Hydrogen-Ion Dependence of Reaction Rates K. S. Gupta a n d Y. K. Gupta' University of Rajasthan, Jaipur, India
The study of reaction rates is one of the most powerful tools available for interpreting reaction mechanism. The information obtained from kinetics experiments sheds light on the transition state, intermediates, and elementary steps, and is usually complementary to results obtained from other techniques. There are several reviews (I)on the mechanistic aspects of the reactions, and the more recent hooks (2,3) give a set of "rules," or rather, "clues," useful in formulating the mechanism. The study of the dependence of rate of the reaction on hydropen-ion concentration is of snecial sienificance because m o g of the reactants in aqueous media are involved directly or indirectlv in eauilihria which are influenced in some way by hydrogen-ion Eoncentration. A discussion of the formulation of the mechanism on the basis of hydrogen-ion dependence may he found in almost all the reviews and hooks on kinetics and mechanism, hut without comprehensive treatment. Wilkins' treatment of kinetics and mechanism (4) is worth mentioning since it complements the present article. The hydrogen-ion dependence of rates in chemical reactions could arise on account of the involvement of one or more of the following types of equilibria: HA + Hz0 + H30+ + A-
(1)
The protonation or deprotonation equilihria may also involve intermediate soecies, such as shown below, where Red is the reducing subslance. One such case is the oxidation by neptunium(VI1) (5). Red + Np(VII),9Intermediate K.
k
Intermediate + H++Intermediate H+ +Products
(4)
T h e other example is provided by the oxidation (6)of Fez+ with PuO;+.
2 PuOa .Fe4+ PuOz .FeOH3f + Hi If two or more forms of a reactant differing in the number of protons, also differ in the reactivity, which is usually the case, the rate of reaction changes with the change of hydrogen-ion concentration. There could he a numher of situations of the hydrogen-ion dependence depending on the nature of species. However, these can he grouped in suitable types and for each type of hydrogen-ion dependence we give the type of plot expected between a suitable function of the observed rate and a suitable function of [H+], a specific example showing
' Corresponding author. 972
Journal of Chemical Education
PLOTS OF RATES VS [H+] OR [H+]-l that kind of hydrogen-ion dependence, and a few references to other, similar situations. In most cases we do not have higher kinetic order in [Hf], hut these can be treated in a manner similar to that for those eiven. In the discussion that follows, the terms protonated and deprotonated species also mean more protonated and less ~rotonatedsnecies differine by one proton.
-
-
Rate Increases with Increase 01 Hydrogen-Ion Concentration The different situations related to increase in rate with the increase of [H+] are discussed below. The rate increases with the increase of [H+],and the plot of rate against [H+] is a straight line passing through the origin (Fig., part A). This is the simplest case of hydrogen-ion dependence and conforms to the rate law hoba = k [H+l (6) This indicates that eauilihrium between the orotonated and deprotnnated forms of a reactant prior to the rate-determining steo is raoid. that the eauilibrium constant for nrotonation is small and is not compiete even a t high aciditiks, and that only the protonated form is reactive. Although the foregoing
explanation appears to be more attractive, but the following mechanistic path would also give the same rate law. However. a distinction between the two mechanisms--one involving the protolytic equilihrium, and eqn. (7)-can be made on the basis of independent evidence for the ~rotolvtic equilibrium or some physical and chemical properties ofthe protonated or deprotonated species. The reduction of nitrous acid by azidopentaaquochrdmium(II1) ions (7) obeys the experimental rate law -d[HNOz]ldt
= kobs[(Hz0)5.CrNat][H+] [HNOz]
(8)
