5. G. Canagarafna University of Ceylon Perodeniya, Ceylon
I I
The Definition of the Rate of a Chemical Reaction
When a chemical reaction takes place in a system, the molecules of the reactants decrease in number while the molecules of the products increase in number. The rate of the reaction should therefore be defined in terms of the rate of destruction of reactants, or, when there are no intermediates or the concentrations of the intermediates is constant (I), in terms of the rate of creation of products. Most texts take i t as obvious that the rate of a reaction may be defined in terms of changes in concentrations. This definition is valid only if the reaction takes place without change of volume. There are reactions, e.g., reactions investigated by dilatometry and reactions carried out in a flow reactor, for which this condition is not true. In such cases there is, in general, no simple connection between the changes in concentrations and the rate of reaction. We shall examine the definition of the rate of reaction which holds in all cases. General Definition: Closed Systems
Rote of reaction= J'
=
1
$dnj/dt)
(3)
The species in eqn. (3) should strictly be one of the reactants. If the concentrations of the intermediates are constant or zero, the species i can also be taken to be any one of the products. J' will be expressed in amount of substance per unit time. Multiplication of J' hy the molar mass of i will give a reaction rate that is expressed in mass per unit time. Now, (dni/dt) is an extensive quantity and will he proportional to the volume of the system. We may therefore write
The quantity J = S / V is the rate of chemical reaction per unit volume and is an intensive quantity. It will therefore involve only the concentrations c,, T,and P. Thus
Consider the reaction
taking place in a closed system. The rate a t which the number of molecules of a substance is being created or destroyed is proportional to its coefficient in the reaction. Thus
For elementary reactions the function f i n eqn. (5) will be the usual mass action expression. If the volume is constant, we have
Under these conditions the rate of change of concentration is a measure of the rate of reaction. Since c, = ni/V, we have
Equation (1) is valid only if the concentrations of the intermediates is constant or zero. If we denote by u the coefficient, including sign, of a substance in the reaction, i.e., -ive for reactants and +ive for products, then
Thus, when V is not constant, there is no simple connection between J and de,/dt and the latter cannot therefore be used as a measure of the rate of reaction. The rate of change of volume may be used as a measure of the rate of reaction when the partial molar volumes u , are constant during the reaction. We then have
Clearly, each of the terms in the above equation is independent of the nature of the reactant or product and depends only on the rate of reaction. We therefore write
and
Suggestions of material suitable for this column and guest columns suitable for publication directly should he sent with as many details as possible, and particularly with reference to modern textbooks, to W. H. Eberhardt, School of Chemistry, Georgia. Institute of Technology, Atlanta, Georgia 30332. Since the purpose af this column is to prevent the spread and continuation of errors and not the evaluation of individual texts, the sources of errors discussed will not be cited. In order to be presented, an error must occur in at least two independent recent standard books.
I t is clear from the foregoing discussion that dci/dt cannot be taken as a rigorous measure of the rate of reaction for systems which change in volume during the reaction. Indeed McGlashan (2) in his monograph recommends against calling the quantity dcildt the rate of reaction. It remains for us to examine actual data to see how significant the difference between J and dci/dt can he. In general, the volume change is about 1% for most reactions taking place in solution. Thus, dlnV/dt is not likely to be large. Also, if we work with dilute solutions eqn. ( 7 ) shows
200
/ Journal ot Chemical Education
that the difference between dc,/dt and vlJ will be small. The case is different, however, when we consider reactions taking place in the absence of an inert solvent. A. D. Jenkins (3) has worked out that the theoretical contraction for monomers at 100% polymer conversion a t 60'C ranges from 16.490 for styrene to 36.1% for acrylonitrile. Polymerization of undiluted monomer should therefore serve as a useful test of the difference between the two definitions. We shall analyze below the data for the polymerization of vinyl acetate. The data yields a 1st-order plot when we use eqn. (4) for the definition of the rate of reaction. Our prohlem will be to determine whether there is a significant deviation when we take the rate of reaction to be defined by eqn. (6).
ever, different as seen from the figure. I t is instructive to find out why eqn. (13) also yields a straight line. If eqn. (12) is correct, then an approximate linear dependence of l n c ~on t implies that ln(V/Vo) is approximately proportional t o t . We recall that
whence for small x we may write
where 6 is a positive correction term. For small values of AVwe may write
Detailed Analysis of an Example Starkweather and Taylor (4) have published the data for the bulk polymerization of vinyl acetate for which the theoretical contraction a t 100% polymer conversion is about 26%. We shall analyze the data according to the two definitions of the rate of reaction. A. The reaction that we consider may for the purposes of the rate-determining step be written as
and similarly
Thus, hecause the 6 and 6' terms tend to cancel out, the equation
M-P
According to eqns. (4) and (5),assuming 1st-order kinetics
whence
which on integration yields
Assuming that the contraction is proportional to the extent of reaction
has approximate validity. A plot of InV vs t is linear until about 30 minutes. Because the proportionality constant is not k hut k(AV,/Vo), the slope of the plot of eqn. (13) will he different from k . The foregoing analysis has shown that the apparent value of k obtained can he significantly different from the true value even when the contraction in volume is not large enough to interfere with the linearity of the graph. Gas Reactions Gas reactions carried out at constant pressure either in static systems or flow systems are good examples where the effect of changes in volume can he quite large (5). Thus the reaction Table 1. Kinetic Data for the Polymerization of Vinyl Acetate
and AV.
