Branching First-Order Reactions

A compound may be able to react in two different ways. An example from coordination chemistry is

C o ( e n )²( H²0 ) ( X )^{2 +} + X "

Co(en)²(X)(Y)+ + H²0 < ^ (15-80)

C o ( e n )²( H²0 ) ( Y )^{2 +} + Y "

where " e n " denotes the bidentate ligand ethylenediamine and X and Y are halogens or pseudohalogens. If both reactions are first order, then the rate of disappearance of reactant A is

d(A)

dt

= - f o + k²)(A), (15-81) or still first order, but with A:^{a p p} = k^t + k². The rates of formation of the alter­

native products Β and C are

t, hr

Thus the ratio of (B) to (C) at any time must be kjk², and their sum, of course, must be (A)⁰ — (A). If, for example, k^x and k² were 0.1 h r^{- 1} and 0.2 h r^{- 1}, respec­

tively, the plots of concentrations versus time would look as shown in Fig. 15-14.

The products Β and C will have some equilibrium ratio themselves, and this in general will not be the same as kjk². The example of Fig. 15-14 is based on the assumption that the B - C equilibration is slow, so that the product ratio (B)/(C) is kinetically determined. The general situation is

Β

(15-82)

C

If k³ and k_³ are large compared to k^x + k², then Β and C will equilibrate as they are produced, and the observed product ratio will be the equilibrium one.

The distinction between these two situations is an important one if the rate data are to be used in mechanistic interpretations.

Example. If X a n d Y i n Eq. (15-80) are CI" a n d N C S ~ , respectively, it turns o u t that k^x is m u c h larger than k², s o the reaction is exclusively

C o ( e n )²( N C S ) ( C l )⁺ + H^aO — C o ( e n )²( H²0 ) ( N C S )^{2 +} + Cl~. (15-83) The equilibration between the t w o possible products is slow, s o that in this case the product

ratio is kinetically determined. T h e equilibrium also strongly favors the product C o ( e n )2( H20 ) ( N C S )^{2 +}, s o that the rate constants parallel the equilibrium constants K^x and K², that is, k^x k² and K^x ^> K². A linear free energy relationship thus appears to be present.

This need not be the case. Continuing with the same example, the complex C o ( e n )²( N C S ) ( C l )⁺, being octahedral, has cis and trans isomers as shown in the accompanying diagram. The cis

SPECIAL TOPICS, SECTION 2 639

isomer is the thermodynamically favored one, and if allowed to aquate according to Eq. (15-83), the product is entirely c w - C o ( e n )²( H²0 ) ( N C S )^{2 +}. However, if the trans isomer is the starting material, then about 6 0 % cis and 4 0 % trans product results. In this last instance, then, the reaction is definitely subject to s o m e kinetic, stereospecific control. If X = C I- and Y = N 02~ the chloride aquation reaction, which again dominates, is 100 % stereospecific, cis starting material giving cis product only and trans starting material giving trans product only.

The reaction scheme of Eq. (15-82) is the classic one of a triangular reaction system. The mathematics was investigated extensively in the 1920's and for a while it appeared that if an equilibrium system were perturbed by, say, the sudden addition of more A, then the concentrations of A, B, and C could undergo oscilla­

tion. While we now know that indefinite oscillation will not occur, systems of coupled reactions can oscillate, although the oscillations damp out and the system finally approaches ordinary equilibrium. Several cases of so-called clock or periodic reaction systems are known.

β. Sequential First-Order Reactions

The reaction sequence

A ^ B ^ C (15-84)

occurs often enough that its mathematical behavior should be explored. Each reaction is first order or pseudo first order and each goes to completion. The equations for (A) are as before,

^ = - ^ ( A ) [Eq. (14-3)], (A) = ( A )⁰e - ^ [Eq.(14-6)].

For (B), however, we have

= *!(A) - fc²(B) = ^(Α\ e-^* - kJB). (15-85)

dR i_ /A\ ι /Ί , χ - C dR

and

~dt = *^{l ( A )}° + <*' ~ ^k>^)R> ^{0 Γ} S fr^l(A)⁰ + - k²)R = '·

Ιη[^(Α)⁰ + (k^x — k²)R] = t + constant.