This is in agreement with the following mechanism K
HNOz + H+ +HzNO:
(9)
Similar rate laws have been found for the reduction of nitrous acid hv hvdrazoic acid (8).hv azidooenmau~m~balttIII) . (9). . .. and h i h~droxylammo~i& ibn (lo( A number of other cases have also been listed (11). The plot of rate versus [H+] yields a straight line with non-zero intercept (Fig., part B). This kind of plot would result from the experimental rate law. koba = kl+ kz[H+]
From this m e of hydrogen-ion dewndence, it can be inferred that there & a rapid pri-equilihri&n between the protonated and deprotonated forms, that the value of the protonation equilihrium constant is great enough that at higher acidities protonation is almost complete, leading to the limiting rate. and that only the protonated form is reactive. A plot of (rntej-' versus IH+l-1 gives a straight line. From the dope and the intercept;theraG constant h a n d protonation equilibrium constant K can be calculated. The oxidation (14) of hypophosphorous acid by chlorothallium(II1) complexes obeys the rate law, eqn. (la), which is of the same form as eqn. (17). This is in accordance with the following mechanism:
The rate increases with the increase of [H+] tending to attain the limiting values a t high acidities, and the plot of rate versus [H+] is a curve of decreasing slope and not passing through the origin (Fig.,part D). For such a case the general rate law
(13)
This indicates that a pre-eauilibrium between protonated and deprotonated f o r m nf a rektnnt is rapid, that ;he protonation equilibrium c o n s m t is small, that hoth the forms we reactive, i d that the protonated form is more reactive. In the aquation of [(HzO)&rFI2+ (12) a rate law of the type of eqn. (13) was obtained and the reaction is believed t o occur through the following mechanism:
holds. This situation has all the features of the previous case (Fig., part C ) and in addition the deprotonated form is also reactive. Thus the rate law has two terms. Acid dissociation of tris-bipyridyl iron(I1) complex (15) into pyridyl and bispyridyl iron(I1) complex with the formation of an intermediate occurs by two paths, one of which involves a proton.
The reaction is more complicated than required by the situation, but a steady-state treatment for the intermediate would lead t o the rate law
The reactions between hyponitrous acid and nitrous acid or hydrogen peroxide or alkyl hydroperoxides are other examples of this type (13). The rate increases with the increase of [H+] tending to attain limiting values a t high acidities, and the plot of rate versus [H+] is a curve of decreasing slope (convex to the rate axis) and passing through the origin (Fig., part C). This situation of hydrogen-ion dependence would conform t o the rate law
and the ohserved first-order rate constant would he given by
A plot of ko versus [H+] would yield the curve shown in Figure 4. The rate expression can be simplified under limiting conditions of [H+]. Thus, when [H+] tends to a value of zero,
ko = klkd(k2 + kd
(24)
and when ka[H+] >> (kz + ks), ko = k~ Volume 61 Number 11 November 1964
973
A general equation conforming to all the ahove-mentioned situations is given by .
te(I1) This is con. . and hexacvanoferrate(II1). . . resnectivelv. . sistent with the following mechanism, and with H ~ P ~ and OR HFe(CN)$- as the reactive species.
.
where k , and k2 are the rate constants for the protonated and denrotmated soecie*. resnectivelv. and K is the de~rotonation co&tant. The four cases discussed above follow from eqn. (25) under different limiting conditions. (1) If K >> [H+]and the protonated form only is reactive, ko = k~lH+]lK (2) If K >> [H+]and both forms are reactive, ko = (kJH+]IK) + kz (3) If K = [H+]and the ~rotonatedform only is reactive,
+
ko = kl[H+]l(K [H+]) (4) If K
[H+]and both the forms are reactive, eqn. (25)will hold. Further, if protonated form is more reactive, a curve of part D of the figure would be obtained by a plot of ko versus [H+].
