Research: Science and Education
Rate of Reaction and Rate Equations S. Le Vent Department of Chemistry, UMIST, Manchester M60 1QD, UK;
[email protected] How Is Rate of Reaction Defined? The current edition and the immediately previous edition of the definitive International Union of Pure and Applied Chemistry (IUPAC) book on physicochemical symbols, terminology, and units, commonly referred to as the “green book” (1) define “rate of reaction (based on amount concentration)” ⋅ ⋅ as “ξ兾V = B1dcB兾dt”, where ξ is dξ兾dt = B1dnB兾dt, ξ is the extent of reaction (dξ = B1dnB), B is the stoichiometric number for any species B in the chemical equation defining the overall reaction (sign of B—positive for products, negative for reactants), nB is the amount of substance of B, cB (subsequently written [B]) is the concentration of B, V is the reaction volume, and t is the time. A footnote in the green book then draws attention to the fact that the two sides of the rate of reaction definition, ξ兾V and B1d[B]/dt, are equivalent only when the volume remains constant during the reaction; on this basis “(based on amount concentration)” is a somewhat puzzling qualification. Constant volume is not an uncommon situation for batch processes, exactly so for constant volume (isochoric) gas reactions, and approximately so for reactions in liquid solution and for many pure liquid reactions.1 It is certainly not the case for isobaric gas-phase reactions when the stoichiometric sum is not equal to zero,
∑ νB ≠ 0 B
The duality of rate-of-reaction definition is strongly avoided in earlier editions of the green book (2) and in another IUPAC recommendation titled “A Glossary of Terms Used in Chemical Kinetics, Including Reaction Dynamics” (3). These earlier editions define rate of reaction exclusively ⋅ as ξ (a practice generally followed by chemical engineers); ⋅ the authors emphasize that ξ兾V (rate of reaction divided by volume), dnB/dt (rate of formation of B) and d[B]/dt (rate of increase of the concentration of B) should not be called rate of reaction. However, reference 3 defines the rate of re⋅ action as ξ兾V . The defining of rate of reaction is, to say the least, confusing but there can be no doubt that the earlier versions of the green book and the glossary of terms (although different from one another) are less ambiguous than the more recent green books. Standard physical chemistry texts (4–7), reflecting the confusion, vary in the definition of the rate of reaction. Generally, the textbooks (4–6) do initially emphasize the need for constant volume if V 1dnB and d(nB兾V) (= d[B]) are to be equated, but then proceed exclusively with B1d[B]兾dt as effective definition of rate of reaction. Following the recent IUPAC recommendation, the unresolved question may be presented in terms of a particular case. For the bimolecular process, 2A → B, is it d[B]兾dt {= 1/2d[A]/dt} or is it V 1dnB兾dt {= (2V)1dnA兾dt} that should universally be called rate of reaction? The author’s contention is that it should be the one which is theoretically equivalent to k[A]2, where k is the rate constant. The pur-
pose of the present paper is to support the latter option, V 1dnB兾dt, on the basis of both the collision theory and the transition state theory of (gas-phase) reaction kinetics. Since both these theories are extensively covered in standard undergraduate physical chemistry texts (7, 8), only the essential aspects of the theories, that is, those relevant to the above question, will be covered here; many equations will simply be quoted. Collision Theory The process to be considered as an example will be the bimolecular, ideal gas process, A + B → products; the process, A + A → products is slightly more complicated because of the need to avoid double counting of bimolecular collisions. Textbook symbolism varies, so care has been taken to define all symbols. Assuming a simple hard-sphere collision model, dZAB, the number of collisions of all A molecules with B molecules in time interval dt is given by,
dZ AB =
π σAB2 sAB NA NB dt V
where σAB is the mean diameter of an A molecule and a B molecule; sAB is the effective mean speed of A and B molecules, equal to (8kbT兾πmr)1/2, where kb is the Boltzmann constant, T is the thermodynamic temperature, mr is the reduced mass of A and B, equal to [mAmB兾( mA + mB)]; and Ni and mi (i = A or B) are, respectively, the number and mass of i molecules. Consequently,
dZ AB N N = π σAB2 sAB A B dt = L2 π σAB2 sAB[ A ][B]dt V V V with L being the Avogadro constant. Then, considering the Arrhenius factor—exp(E兾RT), in which E is the activation energy and R is the gas constant—the number of collisions successful in promoting reaction in time interval dt divided by volume equals the number of A molecules consumed in dt divided by volume, −
dNA dZ AB E = exp − RT V V E = L2 π σAB2 sAB[ A ][B]exp − dt RT
Rearranging −V −1
dNA E = L2 π σAB2 sAB[ A ][B]exp − RT dt
and −V −1
dnA E = Lπ σAB2 sAB[ A ][B]exp − = k[ A ][B] (1) RT dt
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k being a function of T (and the nature of A and B) and identified with the rate constant. The key point of this development of successive relationships is that it produces V 1 dnA兾dt and not d[A]兾dt on the left of eq 1.
the quantity {kbT兾hco}{ZC兾ZAZB} being identified with the rate constant k. The net result is the same for the collision theory, eq 1, that is, V 1dnA兾dt rather than d[A]兾dt is on the left side of the equation.
