Research: Science and Education
A New Look at Reaction Rates Tomislav Cvitaˇ s Physical Chemistry Laboratory, University of Zagreb, POB 163, HR-10001 Zagreb, Croatia;
[email protected] Chemical reaction rates are usually defined in terms of concentration changes with time, corrected by the corresponding stoichiometric coefficients. Rates of other processes, such as radioactive decay, are defined by the differential quotient of the number of events and time. This paper shows that reaction rates can be thought of as the change in the number of chemical transformations per time. This requires a different approach to the recently introduced quantity extent of reaction and a definition of the stoichiometric numbers, which have hitherto only been described by giving examples. The relationship between the conceptual definition of reaction rate as an intensive quantity describing the number of events per time and the practical definition in terms of concentration changes is also given. Historical Introduction Before 1970 most physical chemistry textbooks and some textbooks on chemical kinetics defined the rates of chemical reactions simply as rates of change of concentration of a substance involved in the reaction with a minus or plus sign attached, depending on whether the substance is a reactant or a product (1). Since it is obvious that the concentrations of different reactants do not always change at the same rate, depending on the ratio of the corresponding stoichiometric numbers, it was essential to define exactly what substance was used to define the rate. Alternatively the rates of concentration change for particular substances taking part in the reaction were divided by the corresponding stoichiometric numbers appearing in the reaction equation. In the late 1960s McGlashan compiled a list of symbols for quantities used in physical chemistry, including chemical kinetics. These names and symbols for quantities were accepted by IUPAC and published as official IUPAC recommendations (2). They included a completely new definition of the rate of reaction as the time derivative of the extent of reaction or advancement ξ: dξ ξ= (1) dt The problem of selecting a particular reactant or product was thereby avoided. The extent of reaction was introduced into thermodynamics more than 75 years ago (3). It describes one of the most fundamental concepts studied in chemistry—the progress of a chemical reaction—and is essential in describing energy changes accompanying chemical reactions as well as rates. Modern textbooks on physical chemistry (4–6 ) use it in chapters both on thermodynamics and on chemical kinetics. The usual definition of the advancement is
dξ = ν1 dn B B
(2)
or in integrated form
ξ = ν1 n B – n B,0 B 1574
(3)
where nB is the amount of entities B and νB is the corresponding stoichiometric number. The initial advancement in eq 3 was taken to be zero. In addition, it is usually said that the stoichiometric number is negative for reactants and positive for products. The definition of reaction rate given in eq 1 was quickly included in textbooks of physical chemistry (7–8), but merely as a definition. In 1972, Moore (7) defined the rate of reaction by eq 1 but said a few lines later that the quantity ξ /V, where V is the volume of the system, is often referred to as simply the rate of reaction. Atkins (8) in 1978 called eq 1 the true reaction rate, keeping the name reaction rate for the established usage. Thus the quantity defined by eq 1 itself was used just in definitions, and the perception of reaction rate remained linked to a concentration change, dcB, of a substance B taking part in the reaction corrected by the corresponding stoichiometric number. It did not take long for the kineticists to bring back the old definition for the rate of reaction (9) in an improved form,
v=
1 dξ 1 dc B = ν dt νB dt
(4)
inventing a new name—rate of conversion—for the hardly ever used quantity defined by eq 1. The more recent IUPAC recommendations (10) adopted this change and list quantities in both eq 1 and eq 4. Some of the relevant current recommendations (10, 11) are given in Table 1. Recent textbooks on physical chemistry use the same definitions and it is hoped that general chemistry textbooks will also do so soon. Radioactivity These concepts and definitions can be better understood and integrated if we focus first on a much simpler process, radioactivity. Radioactivity is the rate of radioactive decay when an atomic nucleus R disintegrates into a new nucleus P. R→P
