The Rate Laws tor Reversible Reactions
Downloaded via UNIV OF READING on July 30, 2018 at 10:49:49 (UTC). See https://pubs.acs.org/sharingguidelines for options on how to legitimately share published articles.
Edward L. King University of Colorado, Boulder, CO 80309
In discussion of rate laws for reversible reactions and the relationship of the concentration dependences of the rates of forward and reverse reactions, there are advantages in employing real chemistry rather than A-plus-B-equals-C-plusD chemistry. The disproportionation reactions of nitrous acid and nitrogen dioxide are ideal examples with which to illustrate some aspects of this subject. The potential diagram (Latimer diagram) for nitrogen (oxidation states 2+ through 5+) for acidic solution (1): 0.80 V
_
K,% IH^NOT “no
HONO
3
=
2H++
2
N03“ + NO(g)
H+ + N03“ +
2
(1) (2)
NO(g)
measurable equilibrium under realizable concentration conditions. Reaction (1), a step in the production of nitric acid, is the predominant reaction when nitrogen dioxide is added to nitric acid; reaction (2) is the predominant decomposition reaction for nitrous acid in dilute acidic solution. Kinetic studies of each of these reactions are consistent with the same reaction steps, a rapid equilibrium (reaction (3)) and a slow step (reaction (4)): proceeds to
a
k3 2
HONO
=
N02(g) + NO(g) + H20
(3)
k4
2
N02(g) + H20
H+ + N03“ + HONO
^-4
(4)
Reaction (1) is reaction (3) plus two times reaction (4), and reaction (2) is two times reaction (3) plus reaction (4). In reaction (1) the unstable intermediate is HONO, and in reaction (2) the unstable intermediate is NO2. The empirical rate laws (with activity coefficient factors omitted) (2, 3) that suggest this mechanism are -
^
7T
d[NQ3~] —
*ir[H'][N03
[HONO]4
df
P -*NO
2
(5)
IRno'^-Pno,172
(6)
fc2r[H+][NO,fl[HON01
with ku 1/2 k4, klr 1/2 k-4K3~112, kot K32k4, and k3i k~4. The important feature of these rate laws to be noted at this point is that equating to zero the rate given by eqn. (5) yields the square root of the equilibrium constant for reaction (1) and equating to zero the rate given by eqn. (6) yields the equilibrium constant for reaction (2): =
1/2
yields
ktPm2
=
-
=
1/2 A_4K3_lffi[H+][N03“]Pno1/2Rno,1/2
_
fe_4[H+][NOg_] [HONO]
[HONO]3
reveals that each of the disproportionation reactions =
K™
and
[H+][N03~]Pno2
0.93 V
N02(g) + H20
=
NO,
1.05 V
3
K31/2
3/2
-
0
yields
M6vHoNoMivNO(g)
,
kt
[H+][N03-]Pnos/2
=
k4
_
fe-4
g3
2
Thus reaction (1) provides an example of a rate law for a reversible reaction that upon being equated to zero yields not the conventional equilibrium constant equation, that involving smallest integral exponents, but rather the square root of the conventional equilibrium constant equation. On the other hand, the rate law for reaction (2), eqn. (6), yields the conventional equilibrium constant. The key to the difference between these two situations is the factor by which the equation for the slow step, reaction (4), must be multiplied in the summation of steps that is the net change. In the summation to obtain reaction (1), eqn. (4) is multiplied by two-, in the summation to obtain reaction (2), eqn. (4) is multiplied by one. That is, the stoichiometric number1 for the slow step is two for reaction (1) and is one for reaction (2). As shown in this example, the rate law for a reversible reaction involves rate constants for the forward and reverse of the slow step each raised to the first power. The rate law may involve ratios of additional rate constants, and these ratios may be raised to powers other than one. In this example, we see