Transition State Theory
Composite Mechanisms
The same bimolecular process will be considered here and the activated complex will be designated C. This is regarded as a col on a multidimensional surface plotting potential (electronic) energy against interatomic coordinates of the reaction system. The minimum energy path on the surface between reactants and products is the reaction coordinate. The following equations are based on a modern treatment given by Raff (8), which considers the flux of A,B molecular pairs across a plane multidimensional surface, designated S, drawn through the C col, and perpendicular to the reaction coordinate. (For simplicity, one might visualize the situation when there are hypothetically just two interatomic coordinates, in which case both the energy surface and S are two-dimensional entities in a three-dimensional space; furthermore, the energy surface is representable by a two-dimensional contour diagram on a flat surface.) Within a number of reasonable approximations which need be of no concern for the present purpose, FAB, the number of A,B pairs per time passing S in the direction of the product is given (8) by
The process used to exemplify collision and transition state theories is an elementary (i.e., single stage) process. However, for an overall reaction represented by a single chemical equation
N N k T z * FAB = A B b C zA zB h
(2)
where h is the Planck constant and zi (i = A, B, or C) is the molecular partition function of i, written here for a common energy zero of the potential energy surface; in the case of the C partition function, however, a factor essentially for translational motion along the reaction coordinate is necessarily omitted.2 If, for simplicity, it is assumed that all A,B pairs passing S result in reaction, then to a good approximation FAB = −
dNA dt
(3)
Furthermore, one can replace each partition function zi in eq 2 by a modified partition function Zi, defined as zio兾Ni where zio is the molecular partition function at a standard concentration co. Unlike zi, Zi is independent of V, and is related to zi by zi = Z iVL c o
(4)
and to standard chemical potential µio by µi o = −RT ln Zi Combining eqs 2, 3, and 4, and replacing each Ni by Lni, −
d(LnA ) (Ln )(LnB ) kbT Z C*VL c o = A 2 dt h ZA ZB (VLc o )
simplifying, with each ni兾V replaced by the concentration of i, [i], to
−V −1
90
dnA k T Z * = b o C [ A ][B] = k [ A ][B] h c Z A ZB dt
(5)
aA + - - - → xX + - - where a and x are unsigned stoichiometric numbers, there may be several stages each with its own rate equation (of the type of eq 1, omitting the intermediate collision-theory equality); such an equation can involve any of the chemical species B involved in that stage for the term [(BV)–1dnB兾dt]. The various single stage expressions for [(BV)–1dnB兾dt] can then be combined to form overall expressions for each [V –1dnB兾dt]. By the steady state approximation, all such expressions for unstable intermediate species—that is neither reactants nor products in the overall chemical equation—can be equated to zero. Solution of the resultant set of simultaneous equations will then lead to expression for what may be termed the overall rate of reaction –(aV) –1 dn A 兾dt ⋅ [= (xV)–1dnX兾dt] = ξ兾V in terms of individual-stage rate constants and reactant concentrations. The important point is that the expression generally equates to [–(aV)–1dnA兾dt] and not to [–a–1d[A]兾dt], the two being equivalent only when the process occurs at constant volume. A variant of this situation arises when a single set of reactants yields more than one set of products. Here the composite process is represented by more than one chemical equation—one for each product set—with an overall rate for each, expressed as [(xV)1dnX兾dt]. When intermediates are not unstable species, the concept of overall rate of reaction is inapplicable. Constant Pressure, Ideal Gas Reactions with Nonzero Stoichiometric Sum As a further argument against incautious general involvement of concentrations in rate of reaction definitions, consider the case of gas reactions proceeding at constant pressure and with changing volume as the reaction proceeds; admittedly, this may be an unusual experimental situation. The ideal gas reaction A → B + C, elementary or otherwise, will be used as a simple example. It will always be the case that [dnA兾dt = dnB兾dt = dnC兾dt] and therefore that [V 1dnA兾dt = V 1dnB兾dt = V 1dnC兾dt], but it will not always be true that [d[A]兾dt = d[B]兾dt = d[C]兾dt]. The rate-of-concentrationchange equalities will, of course, be true at constant volume but consider instead the constant pressure situation. At constant temperature and pressure, p, the total concentration by the ideal gas equation [A] + [B] + [C] = p兾RT, is a constant throughout the process, so that
