RATE, ORDER AND MOLECULARITY IN KEITH J. LAIDLER Catholic University, Washington, D. C. SAMUEL GLASSTONE Boston College, Chestnut Hill, Massachusetts
IT
APPEARS from some recent discussion1-' that there rate = ... (1) exists a certain degree of misunderstandig regarding the over-all order of the reaction is simply the fundamentals of chemical kinetics, and the staten= u 8 ... (2) ment has recentlv been made6 that the subiect is in a highly confused siate. I t has accordingly been thought Such a reaction is said to be of the uth order with reworth while to present an outline of the basic principles spect to the substance A, the pth order with respect to which should demonstrate, that contrary to this conten- B, etc. tion, no serious logical inconsistencies exist. In addiSimple or ''elementary" reactions, which occur in a tion to covering matters that are generally agreed upon, single stage, will first be considered. If only one type and which for the most part have found their way into of chemical substance is concerned, e. g., in a thermal the textbooks, the present paper will present certain decomposition or an isomeriaation, the rate equation has points that do not appear to have been treated before in the form an explicit form. ~~
+ +
~
- -dc =
dt
THE RATE OF A CHEMICAL REACTION
kP
(3)
The rate of a reaction, which may also be called its so that the order is simply or. No clear-cut example of velocity or speed, can be d e h e d with relation to the a > 2 seems to have been investigated, but values of u concentration of any of the reacting substances, or to < 2, both integral and nonintegal, have heen found. that of any product of the r e a ~ t i o n . ~If the species Thus, for the decomposition of nitrogen pentoxide a = chosen is a reactant which has a concentration c a t 1, for the decomposition of hydrogen iodide a = 2, time t the rate is - dc/dt, while the rate with reference while for the ortho-para hydro$& conversion a = 1.5. to a product having a concentration x at time t is dx/dt. An order of ae;o ( a = 0) is frequently found in surface Any concentration units may be used for expressing reactions involving a high degree of adsorption. Nonthe rate; thus, if moles per liter are employed for con- integral values are relatively common, and would probcentration and seconds for the time, the units for the ably be more so if the experimental results were not frerate are moles liter-' set.?. The most fundamental quently constrained to follow laws of integral order. units for concentration are molebules per cc., and the An order may vary with the t o t d concentration, excorresponding rate unit is molecules w.-l set.-l; the amples being found in the thermal decomposition of latter unit is usually found to be most useful in the acetaldehyde, and in all unimolecular processes (see application of molecular statistics to chemical processes. later). For gas reactions pressure units are sometimes used in The simplest example of a reaction rate depending place of concentrations, so that legitimate units for the upon the concentiation of two substances is given by rate would be (mm. Hg) sec.-I and at,m. set.-' rate = Ico~cs
THE ORDER OF A RERCTION
The order of a reaction concerns the dependence of the rate upon the concentrations of reacting substances; thus, if the rate is found experimentally to be proportional to the ath power of the concentration of one of the reactants A, to the Bth power of the concentration of a second reactant B, and so forth, via., ANTONOFF, G., J. CHEM.EDUC., 21,420 (1944). *LUDER, W. F.,ibid., 21, 559 (1944). SNTONOFF, G., ibid., 22, 98 (1945). LUDER,W. F., ibid., 22, 201 (1945). GLASSTONE, S., ibid., 22, 201 (1945). ANTONOFF, G., J . Phys. Colloid Chem.,51, 513 (1947). 7 The actual value obtained may differ aooordiug to the subst,ance chosen; this point is discussed in a later sect,ion.
