Consideratinos on the applicability of bodenstein's stationary-state method to a certain problem of chemical kinetics-香川大学学術情報リポジトリ

Vol 11 (No 29) (1959) 275

CONSIDERATIONS ON T H E APPLICABILITY

OF BODENSTEIN'S STATIONARY-STATE

METHOD T O A CERTAIN PROBLEM

OF

CHEMICAL KINETICS

S hir8 MATSUMOTO

(Laboratory of Organic Chemistry)

(Received July 23,

1959)

In chemical kinetics, an approximation called "BODENSTEIN'S stationary-state method" (") i s often

used to derive the rate formula With this simple and valuable method we can easily obtain perspective results which cannot be derived by direct integration of the differential equations corres- ponding to the mechanism I t i s essential, however, to note the limitations of i t s applicability In this approximation the variation of the concentration of every intermediate with time i s put to zero This i s justifiable when all the concentrations of the intermediates are very small compared with those of reactants and products If the last conditions are not fulfilled, however, there may be cases where BODENSTEIN'S method will give incorrect results despite of apparent stationariness of the reaction Such an example is given in the second p r t of this paper Thus, unless the nature of the intermediates i s certain, a s in the cases of chain reactions involving unstable free radicals, some consideration must be exercised as to the applicability of the method to each particular case

In a previous studyC2) on the effect of water content of the medium on the rate of decarboxylation of benzoylacetic acid catalyzed by aniline, an attempt was made to draw some conclusions from the experimental data by deriving the rate equation assuming the following rather general scheme:

ko ki k2 kn. k

s

~

K

~

~ x , ~ A + H ~ O - - - + B + H ~ O ~ C ~ ~ ~ K ~ A C ~ ~ - ~ C O ~

x

~

~

x

~

~

(1)

k'o k 1 k'z k'n.

where S, K, and Acph mean benzoylacetic acid, catalyst aniline, and acetophenone resp A i s a SCHIFF'S base-like intermediate formed by the condensation of S and K, and X's are intermediates to A B i s the decarboxylation product of A Then

(Hz01 = W t (A3

+

IB) (2)

where [ ) means concentration 74' means the concentration of water present before the start of the reaction and [A)-t(BI corresponds to that of water formed by the condensation of S and K In the paper cited above(2' i t was assumed that [A]

+

(BI

was very small compared with W and could be neglected in (2), from the fact that no separation of water was observed when benzoylacetic acid decarboxy- lated in pure aniline. The mathematical treatment was thereby greatly simplified and eventually lead to the justification of the use of BODENSIEIN'S approximation No dir-ct evidence was given there, however, a s to the validity of the assumption in the case of decarboxylation in acetic acid in which the rate was measured

In this paper i t i s shown that such an assumption i s unnecessary to get to the same ~onclusion; only some additional mathematical reasoning is necessary

I.

The verification o f the applicability of BODENSIEIN'S method without a priori assumption ~ ' > > ( A l + i B I

276 Tech Bull Fac Agr Kagawa Univ concentration of the catalyst)>[B)c2), P=0.3 mole/L and W 1 0 . 3 mole/L Hence with close approximation

(H201= W t (A

I

(3)

Let the quantity iS1-t [XI IS .+[XVL)

+

(A) be put to Q I t corresponds to the experimentally deter- mined total concentration of substrate molecules that remain a t time t

The fact that the decarboxylation i s of the first order with respect to the substrate concentration i s expressed by

dQ/dt =

-

hQ

(4)

where h i s the experimental first order rate constant On the other hand ( c f scheme (1)) dQ/db= -k[Al So that (A) = PA& (5) (PA= hlk: const ) and d[A)/dt = d ( p ~ Q ) / d t = $A dQ/dt = -p.dhQ (6) Again, inspection of scheme (1) yields

d [ A ) / d t = k n [ X n ) - k [ A ) - k ' l z [ ~ ] [ H z O l

This i s transformed by virtue of (3),(5), and (6) to

- h p ~ Q = kn(Xn) PA&-kfn( W ~ A Q $ p~2Q2) or :Xm) = Pnc1)Q+ pm(2)Q2 (Pdl), PnC2) : const) Similarly (X,) = p,(')Q

t

pa(2)Q2 (7) (Pd1),PJ2) : co 1st i = 1,2, n

-

1 ) Also [S

1

= Pscl)Q

+

Ps(')Q2 ₍₈₎ (p~('),ps(~) : const

1

for n ( S I = Q - ( ~ ( x t ) t I A J ) 2 = 1 Then

(K1=

F-(C(Xs1 -t I A l ) = P - (Q- ( S l ) = P

t

(Ps(')-1)Qt Pd2)Q2 (9) Again scheme (1) gives

-d[Sl/dt=ko[Sj[Kl -k'o(Xil (10)

If (7),(8) and (9) are inserted into (lo), and (4) is taken into account, both sides of (10) are written i n the form of polynomials of Q with coefficients made up of k's, p's, P and A, which are constants a t leat for each particular experiment By co-mparison of the coefficients before the same power of Q on both sides there are obtained:

Q1 : ps(')h= ko~~(')P-k'oki(') (11) Q2 : 2 p ~ ( ~ ) X = ko {P (2)P-t ps(l)(pscl)- 1)) - k o ' ~ ~ ( ~ ) (12) QS : O= ko {ps(')~s(~)+ PS(~)(PS(')- 1)) (13) Q' : O=ko ps(2)2 (14) (13) and (14) give ps(2)= rJ (15) As I S ) + C ( X L I + I A I = Q

vol 11 (No 29) (1959) or

i t follows t h a t

ps(')+

CPi"'

t PA=

1 psC2)

t

CP%(2)

= 0

The last equation means by virtue of (l5)

2

po(2) = 0 k 1

Explicit forms of p2(2)'s are

k' fin(2) = 2pA2 kn ( 2 ) - -2h

+

k'lt-1

.