A plot of rate versus [H+] may yielda curve concave to the rate axis and as sine throueh the oriein (Fie.. . ....Dart E). This indicates that ihe order in I+] is greater than one and, hence. a lot of ko versus IH7I2 should also be constructed. and that there is more than'ondreactive, protonated species, and the roto on at ion constants are small. Generallv in such cases mono- and diprotonated species are involvid. Oxidation of arsenious acid with acid chromate (16) follows the rate law
In the presence of added hexacyanoferrate(III), the rate law simplifies to
-
-d[HCrOJldt
+
= [As(OH)~][HC~OJ(~I[H+] kz[H+Iz)
(26)
If a plot of (rate/[H+]) versus [H+] is made, a straight line with phsitive slope and non-zero intercept would be obtained. The values of k l and kn can be calculated from the intercept and the slope, respectively. A plot of rate versus [H+] may give a curve concave to the mte axis and making an intercept on the rate axis (Fig., part F). This indicates that there are more than one protonated species and all the species are reactive and that the protonation constants are small. Oxidation of Hz02 with iodate ions (I7,18) follows the rate law
A graphical solution of the rate law can be obtained by plotting (rate) versus [H+] and extrapolating the resulting curve to zero [Hf]. The intercept on the rate axis would yield the value of kl. There should be several data points in the lower [H+] range so that not much error is involved in making the extrapolation. Having determined k, a plot of (k,b. kl)l[H+] versus [H+] can be drawn, where hob, is the experimental second order rate constant. The intercept and slope of this plot would give the values of kz and k3. A plot of rate versus [H+] may yield a sigmoid curve tending to a limiting rate a t high acidities (Fig., part G ) . Such a situation indicates that there are a t least two orotonated forms of a reactant and that only the more proionated form is reactive. The nrotonation constant for the less nrotonated species is large, but that for the more protonatedone is neither large nor small. The oxidation of hexacyanoferrate(I1) with peroxodiphospbate (PDP) in the presence of Cu(EDTA)=- complex (19)has a complicated rate law d[Fe(III)] dt
A variation of [H+] in the presence of Fe(1II) and a plot of rate versus [H+]would yield a curve similar to that of part G of the figure. Rate Decreases wlth Increase of Hydrogen-Ion Concentration
This situation is commonly met in redox reactions involving highly charged cations and the following four types have been noticed. In the simplest case the rate has inverse first-order dependence on [H+], and a plot of rate versus [H+]-1 yields a straight line with zero intercept Fig., part H ) . In such cases rate law ko = kl[H+]
is obeyed. This indicates a protolytic equilibrium with small equilibrium constant and deprotonated species being reactive. Reaction of thallium(II1) with hydrazine in aqueous perchloric acid solution (20) is a redox reaction that occurs by any of three mechanisms. The rate law
holds in all three cases. Mechanism 1
TF+ + N Z H ~
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Journal of Chemical Education
K
++ TINZH:+
TI. N2Hif
(K is small)
Products
Mechanism 2
TIOH2++ NzH: (28) where Fe(I1) and Fe(II1) have been used for hexacyanoferra-
(30)
K +
T1. N2H$++ Hz0 (K is small)
neither small nor large. If Mn+is the metal ion and SH is the reducing substance, the mechanism is:
Mechanism 3 K
+
+
TB+ N2Hd
TI. N2Hi'
+ H+
(K is small)
MOH("-')+ kl of the rate expression, eqn. (311, would be equal to kKKd, i kK&, and kK in each of the above cases, respectively. S hydrogen-ion dependence has been observed in the oxidation of mercurous chloride (21) and arsenious acid (22) with thallium(II1). A olot of rate versus IHfl-I mav vield a straight line with non-zero intercept (Pig.,part I). A large number of metal-ion redox reactions follow this tvoe of hehavior which sienals that there are two terms in the ra'te law, one being independent of hydrogen-ion concentration.
. .