(4)
This corresponds to a second order process which is of the first order with respect to both A and B. Reactions having a total order of three are also known (e. g., 2N0 Brz), but it is doubtful whether reactions of higher order exist in the gas phase. In a number of instances the order of a reaction is less than the over-all order, for the reason that the concentration of one or more of the reacting substances remains sensibly constant during the course of the reaction. This may arise in two ways: (1) One of the reacting substances, e. g., the solvent, may he present in such great excess that its concentration hardly changes as the reaction proceeds. (2) One of the reacting substances, although not
383
+
JOURNAL OF CHEMICAL EDUCATION
384
necessarily present in large excess, is also a product of the reaction; it therefore plays the part of a catalyst, its concentration remaining constant. An example of the first case is the hydrolysis of an alkyl halide in aqueous solution; the rate of this reaction is proportional to the concentration of the halide, and so is of the first order with respect to that reactant. In this case the concentration of the water does not change appreciably as the reaction proceeds. However, if experiments were made in nonaqueous solvents containing small amounts of water, the concentration of which varied during the course of the reaction, they would in all probability indicate the reaction to be of the first order also with respect to the water. It is common to speak of such rersctions as being pseudo-unimolecular, but this nomenclature may lead to confusion and should be avoided. As will be seen below, the molecularity of a reaction is related to its fundamental nmchanism, and consequently terms such as pseudo-unimolecular, pseudo-bimolecular, etc., as generally employed, are not entirely rational. The situation regarding catalyzed reactions is similar. In the presence of a constant amount of catalyst, reactions frequently have an order of unity, the rate being directly proportional to the first power of the concentration of the substrate. However, on varying the concentration of catalyst the reaction is almost invariably found to be of the &st order with respect to the catalyst also, and under these experimental conditions it is therefore of the second over-all order. If the reaction is taking place in solution the possibility still remains that the reaction is also first order with respect to the solvent, a matter which could only be investigated in indifferent solvents. If it were found to be so the reaction would be of the third over-all order. Such is probably the case for the familiar hydrolysis of an ester in aqueous solution when catalyzed by hydrogen ions; the reaction is of the first order with respect to the ester, the water, and the catalyst. The foregoing applies to surface catalysis a8 well as to homogeneous catalysis. For example, the reaction between nitric oxide and oxygen on a glass surface is of the first order with respect to both nitric oxide and dxygen, and from this standpoint would be spoken of as being of the second order; however, if experiments are carried out in which the surface area is changed it is found to be a reaction of the third (over-all) order. THE SPECIFICREACTION RATE
The specific reaction rate, which is also known as the rate constant, is the value of the constant k appearing in equation (1). It is numerically equal to the reaction rate when the reactants are present a t unit concentration. In general its units depend upon those employed for the concentration; thus, if moles per liter are used, thespecific rate has the units (molesperliter)/sec. (moles per liter)", where n = or 3! . . . is the over-all order; this reduces to moles1" litersn-' see.-'. For a second-order reaction, for which n = 2, this becomes liters moles-' set.?, while for a reaction for the 3/2
+ +
order the units would be liters1/' moles-'/' sec.3. For a reaction of the first order (n = 1)the units are sec.-1, i. e., the concentration units are irrelevant. This resnlt has led to the propo~al'~ that it is better to drop the use of concentrations for expressing the rates of such reactions, and to use the number of moles or of molecules instead. According to this suggestion the rate of a &st-order reaction should he expressed as - dN/dt, whereNis the number of molecules of reacting substance present at any time. The rate equation would then be
which can, of course, be derived at once from the more usual equation
by multiplying -~ - each side by the volume. FIRST-ORDER REACTIONS AND RADIOACTIVE CHANGES