K3 (a ratio of rate constants) raised to the onehalf and the second powers. The equilibrium constant expression derived by equating the rate to zero involves the first power of the quotient of rate constants for the forward and reverse of the slow step. And a corollary of this is that the balanced chemical equation that goes with the rate law for a reversible reaction is that which is obtained as a sum of reaction steps including one times the rate-determining step. If to obtain the conventional chemical equation for the reaction, that with smallest integral coefficients, the equation for the slow step must be multiplied by a factor other than one, the equilibrium constant equation based upon the rate law will be the conventional one raised to a power other than unity; it is 1/2 in the present, example. This point has been made by Horiuti and Nakamura (4) who give as the relationship between the equilibrium constant and empirical rate constants for forward and reverse reactions: k{ 7-
=
K'u'
(7)
The term stoichiometric number is used here as in Horiuti and Nakamura (4). The stoichiometric number for a reaction step depends, therefore, upon coefficients in the overall reaction, as will be illustrated. 1
=
0
Volume 63
Number
1
January 1986
21
in which v is the stoichiometric number for the rate-deterK1/2, and for mining step. Thus for reaction (1), kf/k, reaction (2), kf/k, K. One can make the stoichiometric number for the slow step equal to one simply by adjusting the coefficients in the equation for the overall reaction. If reaction (1) is multiplied by one-half, the stoichiometric number for the slow step becomes one. This discussion is, of course, background for the question of predicting concentration dependences of a reverse rate law given the concentration dependences of a forward rate law. Clearly the mechanism must be known, in particular the stoichiometric number of the slow step, to make a certain prediction. In the absence of definitive data, the cautious prediction probably is that based upon the assumption that the stoichiometric number of the slow step is one. Other than the example before us there are few systems for which the cautious prediction would fail. Another point related to the ambiguity in predicting the form of the reverse rate law from the form of the forward rate law can be made with reference to the oxidation of iron(II) by mercury(II), which is the example used in textbooks by Levine (5) and Adamson (6): =
=
2 Hga+ + 2 Fe2+
=
Hg22+ + 2 Fe3+
(8)
a distance from equilibrium the forward reaction is first order in each reactant,
At
o)
=MH&2+1[Fe2+]
Two mechanisms consistent with rate law (9) the reactions
are
made up of
K Hg2+ + Fe2+
Hg+ + Fe3+
4=
k-l
(10)
2Hg+^Hg22+ Hg+ + Fe2+
^
Hg° + Hg2+
(11)
Hg° + Fe3+
(12)
^
(13)
Hg22+
with reaction (10)
a slow reaction and the other reactions ((11), (12), and (13)) postulated to be rapid equilibria. Two times reaction (10) plus reaction (11) is equal to reaction (8); this will be called mechanism A. Reaction (10) plus reaction (12) plus reaction (13) is equal to reaction (8); this will be called mechanism B. Thus for mechanism A, the stoichiometric number of the slow reaction is two; for mechanism B, it is one. For mechanism A, kf (defined by equation (9)) is equal to 1/2 k\; for mechanism B, k[ (defined by eqn. (9)) is equal to k\. The form of the reverse rate law, which would distinguish mechanisms A and B, has not been established, and it will be difficult to study the reverse reaction at a distance from equilibrium since the equilibrium constant for the reaction as written (eqn. (8)) is large (K 105 L mol-1 at 25°C). Perhaps the approach to equilibrium from small displacements can be studied, and, if so, you might expect that such a study would resolve the question. The rate laws associated with the two mechanisms just discussed are:
ing the equations for the relaxation time. The dependences of the relaxation time for the establishment of equilibrium upon equilibrium concentrations, [Fe2+]e, [Hg2+]e, etc., can he written as the product of the rate of the forward reaction and the sum of the reciprocals of the concentrations of reactant and product species of the reaction, each reciprocal concentration being multiplied by the square of the coefficient of the species in the balanced chemical equation (7). Thus the functional dependence relating relaxation time and concentrations is the same, whichever of these rate laws is correct =
MFe2+]e[Hg2+
=
&1[Fe2+][Hg2+]
-
fe,[Fe3+][Hg22+r
(14)
Mechanism B 1 |
2
d[Fe3+] dr
=
MFe2+][Hg2+]-V
[Fe3+]2[Hg22+]
(15)
[Fe2+][Hg2+]
The balanced chemical equation for the overall reaction that goes with rate law (14) is one-half of eqn, (8) and that which goes with rate law (15) is eqn. (8). This is relevant in obtain22
Journal of Chemical Education
+ [Fe3+]E
4[Hg; &22+]e) (16)
T_1
4 =
^i[Fe2+lt[Hg2+]e
+._i_ +
+._4,
[Fe2+]c
[Hg2+]f
[Fe3+]e
[Hg2 h2+h) (17)
for rate law (15). If only the rate of approach to equilibrium from small displacements was studied, eqns. (16) and (17) could not be distinguished. But with the rate constant kf (eqn. (9)) established, they can be distinguished; if mechanism A is correct, the value of k\ derived from the relaxation time by use of eqn. (16) will be two times the value of kf derived from application of eqn. (9) to data for the forward reaction; if mechanism B is correct, the value of kf derived from the relaxation time by use of eqn. (17) will be equal to the value of kf (eqn. (9)). This example brings out points not made in the paper (7) in which the general equation for relaxation time was developed: 1) Study of the approach to equilibrium yields a value of the relaxation time, but 2) this yields a value of Vf,r, the reaction velocity at equilibrium, only if one knows which of the possible balanced chemical equations goes with the complete rate law, i.e. the conventional balanced equation or some multiple or fraction thereof, and, 3) the observed dependence of relaxation time upon concentration will not resolve the ambiguity in settling the form of a reverse rate law given the form of the forward rate law. This is the consequence of the necessary relationships between concentrations at equilibrium. For this example [Fe2+]e[Hg2+]e =
[Fe2+]e[Hg2+]e (ffs-HHg^+letFe^le2)1'2 tfs-MHga^letFe^V/aFe^WHg^U. or
=
Still another facet of this subject can be illustrated by the ligand exchange of tristriphenylphosphine platinum(O) studied by Halpern and Weil (8). The rate of the reversible reaction Pt(PPh3)3 + CH3CCPh
=
Pt(PPh3)2(CH3CCPh) + PPh3 (18)
studied in benzene at 25°C, is governed by the rate law: -
«
-
[API
L
4
(o.050
250°g‘;'|E))»
-1.43 «) (19)
in which A CH3CCPh, C = Pt(PPh3)3, B Pt(PPh3)2(CH3CCPh), D PPh3, and Q [C][D]/([A][B]). This rate law is consistent with concurrent pathways, a direct associative pathway with a second order rate constant for the forward reaction of 0.050 L mol^1s^1 and a dissociative pathway with an intermediate Pt(PPh3)2 which has a formation rate constant from Pt(PPh3)3 of 0.91 s-1 and which, in being competed for by the two ligands, discriminates in favor of triphenylphosphine by factor of 250. The negative term in the rate law corresponds to the reverse reaction, and the numerical factor in this term is K"1 (K 0.70). Halpern and Weil make the interesting observation that a pseudo-first-order study ([A] + [C]) « [B], [D]) under conditions where equilibrium lies far to the right ([B] = 0.30 =