(5)
The definition of this physical quantity can be expressed by the equation
A=
dNt dt
(6)
where Nt is the number of nuclear transformations or decays. Nt can be measured directly, since a highly ionizing particle leaves the system on each event and can be registered by a suitable counter. The number of such events occurring in a time interval dt is proportional to the number of radioactive nuclei R in the sample A = λ NR
(7)
where the proportionality constant is called the decay constant and is characteristic for the decaying nuclei R. Equation 7 states that a large sample will display a higher radio-
Journal of Chemical Education • Vol. 76 No. 11 November 1999 • JChemEd.chem.wisc.edu
Research: Science and Education
activity than a small sample; that is, the radioactivity A is an extensive quantity. It can be converted to an intensive quantity if we divide it by another extensive quantity, such as mass. Thus the specific radioactivity a = A/m
(8)
does not depend on the size of the investigated system and is characteristic of the radioactive substance or of the decay process itself. Uranium-238 has a lower specific radioactivity than radium-226; that is, the α-decay of uranium-238 is slower than the α -decay of radium-226. In order to study the behavior of radioactive samples in time, eqs 6 and 7 have to be combined:
dNt = λ NR dt
(10)
By inserting this equation into eq 7 we obtain the familiar law of radioactive decay in differential form,
{
dNR = λ NR dt
(12)
where NR,0 is the initial number of atoms R at t = 0. Chemical Reactions The simplest form of a chemical reaction, such as an isomerization, can also be represented by the general equation (eq 5), and there is no reason why we should not define the rate of such a process in the same way as for radioactive decay: as the differential quotient of the number of chemical transformations and time, eq 6. If the chemical reaction were a stochastic process such as radioactive decay, this rate would be proportional to the number of reactant molecules R as given by eq 7.
dNt = k NR dt
Definition SI Unit ∆ξ = (1/νB )∆nB mol — ?
1
ξ = d ξ /d t
mol s {1
Rate of concentration change (due to chemical reaction)
vB
vB = dcB /d t
mol m {3 s {1
Rate of consumption of reactant R
vR
vR = { dcR /d t
mol m {3 s {1
Rate of formation of product P
vP
Rate of reaction (based on amount concentration)
v
vP = d cP /d t mol m {3 s {1 ? v = ξ /V = mol m {3 s {1 (1/ν B)(dc B/dt )
teristic of the reaction and is called the rate of reaction (based on number concentration). More commonly in chemistry, one uses amount-of-substance concentrations, c = C/L, where L is the Avogadro constant, and the corresponding rate of reaction (based on amount concentration) can then be obtained from eq 14 by dividing by the Avogadro constant
dξ vc = 1 = k cR V dt
(14)
where CR = NR/V is the number concentration of molecules R. Just like specific radioactivity (eq 8), this rate is charac-
(15)
The advancement ξ in this equation is given by
ξ=
Nt L
(16)
This is a new conceptual definition of the quantity defined operationally by eq 2. There is an important difference with respect to radioactive decays: we cannot measure the number of individual chemical reactions. There is no counter for such events. We can only measure the number of molecules R or P or their amounts. Thus in practice the extent of reaction is determined from the change in amount of molecules taking part in the studied reaction as given by eqs 2 and 3. There is no reason, however, not to think of it conceptually as the amount of chemical transformations (12) as defined by eq 16. To establish the time dependence of concentration during a chemical reaction (i.e., to integrate eq 14), we have to use relation 10 connecting the change in the number of molecules and the individual chemical transformations. Thus by substituting {dNR for dNt into eq 14 we obtain the familiar first-order rate law of chemical kinetics
(13)
where the proportionality constant k is called the rate coefficient. Like radioactivity (eq 7) the rate defined by eq 13 is an extensive quantity. To make it intensive it is convenient in chemical kinetics to divide it by volume, as long as the volume is constant during the reaction:
dNt vC = 1 = k CR V dt
Symbol ξ Extent of reaction, advancement ν Stoichiometric number, stoichiometric coefficient ? ξ Rate of conversion
(11)
which can be integrated to NR = NR,0 e {λt
Name of Quantity
(9)
In addition, we need the relationship between the number of radioactive nuclei NR and the number of disintegration events (decays) Nt . The change in the number of nuclei R with each event is {1 because we lose one atom on each decay. This can be expressed mathematically by
∆ NR = {1 ∆ Nt
Table 1. Some Recommendations by the IUPAC Division of Physical Chemistry (10, 11)
{
dC R = kC R dt
(17)
On integration we obtain
C R = C R ,0 e{ k t
(18)
Stoichiometric Numbers A general chemical reaction is much more complicated than that given by eq 5. We can write it in the form |ν R1| R1 + |ν R2| R2 + … → ν P1 P1 + νP2 P2 + …
JChemEd.chem.wisc.edu • Vol. 76 No. 11 November 1999 • Journal of Chemical Education
(19) 1575
Research: Science and Education
This equation tells us that |ν R1| molecules R1 react with |ν R2| molecules R2 and … to yield νP 1 molecules P1, νP 2 molecules P1, and …. We shall call such an event a chemical transformation. In IUPAC recommendations (10, 11) they are described with an example, rather than formally defined as are all the other physical quantities. They are negative for the reactants and positive for products. Thus in eq 5 the stoichiometric coefficient of R is ν R = {1, but the formal definition is given by eq 10. Indeed, for any molecule B taking part in the reaction, whether reactant or product, the stoichiometric number can be defined by the same equation:
νB =
∆ NB ∆ Nt
(20)
From this definition it is obvious why the stoichiometric numbers of the reactants have to be negative and why they are positive for products, since the number of reactant molecules decreases and the number of product molecules increases as the reaction advances. The stoichiometric numbers are thus the changes in the number of molecules per chemical transformation as indicated by the chemical reaction equation. Chemical transformations, as given by reaction equations of the type 19, can be treated as countable objects, and these numbers can be converted to amounts by dividing by the Avogadro constant:
νB =
∆ NB /L ∆n = B ∆ξ ∆ Nt /L
(21)
With radioactivity this is more obvious, probably because the measurement process relies directly on counting disintegration events. In my experience beginning students of chemistry do not have difficulties in perceiving chemical transformations as countable objects, and it is no problem to introduce the concept of the number of chemical transformations and the extent of reaction early in a general chemistry course. No physicist would talk about the velocity before defining the path, or about the volume flow rate before defining the volume, so why should we talk about reaction rates without defining the advancement or extent of a reaction? In my opinion the simplest way to do this is in terms of amount of chemical transformations as given by eq 16 above. Equation 21 is the same as eq 2 and is usually taken as the definition of the extent of reaction, although strictly speaking the stoichiometric number has never been properly defined. It is an operational definition, in that it provides a means of determining the advancement on the basis of measurable changes in the amount of a particular reactant or product. Conceptually, however, definition 16 is much simpler and enables a proper definition of the stoichiometric number by eq 20 or, more conveniently in practice, by eq 21.
In eq 15 we have divided the rate of conversion by volume to obtain an intensive quantity. It is convenient to give the quantity ξ/V a special symbol (13):
1576
ξ V
v c = dx (23) dt By dividing the numerator and denominator in eq 21 by the volume we obtain a third expression for the stoichiometric number: ∆c νB = B (24) ∆x This is an equation linking the change in concentration of any substance taking part in the reaction to the concentration of chemical transformations that took place in the system. In integrated form we obtain cB = cB,0 + ν B x
(22)
(25)
for any substance B taking part in the reaction or cR = cR,0 – |ν R| x
(26)
for the reactants. Here again it was assumed that the initial advancement was ξ 0 = 0. The rate law of an nth order reaction could then be written dx = k c n = k c – ν x n (27) R R ,0 R dt and integrated to kt =