We now replace R by (B) e^klt and solve for (B), evaluating the constant of inte­

gration by setting (B) = 0 at t = 0, to obtain

(B) = y f c l ( A ) 7° - (15-86)

k² — k^x

If ( C ) is the final product, then, by material balance,

(C) = (A)⁰ - (A) - (B). (15-87)

The behavior of the reaction sequence of Eq. (15-84) is somewhat complex since, although (A) is always given by Eq. (14-6), there is a qualitative difference in the behavior of (B) and hence of ( C ) for k² > k^x versus k² < k^x.

CA S E 1. k²> k^x. Equation (15-86) may be put in the form

( B ) = *l(A)° - *-<*.-*i>«) = /^{l (A )} (1 - ^e-^{{ k}^^u) . (15-88) fC2 Κι K2 Κι

The second exponential term of Eq. (15-88) goes to zero at times long compared to l/(k² — fci), and in this limit

or <B>

"» = τ-τ^"-""- <

¹⁵

-

⁸⁹

>

Thus (B) eventually must parallel (A), differing from it by the constant factor k\l(k² — &i). This limiting condition is known as one of transient equilibrium, a name given by early radiochemists; sequences such as Eq. (15-84) are especially common with radioactive species (see Section 21-5). Since (B) = 0 at t = 0 and returns toward zero as (A) goes to zero, we conclude that (B) must have a maximum value, as is indeed the case. At the maximum d(B)Jdt = 0, and so, by Eq. (15-85), ki(A) = k²(B). Insertion of this relationship into Eq. (15-88) gives

e - i ^ W ⁼ *L F ,^{m a x =} tofa/y ^{m ( 1 5}_ 9 0 ) K2 K2 fCi

Figure 15-15(a) shows the variation of (A), (B), and ( C ) with time for the case of k^x = 0.1 h r^{- 1} and k² = 0.3 h r^{- 1}. Note that (B) never exceeds (A); in general, Equation (15-85) may be put in integrable form by a change of variable,

R = (B) e^ or (B) = Re~^ and ^ = 4* - k.Re^.

Substitution into Eq. (15-85) introduces e~^klt into each term, so that it may be canceled out, leaving, on rearrangement,

SPECIAL TOPICS, SECTION 2 641

0 10 2 0 3 0 / , hr

0 10 2 0 3 0 4 0 5 0 6 0 /, hr

( b )

F I G . 15-15. Sequential first-order reactions, (a) The case of transient equilibrium, k2 > kx. (b) The case of no equilibrium, k2 < kx.

the mathematics of this case does not allow Β ever to be the majority species.

Note also that ( C ) shows a time lag or induction period before beginning to rise rapidly; the inflection point is at i^{m a x} of Eq. (15-90).

CA S E 2 k^x > k². We now write Eq. (15-86) in the form

(Β) = ^^{l ( A )}° *-*»'(l - *-<*!-*»>«). (15-91) Κι Ar²

The second exponential term goes to zero at times long compared to 1/(£χ — k²), to give

( B ) i^{i m} = ^T ^ - ^v (A)⁰ e-^k>\ (15-92) The previous condition of transient equilibrium is not present—(B) simply disap­

pears according to its own rate constant k². If k^Y ^>k², (A) disappears very quickly, leaving only (B), which then reacts at its own slower rate:

( B ) i^{l m} = (A)o

An illustrative set of curves is shown in Fig. 15-15(b) for k^x = 0.1 h r^{- 1} as before, but now with k² = 0.0333 h r^{- 1}. The concentration (B) goes through a maximum as before, with /^{m a x} again given by Eq. (15-90), and for a time Β is the dominant species. Again, there is an induction period for (C).

The above represent two important special cases. There is, however, a general solution for any scheme of coupled first-order reactions [see Benson (I960)].