--
-
This means that the two rate-controlling steps are preceeded by a rapid deprotonation equilibrium for which the equilibrium constant is small and both the forms, protonated and deprotonated, are reactive. The reaction between vanadium(II1) and neptunium(V1) has the stoichiometry (23) V(II1) + 2Np(VI)
-
V(V) + 2Np(V)
(33)
and the rate law is
+
+-=
(kl kzKd[H+])[Np(VI)][V(In)] (34) dt Np(V1) exists in its hydrolyzed form NpOf', but V(II1) is partially hydrolyzed, and both the species of vanadium(II1) are reactive. The two paths corresponding to the above rate law are:
k + SH +Products
The corresponding rate law is -d[Mlldt = kKh[M][SH]I([HC]+Kh)
(35)
However, if the metal ion forms a complex with the substrate and the complex decomposes to yield the products, the following mechanism would operate:
From this mechanism, the rate law -d[M]/dt = kK[MI[SH]l([Ht]+K[SH])
(36)
can be derived This rate law has the same form as that of eqn. (35). The values of K or Kh can he calculated from the slope and intercept of the straight . line obtained from the plot of (rate)-1 versus [H+]. Oxidation of methylhydrazine with cerium(1V) is an example of the former situation and has the following stoichiometry (26):
The rate law
Mechanism 1 V3+
k, + NpO:+ -+ Products
is consistent with the following mechanism:
Mechanism 2
Oxidation of H3P03 with thallium(II1) (27) is an example which is a combination of the two equilibria cited above. Thallinm(II1) exists as TI3+, TlOHZ+, and a complex with H3P03, but only T10H2+ is reactive.
A plot of rate versus [H+]-1 would give a straight line with a non-zero intercept, and, from the slope and the intercept, k l and kz can be calculated. The reaction between V(II1) and U(N) (24) and that of Co(II1) and hydrazoic acid (25)have similar rate laws, both the hydrolyzed and unhydrolyzed forms of V(II1) and Co(II1) being reactive. The third case of the decrease of rate by increase of [H+] is represented by aplot of rate versw [H+]-1 in part J o f the figure. In such cases the rate has a limiting value a t low [H+] tending to a zero value at large [H+]. In all such cases a plot of (rate)-' versus [H+] would yield a straight line with a non-zero intercept. Such a situation in general would arise when a particular reactant exists in two or more forms which are in equilibrium involving hydrogen ion (hydrolytic equilibrium, complex formation with proton liberation) and the deprotonated form is reactive. The equilibrium constant is
+
[TKIII)], = [TI3+] [TIOHZ+]+ [TIH2PO:+]
+
+
= [T13+](l Kd[H+] K[H3POdI[H+])
The rate law
can be easily derived from this. This gives the hydrogen-ion dependence as shown in part J of the figure. The reaction between vanadium(II1) and chromium(I1) (28), and the copper(I1)-catalyzed,free-radical chain oxidation (29) of H8P02with peroxydisulfate are other redox processes which conform to the rate lawpf eqn. (36). Volume 61 Number 11 November 1984
975
A plot of rate versus [Htl-' may yidd a curve intercepting . indicates t h e r o l p a ~ ias ~ s h o w nrn Dart K of the f i ~ u r eThis that both forms of the reactant in the p&nation equilihrium are reactive, but the protonated form is less reactive. Also the protonation constant has a value comparable to [H+]. Oxidation of hydroxylamine with manganese(II1) (30) has the stoichiometry,
+
GMn(II1) NH30HC + 2H20
-
GMn(I1) +NO;
+ 8H+
(40)
Concentratton of Hydrogen Ions for the Mlnlrnurn Calculated and Obmrved Values [Hflrnm
Reaction
Em.
Aquation of (H2O)~CrSO. 0.19M Aquation of (H~O)~C~SCH.CH~NH~+0.0095 (NH3)sC002CO(NHs)s + Fe(ll) 0.3 V(V) Fe(li) 0.07
+
[HcImin
-
Cab.