Although equation (5) is a mathematically correct formulation of the rate of a first-order reaction, its use is open to serious objection. In the first place, it suggests that first-order reactions, and by implication all reactions of integral order, have a different status from reactions of noniutegral order; in actual fact, as will be shown later in connection with the molecularity of a reaction, first-order kinetics arise as the limiting case of a more general law, covering the behavior of certain reactions, which gives rise to a-gradation of order from first to second. A further ebjection to this suggestion stems from one of the very arguments used in favor of the proposal, namely, the analogy with radioactive transformations; this will now be considered. Since radioactive processes are usually treated in terms of the law expressed by equation (5), the suggestion was made that the same fo~mulationshould be applied to chemical reactions of the first order. However, the analogy between the two processes should not be pressed beyond the statement that both follow equations (5) and (6); the mechanisms are entirely different. The number of molecules decomposing per second in a radioactive process involving a given amount of m e terial is the same whether the substance is densely ~ a c k e dor is in a hiahlv " diffuse form where the molecules are at enormous distances from one another. The same is not, however, true of chemical reactions, which depend for activation upon transfer of energy in intermolecular collisions. In a first-order reaction the number of molecules of gas which decompose per second, i. e., the "rate" according to equation (5), may remain constant up to a certain point as the given amount of gas is expanded, but sooner or later it will begin to fall off, the number of collisions being insufficient to maintain the required supply of active molecules. A radioactive change in a given atom, on the other hand, is a process which does not depend upon the presence of other atoms. The apparent analogy he-
-
JULY, 1948
385
tween the two types of change, therefore, offers no argument in favor of employing similar units in expressing the rates. THEABSOLUTEVALUEOFTHEFSTEOFREACTION
One point that may sometimes be a source of confusion t o students of chemical kinet.ics is that even when one set of concentration units is used, the absolute values of the rate of reaction and of the specific rate may depend upon the substance of which the formation or disappearance is under consideration. Thus, consider the third-order gas reaction between nitric oxide and bromine, for which the stoichiometric equation is The rate of this reaction may be expressed in three different ways as follows:
reaction is directly deduced from the experimental results, the molecularity may only be determined on the basis of additional arguments about which there is sometimes some uncertainty. The case of unimolecular gas reactions will be considered first. According t o Lindemann's theorys, which is generally accepted, the activated complex which gives rise to the products is a single reactant molecule which has become excited by colliding with another molecnle; the formation of the complex A* from two molerules of the reactant A may therefore be represented hy A+A*A*+A
(13)
while its decomposition to give products R and C may be indicated hy
(1) The rate of disappearance of nitric oxide, If c, is the concent.rat,ionof A molecules, the rat.e of formation of active molecules by collision is given by '-dc~o/dt. (2) The rate of disappearance of bronline, -dc~,,/dt. klc:, where kl is the specific rate of the process. The (3) The rate of formation of nitrosyl bromide, complex A* may be deactivated by collision wit,h A (i. e., reverse of react,ion (13)), the rate of which is ~ C N O B ~ / ~ ~ . k-lcA*c., where k-I is the corresponding specific rate. The kinetic results, which agree with the stoichiometric The rate of formation of products, by (14), is equal to equation in this case, show that the disappearance of kzc,*, where k, is the specific rate for this process. The one molecule of bromine corresponds t o that of two net rate of increase of c,* is therefore given by molecules of nitric oxide and to the appearance of two molecules of nitrosyl bromide. The three expressions for the rate of reaction are therefore not equivalent, but are related by Since c,* is small its rate of change is small, and to a good approximation may be set equal to zero; henre, k,e:
The reaction rates are related to the specific rates as follows: .
- k-,eA*c*
knc** = 0
(16)
so that
The observed rate of reaction, whichis equal to the rate of decomposition of the activated complex A*, is therefore given by
and dclros* -= dt
k~os&ocs~
(11)
where k ~ o k~,,, , and ~ N O Bare~ the specific rates eorresponding to the three different methods of expressing the reaction rates. Comparison of equations (8) to (11) shows that krro = Zkm, = klrom
(12)
In stating an actual or a specific reaction rate it is therefore necessary, at least in ambiguous cases, t o state not only the concentration units but also the substance to which the rate refers. THE MOLECULARITY OF A REACTION
The molecularity of a chemical reaction is best defmed as the number of reactant molecules which come together to form a collision complex, commonly known as an "activated complex," which directly gives rise t o the products of the reaction. Whereas the order of a
which is the general equation for a reaction of ord& between 1 and 2. At sufficiently-high pressures, when c, is large, the rate of deactivation k-,c,c,* is large in comparison with the rate of decomposition k2cA*; the constant kz may then be neglected in comparison with k-,c, and equation (18) becomes
which is the equation for a reaction of the first order. At low pressures, on the other hand, keBk-1cA, and equation (18) reduces to
so that the reaction becomes of the second order.
