inst. Tech., 1969.
618
/
laurnal o f Chemical Education
An E.A.I. 180 Hyhrid/Analog computer with 6 integrators, 6 summers and 2 four-quadrant multipliers was used. Additional input resistors were added. The multipliers had a specified accuracy of 1 t o 2% and were the least reliable components. Kinetic schemes requiring additional multipliers would need a better quality unit. Tho output was displayed on a. small oscilloscope or 2-?I recorder.
As a general example let us again take the Lindemann Mechanism.
equations which are not independent is the set (6), (7), (9) and (lo), because (6) and (7) can yield (10). There arc in fact four such sets. Thus, only 26 of the 30
There are three specics for which the concentrations are unknown and three independent equations. Thr minimum number of differential equations (S - C) is two in this example and therefore the minimum number of integrators needed is also two. One of the four possible programs is givcn in Figure 1. It uses eqns. (I),
-dl
Figwe 1.
A" ."dog
program for the Lindemonn Mechanism.
(3), and (4). The concentration of A, A*, and B are obtained simultaneously and continuously at the designated points in the circuit. Of interest is the behavior of [A*] for diffrrent initial concentrations A and the change from first to second order as A. is decreased. The whole of the A. range is accessible to accurate simulation (scr Fig. 2). As an example of a system where certain combinations of equations may contain both a conservation equation and the set of differential equations from which it could hc derived and thus be insoluble, we select homogencous catalysis (hlichaelis-Menten kinetics). A
-
kt k? +Bs An C +B k -1
[AB]
=
Figure 2. Outputs from the program in Figure 1 showing the change from unimoleculor to birnolecular mechonim or the pressure decreases. k~ = k-, ks and A. equals lo) 4; lbl 2; Icl 1 "nib