Reference
0.182 M 0.0085 0.3
(33) (31) (361 ( 34
0.06
The rate law is solution (33), oxidation of iron(I1) with V(V) (34),oxidation of iron(I1) with Ce(IV) (35), oxidation of iron(I1) with (NH3)sCo.02-Co(NH)3)s (36), exchange reaction between V(V) and V(II1) (37). etc. Equation (44) can he used for obtaining the condition of maximum or minimum by successive differentiation with respect of hydrogen ion. The first differentiation gives
The corresponding mechanism is
Mn3+ + NH30HC+Products h1
Setting this equal to zero for maximum or minimum Differentiating eqn. (45) again A plot of ko versus [H+]-' would give a curve as shown in part K of the figure, but a plot of ko ([H+] Kh) versus]H+] would give a straight line, and the values of kl and kz can be calculated from the intercept and slope of the straight line. However, knowledge of Kh would he necessary for this procedure. A general rate law covering the ahove four cases of decrease of rate with the increase of [H+], is given by
+
where Kh is the equilihrium constant for the hydrolysis or deprotonation, and kl and kz are the rate constants for the protonated and deprotonated species, respectively. The four situations discussed ahove are the limiting cases of this equation. (1) If [H+]>> Kh and the less protonated species is reactive, ko = k&hl[H+] (2)
Since the second derivative is always positive, a minimum only would be ohtained, and hence [H+lmi. = dmii
(48)
In a few cases, the hydrogen-ion concentration correspondingto the minimum has been calculated from k l and ks by this procedure, and such values agree remarkably well with the experimental values (see table). Apart from what has been described above, such type of hvdroeen-ion deuendence will also be noted in cases where (1) there are more than two species governed by protolytic equilibria. (2)all of them are Dresent in com~arahleconcentration, (3) ali bf them are reactiGe, and (4) themiddle species is least reactive. In the oxidation of iron (11) with Ce(1V) (35), three species of the oxidant, Ce4+, CeOH3+, and Ce(OH)i+, are present in comparahle concentrations and all of them are reactive as shown below: - ~ d
~~
~-
If [H*] >> Kh and both forms are reactive, ko = k l + k&d[HC]
(3) If [H+]a Kh and the less protonated form is reactive, ko = k&d([H+I
+ Kh)
k~
Ce4++ Fe(I1)+Products
(4) If [HC]a Kb and both formsare reactive, rate eqn. (43) will hold. Further, if the less protonated form is more reactive, the curve
of part K of the figure will be obtained by a plot of rate versus [HC]-'. Rate Decreases, Passes through a Minimum, and then Increases with Increase of [H+] For this category of hydrogen-ion dependence, a plot of rate versus [H+] gives a curve as shown in part L of the figure, and the simplest rate law conforming to this situation is ko =
kl -+ [H+1
kz + ka[H+]
(44)
This,in essence, means that the involvement of two equilibria with small equilihrium constants for the protonation and deprotonation and that all the three species are reactive. There are several systems showing this behavior, e.g., aquation of pentaaquo(2-mercapto-ethylammonium-5)chromium(II1) ion (31),oxidation of iodide with V(V)-HzOz system (32), aquation of sulfatopentaaquochromium(III) ion in acidic 976
Journal of Chemical Education
The overall rate is first order each in Ce(IV) and Fe(I1) and the apparent second-order rate constant is given by,
The rate constants (M-' sec-' a t 25") are K1 = 5865, Kz = 1000, and Kg = 4830, and hence the variation of rate with the variation in [H+] as shown in part L of the figure is ohtained. Rate Increases, Attains a Maximum, and then Decreases with the increase of [H+] A plot of rate versus [H+]yields a curve as shown in part M of the figure. Such a situation arises if a system of three or
more species of a reactant with varying degree of protonation is governed bv nrotolvtic eauilibria and the middle specie3 is reactive, andthe m&imum occurs where the relatiie population of the most reactive species is largest. The apparent' second-order rate constant in such a case is given by
where K1, Kz, and K3 are some constants related to the protonation constants. The maximum would be obtained a t [H+] = K1/K3. The oxidation of Fe(CN)d- with H4P208 in the presence of Cu(I1) EDTAZ- (19) shows this behavior under certain conditions. The mechanism and the rate law have already been described (eqn. (28)). When no Fe(CN)g-is added or in the initial stages of the reaction, kz[Fe(III)]