1 n – 1 νR
–
1 = c R ,0n–1
1 n – 1 νR
1 – n–1
1 c R ,0 – νR x
n–1
cR
1
(28)
c R ,0n–1
which holds for all n ≠ 1. Conclusion There is a close relationship between rates of radioactive decays and rates of chemical reactions. Both can be thought of as numbers of transformations per time. Intensive quantities characterizing the process can be obtained by dividing by mass, as is usual in the case of radioactivity, or by dividing by the constant volume, which is more convenient in the case of chemical reactions. By converting the number of transformations into amounts, eq 16, the definition of reaction rate is obtained. This can be converted to a practical form in terms of measurable concentrations, eq 4. The line of thought is shown in the scheme with the numbers of equations appearing in the text. 1 dNt V dt
Concentration of Chemical Transformations
x=
It represents the amount concentration of chemical transformations. Frost and Pearson (1) and later Moore and Pearson (13) called it reaction variable after the German word Umsatzvariable (14). The rate of reaction is then simply
(16)
1 dξ chain rule V dt
1 d ξ dnB ⋅ ⋅ V dnB dt
(21
)
(22) dx dt
chain rule
dx dc B ⋅ dcB dt
(24)
1 dcB νB dt
Scheme From the conceptual to the practical definition of the reaction rate
Journal of Chemical Education • Vol. 76 No. 11 November 1999 • JChemEd.chem.wisc.edu
Research: Science and Education
The scheme shows also that in the practical definition of the reaction rate (on the right) the division by the stoichiometric number is not merely a correction factor taking into account the different rates of concentration changes for different reactants or products, but a conversion from the amount of transformations to the amount of entities B as clearly shown by the chain rule of differentiation. The stoichiometric numbers or stoichiometric coefficients are physical quantities, but no proper definition was ever given in IUPAC recommendations or common textbooks. I propose here to define them by eq 20 as the change in the number of molecules B per chemical transformation indicated by the reaction equation. The sign is then taken care of automatically. I propose that the reaction variable of Moore and Pearson (13) be treated as the concentration of chemical transformations. The concepts of amount and concentration of chemical transformations are sufficiently simple to be taught in general chemistry courses. They enable a consistent approach to reaction quantities (10) such as reaction enthalpies ∆rH, as well as to reaction rates. There is an important limitation. The concept of a chemical transformation as a countable entity is meaningful only when the reaction equation is stated and constant over time. The same argument applies then to the advancement and the rate of reaction. When the reaction cannot be speci-
fied, neither can its advancement or rate be. In such a case only rates of consumption and rates of formation as given in Table 1 are meaningful quantities. Literature Cited 1. Frost, A. A.; Pearson, R. G. Kinetics and Mechanism; Wiley: New York, 1953. 2. McGlashan, M. L. Manual of Symbols and Terminology for Physicochemical Quantities and Units, 1st ed.; Pure Appl. Chem. 1970, 21, 1. 3. De Donder, T. Bull. Acad. Belg. Cl. Sci. 1922, 8, 197. 4. Barrow, G. M. Physical Chemistry, 6th ed.; McGraw Hill: New York, 1996. 5. Alberty, R. A.; Silbey, R. J. Physical Chemistry, 2nd ed.; Wiley: New York, 1997. 6. Atkins, P. W. Physical Chemistry, 6th ed.; Oxford University Press: Oxford, UK, 1998. 7. Moore, W. J. Physical Chemistry, 5th ed.; Longman: London 1972. 8. Atkins, P. W. Physical Chemistry, 1st ed.; Oxford University Press: Oxford, UK, 1978. 9. Laidler, K. J. Pure Appl. Chem. 1981, 53, 753. 10. Mills, I.; Cvitaˇs, T.; Homann, K.; Kallay, N.; Kuchitsu, K. Quantities, Units and Symbols in Physical Chemistry, 1st ed.; Blackwell Scientific: Oxford, UK, 1988; 2nd ed., 1993. 11. Laidler, K. J. Pure Appl. Chem. 1996, 68, 149. 12. Cvitaˇs, T.; Kallay, N. Educ. Chem. 1980, 17, 166. 13. Moore, J. W.; Pearson, R. G. Kinetics and Mechanism; Wiley: New York, 1981. 14. Skrabal, A. Homogenkinetik; Steinkopf: Dresden, 1938; cited in ref 1.
JChemEd.chem.wisc.edu • Vol. 76 No. 11 November 1999 • Journal of Chemical Education
1577