C . Reversible Second-Order Reactions

We consider the simple reaction

A + B ^ C + ^{D ,}

for which the rate law is d(A)

dt = - ^ ( Α ) ( Β ) + *,(C)(D).

If a and b denote the initial concentrations of A and B, respectively, and χ denotes the concentration of products, so that (A) = (a — x) and (B) = (b — x), we have

dx

— = k^x(a — x)(b — x) — k²x²

(supposing no C or D to be present initially), or dx

oc

+

βχ + γχ² = dt, (15-93)

where oc = k^xab, β = —k^x(a + b), and γ = k^x — k². Equation (15-93) integrates to give

ν* + ί(β + ?

^1/a

)

F I G . 1 5 - 1 6 . The case A + Β % C + D ; illustration of Eq. (15-94).

SPECIAL TOPICS, SECTION 2 643 where q = β² — 4<χγ and δ is the constant of integration, determined by setting χ = 0 and t = 0. Equation (15-94) is difficult to use experimentally. A problem is that, given some experimental error in the data points and a limited range of t⁹ it may be possible to find choices of k^x and k² that appear to fit the results even though some other rate law is actually the correct one. We should either choose experimental conditions such that the system becomes a reversible first-order one (such as by having excess Β and D present) or, at least, obtain an independent relation between k^x and k² (such as from the equilibrium constant). Figure 15-16 illustrates the relatively uninformative shape of an (A)/(A)⁰ versus time plot given by Eq. (15-94); it is calculated for a = 0.1 Μ and b = 0.2 M.

D . Rate Law for Isotopic Exchange

A type of kinetic approach that has been widely deployed in the study of reaction mechanisms is isotopic labeling to follow the exchange of an atom or group between two chemical states. An example of an exchange reaction is

RBr + 8 0B r - = R8 0B r + Br", (15-95)

where R might be an alkyl group, and radioactive ^ B r is used as a label. Other examples are

SO!" + H^{2 l 8}0 = S 0^{3 1 8}0²" + H^aO, (15-96) where oxygen exchange is studied with the use of the stable isotope ^{1 8}0 and

Ni(CN)l" + 1 4C N " = N i ( C N )3(1 4C N )2" + C N " , (15-97)

553+^eF

_| _

^Fe

2

^{+ =}

55 2+ _|

^eF

_

^(15-98)

in which radiocarbon and iron are tracers.

The normal exchange procedure consists of establishing an equilibrium system containing the two chemical species A X and BX, having atom X in common and exchange between which is to be studied, and following the rate of appearance of labeled BX if the labeling was originally present in AX. That is, the reaction type is

A X * + B X ^ A X + B X * , (15-99) where the asterisk denotes the presence of a radioactive atom or an excess of

some stable isotope of the atom. Samples of the system are taken periodically, compounds A and Β are physically separated by some procedure (such as precipi­

tation or solvent extraction), and the content of labeled atom in each is determined.

It is ordinarily assumed that different isotopes of the same atom have essentially identical chemistry, which means that at exchange equilibrium the proportion of labeled X atoms must be the same in compound A X as in compound BX.

It turns out that these conditions imply that the kinetics of an exchange reaction will always be of the first-order reversible type, regardless of the actual mechanism whereby the exchange occurs. The demonstration of this conclusion is as follows.

Since the system of Eq. (15-99) is at chemical equilibrium—that is, the amounts of AX and BX are not changing with time—it follows that Rt, the rate at which

chemical species AX forms chemical species BX, is just equal to i?b , the rate of the back reaction. We denote the total amounts of A X and of BX by a and b, respectively, and the amount of AX* and BX* by χ and y, respectively,