=
=
+
[Hg2+]e
for rate law (14), and
~
Mechanism A
+
+
[Fe22+1
=
=
9861 'Oenuer
zz
j.
jaqiunN
£9 0iun|OA
mnuqtjinba jb saxajduioa omj aqj jo suoijBjjuaauoa aqj jo oijbj IBjnaijjEd b oj spuodsajjoa auq qoBa puB ‘uinuqqmba ‘aq ‘j = yj/b JOJ u^Bip ajB ainSij siqj ui x ado|s jo sauq [BuoSsip aqj, •uinijqqmba oj spaaaoid uotjaBaj SB saSusqa sABMqjed omj aqj jo aauBjioduii aAijBjaj aqj sjuauiuadxa aqj jo qaea joj jBqj SMoqs qaiqM ‘ainSij aqj jo g jiud ui paaBJj aq suoijaeaj omj asaqj jo asjnoa aqj ‘(T_g jora XOO'O) auiBs aqj sjanpojd puB (T_g joui OOE'O) auras aqj sjuBjasai jo suoij -Bjjuaauoa jbijiui aqj qjiM unj ajaM ainSij aqj jo y jjad ui q puB b sjuiod Aq paquasap asoqj aqq sjuauiuadxa omj jj paquasap ApBaip sjuauiuadxa omj aqj Moqs ainSy aqj ui q puB b sjuiod snqj, •qdBjS siqj uo juiod b Aq pajuasaidai si umtiqqinba oj jjbjs uiojj uotjaBaj e jo asjnoa aiijua aqj ‘Ajjeijiui juasaid spuBSq qjoq qjiM ‘saioads puBSq aqj jo suoijBjjuaauoa oj paiBd -iiiod moj .via a uoijBjjuaauoa (o)umuijBjd jbjoj aqj qjiM suoij -ipuoa japio-jsiy-opnasd japun jno paiuoa si uotjaeaj b jj ( ainSij aqj ui a juiod sb pajBuSisap aiB suoijipuoa asaqj Ijx’O X/b PUB ‘6'0I = u ‘i-i iou* 08'0 = [a] pue ‘t_q ioui E_ox x cri = fo] Vi ]Oui ox'O = [s] ‘i-T jora 01'0 = [V] suoijipuoa uoijBjjuaauoa aqj .roq) ‘uotjaeaj pjbmjoj aqj joj jueuiraop aq oj (mbj ojej aqj ui uuaj japjo-puoaas Ajaind aqj) ABMqjBd aAijeiaosse aqj joj paiinbaj aiB suoijipuoa amajjxa (g puE ‘uoijaeai jau aqj jo uoijaajip aqj jo ssajpieSai jtreuiuiop si ABMqjBd aAijeia -ossip aqj g jo suoijBjjuaauoa moj jb (g ‘uoijaBai asiaAaj aqj jo aauajinaao joabj oj spuaj qaiqM uoijipuoa b si siqj (q§iq jsnra aq ‘auiqdsoqdjAuaqduj ‘g jo uoijBjjuaauoa aqj jubu -iraop aq oj AeMqjed aAijBiaossB aqj ioj (x :aJE asaqj, 'jeaja sjuiod uiBjjaa saqeui qdeiS aqj 'ainSy aqj jo y jiad ui q puB b sjuiod sb pajEuSisap aiB jaqiEa paquasap puB (g) ui pauoij -uara suoijipuoa jejuaimiadxa jo sjas omj aqj '[g] [g]093 oj spuodsauoa auq siqj 'juejioduii Ajjenba aiB ABMqjBd aAijBiaossip aqj ioj mbj bjbi aqj ui suuaj jojeuiuiouap omj aqj qaiqM jb g puB g jo suoijBjjuaauoa aAijejaj aqj sauyap
qaiqM jBqj si ‘auq paqsBp aqj ‘x adojs jo auij jaqjouy -jjaj aqj oj saq uiniiqijmba qaiqM ui suoijBnjis oj puodsauoa oi = yj/(y qjiM asoqj -jqSii aqj oj saq umuqijinba ‘suoijipuoa 3utjsixa aqj oj aAijejai ‘qaiqM ui suoijBnjis oj puodsauoa j'0 S0U'I asoHJj 'fol/ivl Pu®XC/b J° S9nIBA JBjnaijiBd X/b oj puodsauoa jojd 3oj-Soj siqj ui x adojs qjiM sauq aqj =
ubo
=
=
j-j ioui aero = *[a] = B[o] iolu 69i ‘o = B[a] = B[v] oisaoeipww i_tiouj os o = °[a] = °[o] ‘[_j i°lu e_oj xoj =°[a] =°[v] (q ^ x iolu bei o = 8[a] =B[o] \-iiowg9ro = |0W S_(H X 0U = °[a] = °[0] ‘i-H ioui OE'O = B[a] = B[V] oisaoBiioiqM °[g] = °[v] (b rumijqiiinbs omoeojdds eq) 0dejj(o puE(Esiu!odie6u!lEU!6iJO sun (-01 (L 'zOl (9 (S iJ (fr :t-0J (£ ^-CH (Z ‘E-01. (l :°([o]/[V]) opej p0)EO!pu! puE(j =>l/0) wmiqnmbo oi spuodspjjoo IB0U|| qoEgJg) = =
'0JI =H ‘x-liorn 08 0 [a] X-1IOLU0J 0 [8] (oq fr =y x_tiouioe o= [a] J-Jiouij.oi x o i [a] (q :oeo’o y j-i iouu E_ot x o'j [a] ‘x_n iolu oe'o [a] (e siuiod =
=
=
=
(IE)
Mx
(f 0i’°)x [al
[Ql
=
:EO]/[v] saioads ranuijejd jo suoijeijuaa aqj jo oijbj aqj puB yj/fi jo anjBA aqj oj saiaads pueSq jo suoijBjjuaauoa aqj Suijejai osjb ubjjijm aq ubo uoijenba U JOJ sauij aqj SMoqs ain3y ny 'O'OI PUB ‘00' T ‘ooro aqj jo y JjBd puB ‘sABMqjBd aAijBiaossip puE aAijeiaosse aqj jo ajBj uoijacai aqj oj suoijnqijjuoa aqj jo oijbj aqj ‘;.j joj -uoa