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It is thus a characteristic of unimolecular reactions that they should be of the second order at low pressures and the &st order at high pressures. Smce reactions of higher molecularity do not have this property it is reasonable to classify as unimolecular those reactions which show this change of order, and a few such reactions are known. It is evident that the existence of this gradation of order constitutes an overwhelming objection to Antonoff's proposal that diierent units should be employed for the two types of reactions. His formulation would require either a sudden transition from one set of units to the other, or the use of a function which becomes a concentration a t low pressures and is dimensionless a t higher ones; the first alternative is arbitrary and the second exceedingly clumsy. The use of concentration units in all cases leads to no such difficulties. When a reaction is biiolecular, and neither reactant is present in large excess9 the order is always the second. For a reaction involving one type of reacting molecule A, the activated complex is now the double molecule A?,
and this decomposes into the products by
Using the specific rates indicated, the rate of increase of A? is
and if this is set equal to zero, it follows that
The rate of reaction is therefore
. . and since c, does not occur in the denominator this represents second-order kinetics at all concentrations. Agreement between the molecularity and the over-all order is also obtained for elementary reactions involving two or more different kinds of molecules. and for reactions of higher molecularity. A discrepancy between molecularity and order therefore only exists in the case of unimolecular reactions. After the over-all order of an elementary reaction has been determined over a range of concentrations, conelusions may be drawn with regard to its molecularity on the basis of the following rules, which are valid provided that the substances reacting are present in similar amounts: (1) If the reaction has an order of one or more with respect to two or more different species the molecularity is the same as the over-all order, all the reactants (including catalysts) being included. 9
The significance of this condition was considered earlier.
(2) If the molecule h a s an order with respect to only one species the situation is as follows: (a) 'If the order is the first under any circumstances the reaction is unimolecular. In such a case the order will necessarily become the second at sufficiently low pressures. (b) If it is of the second order at all concentrations investigated there remains some doubt about the molecularity, since there is always a possibility that a t higher pressures than those studied the order would fall to unity. The difficulty may sometimes, however, be resolved on the basis of further theoretical arguments. (c) Examples of higher order for reactions of this t,ype do not appear to be known, and the case will therefore not be discussed. The above remarks refer to simple or "elementary" reactions; if a reaction is complex, i. e., it involves a number of successive processes or stages, the couclusions can apply only to the elementary processes taking place. The situation regarding the over-all order of such reactions will now be discussed. COMPLEX REACTIONS
One of the most significant results that has emerged from a kinetic study of chemical reactions is that only a very small number of the processes that have been investigated take plake by a simple rearrangement of the atoms in the reacting molecules with the formation of the final products. Even such apparently simple reactions as the combination of Hz and Oz to' form water, and of Ha and Brz to give HBr are actually very complex. In fact, about t&. only gas reactions now believed to be simpkare *he reactions between hydrogen and iodine, the decomposition of HI, and certain isomerizations. Simple reactions occurring on surfaces and in solution may, however, be more common. If one of the stages of a complex reaction is much slower than any of the others4his slow process, often referred to as the "rate-determining" stage, determines the observed reaction kinetics. The over-all order obtained in the usual manner may then be a simple integer, as in the reaction between nitric oxide and hydrogen. The slow, rate-determining process is apparently the third-order reaction 2N0 H* = Nz Hx02 (26)