d[A] d[B] d[C] + + = 0 dt dt dt
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Research: Science and Education
Since, inevitably, [B]= [C], eq 6 must become −
d ([ A ] + [B] + [C]) dt
d[A] d[B] d[C] = 2 = 2 dt dt dt
Individual relationships for these separate time derivatives of concentration in terms of corresponding derivatives of amount of substance can be illustrated for A as follows
d[A] d(nA /V ) dV dn = = V −2 V A − nA dt dt dt dt dV dn = V −1 A − [A] dt dt
p ; RT nB = nC = nAo − nA
(7)
(8)
where nAo is the initial amount of substance of A and RT RT V = (nA + nB + nC ) = (2nAo − nA ) (9) p p
So, by combining eqs 8 and 9, RT dnA 1 dnA dV = − = − [A]o dt dt p dt
(10)
Further combining eqs 7 and 10,
d[A] dn = V −1 A dt dt
[A] 1 + [A] o
(11)
ranging from [2V 1dnA兾dt] at the beginning of the reaction to [V 1dnA兾dt] at its completion. Similar arguments for B yield, d[B] dn [B] = V −1 B 1 − dt dt [A]o
dnA [ A ] + [B] + [C ] = 0 −1 + dt [ A ]o
The situation is rather more complicated when the initial amount of substance of B or C is not zero. Furthermore, if the process is neither isochoric nor isobaric or does not involve ideal gases, the increased complexity of the situation will prevent the straightforward development of equations as above. Summary
using the quotient differentiation rule. Now, assuming no B or C present at the start of the reaction, [A]o, the initial concentration of A,
[A]o =
= V −1
(12)
the negative sign in the parentheses arising from the replacement in eq 10 of dnA by dnB. Because {[A] + [B] + [C] = [A] + 2[B]} is constant, for complete reaction, the final concentration of B, [B]∞, is equal to 1/2[A]o. This means that d[B]兾dt ranges from [V 1 dnB兾dt] at the beginning of the reaction to [1/2V 1dnB兾dt] at its completion. The situation is identical for C; in eq 12, C replaces B where the latter appears. Recognizing that [dnB = dnC = dnA], eqs 11, 12, and the C analogue of the latter can be added together to give, as expected and supporting the separate equations,
This paper demonstrates (i) the need for care when using the term rate of reaction, (ii) the general superiority of [(BV )1dnB兾dt] over [B1d[B]兾dt], and (iii) the dangers of always assuming that [B1d[B]兾dt] is the same for all species B involved in a reaction. When reactions are isochoric, it would seem sensible (i) to stress these points, and (ii) to start with the general [(BV )1dnB兾dt] and then progress to its equivalent [B1d[B]/dt]. Notes 1. The situation is rather different in flow processes where, within a stirred tank or an infinitesimal element of a plug-flow reactor, the volume is absolutely constant. Within such constant volumes, finite or infinitesimal, there is both reaction and flow (both in and out); the reaction component of the composite process is unquestionably isochoric (and isothermal). 2. In the equation corresponding to eq 2 in reference 8, and unlike here: (a) NA and NB represent the numbers of A and B molecules per volume so there are then compensatory volume terms— a left side denominator V and a right side numerator V; (b) the zi are referred to specific energy zeros for A, B, C, resulting in the added presence of an exponential shift term.
Literature Cited 1. Mills, I.; Cvita˘s, T.; Homann K.; Kallay, N.; Kuchitsu, K. Quantities, Units and Symbols in Physical Chemistry, 2nd ed; Blackwell Scientific Publications: Oxford, 1993; 1st ed.; 1988. 2. Manual of Symbols and Terminology for Physicochemical Quantities and Units; McGlashan, M. L., Ed., Pergamon: Oxford, 1973. ibid. Whiffen, D. H., Ed.; Pergamon: Oxford, 1979. 3. Laidler, K. J. Pure Appl. Chem. 1996, 68, 149–192. 4. Levine, I. N. Physical Chemistry, 4th ed.; McGraw-Hill: New York, 1995; p 494. 5. Castellan, G. W. Physical Chemistry, 3rd ed.; Addison Wesley: Reading, MA, 1983: pp 800–803. 6. Moore, W. J. Physical Chemistry, 5th ed.; Longman: London, 1972: pp 324–325. 7. Atkins, P.; de Paula, J. Atkins’ Physical Chemistry, 7th ed.; Oxford University Press: New York, 2002, Chapter 27. 8. Raff, L. M. Principles of Physical Chemistry; Prentice Hall: Upper Saddle River, 2001; pp 1197–1211.
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