=
(os)
[g] x-i0™ i ssao + [a] ,-ioni i act
=
y
jo Suijijm aqj smojje gj uoijenbg mbj ajBj siqj jnoqB sjuiod auios uuoj jBoiqdeiS ui sjuasajd ainSij aqx 'aSuEqa [Bairaaqa jau aqj jo uoijaajip aqj sauiuuajap jBqj (Oi'O)}! °J aAijBjaj (y jo anjBA aqj si ji puB ‘asiaAaj puB pjbmjoj qjoq ‘uotjaBaj aqj oj ABMqjBd qaBa jo suoijnqijjuoa aAijBjaj aqj auiuuajap jBqj (£qgg)g puB (HdDO^HOJa saiaads puBSq jo suoijBjjuaauoa jBnjas aqj si ji jsqj isaja saijBui ‘(6j) -uba ut paABjdsip si ji sb ‘mbj ajBj aqj jo Suuojobj aqj, ‘AjijiqisiaAaj aidoas -ojoira jo ajdiauiid aqj qjiM juajsisuooui si jBqj ‘(61) ‘uba ajBi ‘mbj ajajduioa aqj ui Suiqjou ‘asjnoa jo ‘si aiaqj 'Ajijiqi -siaAaj oidoasojoiui jo ajdiauiid aqj qjiM juajsisuoaui sjjnsai 9ai3 oj jBaddB pjnoa suoijaBai asjaAai puB pjbmjoj jo seipnjs ajajduiooui snqj juButuiop ABMqjBd aAijBiaossB aqj saqBiu (I « b eri U-T 1 on 08'0 [a] ‘i-T (OUI e-0T X O'I [a]) jjaj aqj oj jbj saij umuqijinba aiaqM suoijipuoa japun Apnjs japio-jsjy-opnasd b pue 'juBuiraop ABMqjBd aAijBiaossip aqj saqEiu (j » ^ £f I -i-l (Ora e_0I X O'I [d] ‘i-l lOUI =
=
=
JEqioue guo o;|Enb0 0JB me| siej jo uuojpuoops ui sluJ0j jojeuimouop jo ‘[g] [Q]0SE :0U1I psqsEQ 'j. = [0]/[V] PUE0I =X/OJO'00I = [0]/tv] pue p o = X/0 :E auij d'O = [0]/[V] PUB01 =X/0 JO '01 = [0]/[V] Pub 10 =X/D z aun 'to o = [o]/[v] pub oj =x/o jo q = [o]/[v] pub ro =x/o q aun [0]/[v] Pue >I/O 1° en|B/\ lEinopjBd oj spuodsauoo 0g> jb aun qaeg (y) =
'l
—
y 'suojGaj assLji uasMjaq
0U|| |BO!JJaA jsouijs Jj o > y Jjai jb uoiBaj !oi < y JqSij jb uo|60g Aew -qjEd 0aj jEjOossjp Aq uoiJOBjj/ABmqjBd OAgBioossE Aq uoijoejj = y -0U0|Aj0oe
|Aqjsuj|Au0qd qj|« (o)ujnu!jE|d (aujqdsoqd lAusqdpj) uj jo uoijobbj aiqisjdAej aqj joj sAeMqjEd eAjjBpossjp puE aAijEpossE jo aouEjjodui! baijeioj aqj
i-q puj/(a)6oi
W
\
3 r-
Some points that can be made in summarizing this discussion of the rate laws for reversible reactions are: 1) Although prediction of the form of the rate law for a reverse reaction given the rate law for the forward reaction is not certain, the number of possibilities is very limited because of the relationships that have been described. 2) The ambiguity in resolving the point just raised is not settled by study only of the approach to equilibrium from small displacements. 3) Although the function of concentration conditions that determines which of several reaction pathways is dominant is different from the function of concentration conditions that determine the distance of a system from equilibrium, there may be a bias which favors one pathway for the forward reaction and another pathway for the reverse reaction.
24
Journal of Chemical Education
Literature Cited
(1) Wagman, D. D., et al. J. Phys. Chem. Ref. Data, 11, Suppl. 2 (1982). (2) Denbigh, K. G., and Prince, A. J., J. Chem. Soc., 790 (1947). See also Hill, C. G., Jr., “An Introduction to Chemical Engineering Kinetics and Reactor Design,” Wiley, New York, 1977, pp. 137-8. (3) Bray, W. C., Chem. Rev., 10, 161 (1932) (reviews work of Abel, E-, and Schmidt, H., Z. Physik. Chem., 132, 56 (1928)). (4) Horiuti, J., and Nakamura, T., Adu. Catalysis, 17, 47 (1967). (5) Levine, I. N., “Physical Chemistry,” McGraw-Hill Book Co., New York, 1971, pp.
502-503. (6) Adamson, A. W., “A Textbook of Physical Chemistry,” Academic Press, New York, 1973, pp. 715-717. (7) King, E. L., J. CHEM. EDUC., 56, 580 (1979). (8) Halpern, J., and Weil, T. A. Chem. Comm., 631 (1973).