+
+
f&owed by
+ Ha = 2H20
H20*
(27)
which takes place rapidly. ~h~ value of the molecularity derived from the considerations developed in the preceding section is then the molecularity of the slow step in the nitric oxidestage, The rate-dete-ning hydrogen reaction is thus termolecular, although the complete process involves four molecules, viz., 2NO
+ 2Hn = N1 + 2H20
(28)
when the successive stages involve reactions which do not take place a t markedly different rates, the complexity of the process is sometimes, but b~ no means
JULY, 1948
always, revealed by a complexity in the kinetic laws. Thus the rate of formation of hydrogen bromine from hydrogen and bromine does not obey the simple law
dcas. = kc~cs, dt
(29)
although the analogous law is followed by the hydrogeniodine reaction, but is represented by the expression
mechanism but still show simple kinetics; the decomposition of ethane, for example, follows first-order kinetics, but certainly involves a number of elementary processes. The detailed mechanisms of organic reactions of this kind have not been elucidated, but Rice and Herzfeldl1 have proposed schemes which account for the observed kinetic behavior in a number of cases. Thus, the first-order law for the decomposition of ethane can be interpreted in terms of the simplified scheme
where k and k' are.constants at a given temperature. This result was interpreted10 by postulating that the following elementary processes are involved: . Br2 = 2Br Br + H z = HBr H Bra = HBr H HBr = HI
+ +
+H + Br + Br
(31) (32)
H
Application of the usual steady-state treatment, as applied to the concentration of hydrogen atoms and of the free methyl radicals, gives for the rate
The rate law corresponding t o this scheme is deduced on the assumption, which has been shown t o be justified, which corresponds to a reaction of the first order. that the concentrations of hydrogen atoms and of However, it is certain that reactions other than those bromine atoms are constant, i. e., that dc,/dt = 0, and mentioned are involved in the decomposition;. in pardc../dt = 0. If expressions for thse two rates are ticular, it appears that a significant fraction of the written down and equated to zero, the "steady-state" ethane molecules decompose by the direct reaction concentrations of H and Br can be calculated, and hence C2Hs = C;14 HB (42) the over-all rate, which is found t o be of the form of The elucidation of the detailed mechanisms of complex equation (30), can be derived. Further justification for reactions plays an important part in kinetic studies at this scheme of reactions has been given by the results the present day. of investigations on the rates of all of the individual processes postulated. This reaction between hydrogen SUMMARY and bromine is of considerable importance in that it is The elementary treatment of chemical reactions is the only complex one t o which a quantitative theory reviewed with special reference to (1) the rate, and the has been applied successfully. units in which is expressed, (2) the order, (3) the specific When only one substance enters into reaction, as in a rate and its &its, and (4) the molecularity. The reladecomposition, the process may proceed by a complex tionship between order and molecularity is discussed, J.EA., CEEIS~AN~ N , Kongel. Danske Videidenskab. Selskab. and the criteria for determining molecularity in terms 2. Elekt., of order are defined. The kinetic laws for complex Math.-fysi. Medd., 1, (14) (1919); K. F. HERZPELD, Z. reactions are discussed brieflv. 25, 301 (1919); Ann. Phypik., 59, 635 (1919); M. POLANYI, " . "Textbook of Physical Elekt., 26, 50 (1920); cf. 5. GLASSTOIPE, " RICE,F. 0..AND K. F. HERZFELD,?. A m r . Chem. Soc., 56, Chemistry," 2nd ed., D. Van Nostrand C%, New York, 1946,
+
p. 1079.
284 (1934).
LITTLE ION "Little Ion in myflask Do you mind much if I ask What your name is, Little Ion, Can't you see you'ue got me cryin'? "Can't you see I'm growing weaker As you hide there in my beaker Ain't you got no heart at d l ? Don't you care if Iflunk Qual? You could stop my endless tryin' Tojind your name out, Little Ion. You could end all my confusion If you'd come out of solution." -Jerry Wellins Connecticut College Pharmacy, New Hauen. Connecticut