=
V,
- V- =
tlmin An;
lW(Vo -w/Vo
1V..
-AV
log(4V-nW
VIVO
b V m - AW (VIVOI
log a
(U)
whereA is a constant. We therefore have
Substitution in eqn. (9) gives
A graph of in(AV.. - Av) versus t should be a straight line with slope - k . B. If we take d c ~ l dto t he the rate of reaction we have
A quantity proportional to C M can he computed from nM by dividing by V/Vo, which itself may be calculated from the percentage contraction. Thus, if eqn. (13) is true, a waph of l n c ~versus t should give a straight line with (AV, slope - k . Instead of c~ we use the Av)/(VlVo). Table 1 sets out the ~ r i m a r vdata and the calculated quantities. We know that eqn.-(12) and eqn. (13) cannot both be rigorously valid. Plots according to both equations yield straight lines for small times. The slopes are, how-
Rate data for poiyrnerizalion of vinyl acetate; a = log [ L V A . a = l o g (AV, - AV)/[v/V,) in B.
-
Volume 50. Number 3.March 1973
A V ) in
/
201
HC@'~' I HC\ CH,
+
2 HC' 'CH,
Ha!
Hc,
+
II
HC,
CH=CH,
Table 2.
Values of the Terms in Equation (14) for v = 2
I
,CH-CH=CH,
C H2
shows a 100% change in volume a t constant pressure. Let us consider the reaction
taking place at constant pressure in a static system. Then, since V = (nA+ ne)RT/P
From eqn. (7) we have dc,/dt
= = =
-kc, -kcA -kc&
- c,dlnV/dt - kzAcA(v- 1) - xA(u- 111
ume of the system the amount of species i in this volume changes due to (1) chemical reaction, (2) mass flow, and (3) diffusion. Thus the total rate of change of c, is given.
If the reaction vessel is a straight tube and the flow velocity is u, then the rate at which substance passes through an area A is given by Auci. The change of this quantity over a distance dx viz., A(auci/dx)dx = Au(aci/ax)dx gives the rate of loss of substance in a volume Adx due to mass flow
Since XA varies with time, it is clear that for gas reactions taking place a t constant pressure, equations derived from
will lead to erroneous results if v # 1. Of more practical interest are gas reactions taking place in flow systems. A large number of gas reactions have heen investigated by this method. Still considering the ,B, the relevant equation for a 1st-order rate reaction A constant is (5)
Similarly, the rate of flow due to diffusion is, by Fick's first law, -D(aci/ax). By arguments similar to the ahove, (or, which is the same, hy Fick's 2nd law)
-
where V' = volume of gas entering per second; Vn = volume of reaction space; and F = fraction of A reacted. Each value of V' will yield a different value of F thus enabling us to verify the constancy of the calculated values of k. Neglect of changes in volume will mean that the (v - l ) F term will drop out. Table 2 gives the magnitudes of the two terms in the RHS of eqn. (14) for u = 2. We see that the (U - l ) F term ranges from about 20 to 30% of the total and cannot therefore be neglected in accurate work. Open Systems We have already seen that the rigorous definition of the rate of reaction must he in terms of the rate of creation or destruction of substance. In closed systems the changes in amount of any species is due only to creation or destruction by the reaction. Thus for clcmed systems the rate of reaction can be equated to the rate of change of the amount of any given species and is thus a directly determinable experimental quantity. In open systems, however, the changes in amount of a given species in a given part of the system is due not only to creation or destruction by chemical reaction but also to (1) mass flow of the given species into or out of the part of the system considered, and (2) diffusion. Thus, in open systems the rate of creation or destruction by chemical reaction is not a directly determinable quantity hut has to be calculated from the measured rate of change of amounts of a given species by eliminating the effects of mass flow and diffusion by the use of the law of conservation of mass. This is most conveniently done for the general case in terms of vector notation, but the general principles involved in calculating the rate of reaction for an open system from experimental quantities are best illustrated by considering a simple example. If we consider a unit vol202 1 Journal of Chemical Education
Therefore
Equation (15) shows how the rate of reaction can in principle be calculated from experimental quantities. When a steady state has been attained (dcildt) = 0, and
If the diffusion term is negligible with respect to the mass flow term, eqn. (16) reduces to
If the order of the reaction is known, the ahove equation can be integrated. For example, assuming a first-order reaction (dc,/dt),,
= =
-kc, u(dc,/bx)
Integration gives
where 1 is the length of the reaction tube and ct and CI are the final and initial concentrations. No new principles are involved in treating the three dimensional open system. Literature Cited (1) R a i d , Y., and Freeman,W. A., J. CHEM. EDUC. 17,159 (1970). 121 McGlashan. M. L.. "Phvsieo-chemical Quantities and Units," The Royal lnatifute af memint&, L O ~ ~ 1368. O ~ .p. 39. (31 Jenkin~.A. D.. "Symposium on Techniques of Polymer Science," Repoct of the Royalln~fituted~himistri, No.5. 1956. (4) Sfarkwesfher, H. W..andTsyln. G.B..JAmar.Chsm. Soe.. 52.4708(19301. (6) Marmll A "Teehnioue af O~eanicChemistrv." Yol. 8. Part I. [Editor: Weisberzer.