A X * + B X ^ A X + B X * , χ b a y

and suppose that x⁰ is the initial amount of A X * (and y⁰ = 0 ) . Then the rate of appearance of BX* must be

where R = Rt = Rn . That is, the rate of appearance of BX* is the overall chemical rate R times the fraction of atoms X in compound A X that are labeled, and the rate of back exchange is R times the fraction of atoms X in compound BX that are labeled. Since x⁰ = χ + y, Eq. ( 1 5 - 1 0 0 ) can be written

dt a \a br

Separation of variables and integration gives

~~~ -cy = (constant) e~^c\ where c = R{^ + J ) . ( 1 5 - 1 0 1 )

The constant of integration is evaluated from y = 0 at t = 0 , and rearrangement leads to

cy

= **L

^(l

-

^e

-cty

a

However at exchange equilibrium, yⁿ = xjbjia + b\ so

* * o ^{= R}y* - ^{c v} a ab/(a + b) C y c o 9

and the final form of the exchange rate equation is

1 - = e~c\ ( 1 5 - 1 0 2 )

y<x>

which is a special case of Eq. ( 1 5 - 5 ) . Thus a plot of ln[l —

(y/y<x>)]

versus / should give a straight line of slope — c.

Each exchange experiment provides a value of the exchange rate constant c for some particular set of values of a and b. Use of Eq. ( 1 5 - 1 0 1 ) then gives the rate R of the chemical reaction responsible for the exchange. The rate law for

SPECIAL TOPICS, SECTION 2 645

U.l 1 ι ι , , , , 0 0.2 0.4 0.6 0.8 1.0 1.2

/, hr

FIG. 15-17. Exchange o / / 2 - C⁴H⁹I ( R I ) with I " . ( / ) (RI) = 0.1 Af, ( I ~ ) = 0.2 M , or (RI) = 0.2 M , (I") = 0.1 M. (2) (RI) = ( I " ) = 0.2 M.

the chemical reaction is then explored by repetition of the exchange study with varying concentrations of (AX) and (BX).

Example. T h e exchange of / z - C⁴H⁹I with I " w a s f o l l o w e d at 50°C with the use o f radioiodide i o n as tracer. Samples were withdrawn periodically a n d the a m o u n t o f radioactivity associated with the butyl iodide determined. T h e plots o f [1 — (yly*)] versus t for three sets o f concentra­

tions are s h o w n i n Fig. 15-17. T h u s for ( R X ) = 0.1 Μ a n d ( I ~ ) = 0.2 M , the exchange half-life is 0.588 h r , o r c = 1.179 h r "¹. F r o m Eq. (15-101), w e find the rate o f reaction R = 1.179/[(1/0.1)

+ (1 /0.2)] = 0.0786 liter m o l e "¹ hr " \ T h e s a m e slope and R value are obtained if ( R X ) = 0.2 and (I") = 0.1. If both concentrations are 0.2 M , then the exchange half-life becomes 0.441 hr, which leads to an R value of 0.157. Thus doubling ( R X ) at constant ( I^-) doubles the rate, in­

dicating the rate law to be first order in ( R X ) , and doubling ( R X ) while halving (I~) does not change the rate, implying first-order dependence o n (I~) as well.

T h e rate then appears t o be given by R = ^ ( R I ) ( I ~ ) , with k = 0.0786/(0.1)(0.2) = 3.93 liter m o l e "¹ h r "¹ [or k = 0.157/(0.2)(0.2) = 3.93 liter m o l e "¹ h r "¹] . T h e exchange o f o n e h a l o g e n for another i n alkyl iodides is believed t o occur by the bimolecular process

C H R ^ I + Χ " — CHRxRaX + I".

Exchange studies have, in fact, provided strong confirmatory support for this Walden inversion mechanism.

Some further points are the following. The quantities a and b refer to the amounts (or concentrations) of labeled element, so that in a case such as that of Eq. (15-97), a = 4[Ni(CN)J~] and b = (CN~). The rate law, or expression for R⁹ remains, of course, in terms of the concentrations of the molecular species, [Ni(CN)J-] and (CN~).

It was assumed in the derivation that χ <^ a and y <^ b, that is, that the amounts of labeled elements were small compared to those of the normal ones. This is generally true in the case of radioactive labeling but is usually not so if labeling is by means of a stable isotope. The writing of Eq. (15-100) is altered, but the final result is still Eq. (15-102).

15-ST-3 Effect of

In document CHAPTER FIFTEEN KINETICS OF REACTIONS IN SOLUTION (Pldal 35-44)