&

~~2 Pn-1- _{kn-1 kn} ( 2 ) -4~2-2h(k1n-2-t k'n-I+ k n - I )

t

k'n-2 k'n-I

.

G.

Pn-2- _{kn - 2 kn-1 kn}

PA'

Hence

5

* & ( 2 ) = -pA2

{

( 1 + k : - ~ ~ + k fkn-2 k'n-l ~ - 2kn-1

+

9,=1 kn. ) 1 k'n- 2+ k'n-14- kn-1 -2X(---

+

kn-I kn-2 kn-1

+

1 1 k'n-st k'n-n+ k'n-I+ kn-s+kn-I-+ + 4 X 2 ( k n - z * kn-1

+

kn-3 kn-2 &-I

1

Consequently t h e last expression must be zero, although A, being approximately proportional to the catalyst concentration, varies considerably with experimental conditions Then every coefficient of any power of h must be zero This i s the case only when all t h e quantities:

6 .

pA2, -2. k' pA2

.

-

k'n-1

-.

k'n pA2

.

k'n-2 e kfn-I

kn kn k n - I ' kn kn-2 * kn-1 '

k' 1

-2. pA2

.

-

-.

k'n PA2

.

---, k'n-n+kln-I+ kn-1

L

kn kn-1 ' kn-2 kn-1

k!E. p A 2 . . - I p 2. k' pA2 , k'n-3t k'n-2+ k'n-l+kn.-fhkn-1

kn kn-n * kn-I' kn. kn-a k - 2 0 kni-1

a r e zero or vanishingly small, a s they cannot be negative I t follows then, that every

Pi(2)

itself n u , t be zero, because i t i s expressed by a polynomial of which has above quantities a s coefficients Inversely, if every pic2) i s zero then

P s ( ~ ) = O

by virtue of 1161 Hence these two statements a r e equivalent. More strictly speaking, equation (14) means that t h e differencs between ko P s ( ~ ) ~ & ~ (or i t s square root) and zero cannot be detected by the experiments performed SO i t inay be concluded that t h e neglection of p , ( 2 ) ~ 2 in every [,x6)=pa(')Q+pi(2)Q2 is justified within the limit of t h e exp2rimentai evidence.

Th,n circumstance zs just the sums as what occurs when the tsvm [ A ) I S ojnlttgd zn ihs ~ e l a t ~ o n (3): [ H ~ O ] &

W+

[A]

at the start

I t may be critisized here that

(I) equation (13) or (14 will give mathematically ko=O which has vague meanin?. and

270 Tech I3ull Fac Agf Kagawa Univ

dQldt = -hQ which i s transformed many times without t h e criticism of propagation of error 2

These objections may be avoided by t h e introduction of a small parametric correction term sQ i n A:

dQ/dt= - ( h o t sQ)Q (18)

Calcull+ion is then pushed forward a s b ~ f o r e with retention of t h e terms that contain E At some later stage E&/>o is made ---+ 0, and the consequences are examined

I n this manner of calculation all the expressions are naturally more complex than the cor~esponding expressions in the above simphr consideration, i,e,:

( A ) =

(ho/k)&$ ( ~ / k ) & ~ = p ~ ( ' ) & f ~ A ( ~ ) Q ~

11-2+4

( X I )=

C

p,c7)QT ? = I

where p's a r e constants constructed with k's, ho, and E .

By these equations and the relation

( K I = P - Q f ( S j

both sides of equation (10) are transformed to polynomials of Q, (18) being taken into account. Then t h e coefficients before t h e same power of Q are equated a s before There are thus obtained equations which correspond t o (11)

,

(12) e t c

koPpscl)

-

hops(')= kfopl(') 09)

(cor r esp (11)

1

ko {psc3)P$ psc2)(psc1)-1)

+-

p s ( l ) ~ , ( ~ ) )

-

(3ps(')ho+ 2Ps(')&) = k ' ~ p ~ ( ~ )

₄₀₎

(cor resp (X3))

At this stage let sQ/ho---+O Then by some lengthy calculation it is shown that

1

p,cS)Q2/ptc1)

1,

p ~ ( ~ ) Q V p t ( ~ )

1, I

p~(~)Q*/pt(l) j , and

I

p ~ ( ~ ) Q ~ / p ~ ( l ) l ,

1

PS(~)Q~/PS(~)I

I

@S(~)Q~/PS(')I all become-0.

Thus it is shown from (20), with reference to U9), that

/

psc2)Q/ps(')l also becomes-0. As

psc2)+ Cpic2)$ PA(^)= 0 and l>ps(')>O i t follows tnat

I

& ( C p i c 2 ) , f ~ ~ ( 2 ) ) ( -0

I

Q

,2

$6'2)

I

-0

%=I & 1)

for

1

~A(~)Q/PA(')I -0 and I. >PA(')>O

(21) is formally identical with U'I) but may involve terms containing E. These terms are shown, however, t o be infinitesimal quantities and can be omitted in $1). Thus the final concluszon zs just the same as in the simiplif'ied treatment in which 8 is put to zero at the start

11. Simple example of reactions for which the application of BODENSTEIN'S stationary-state

method is invalid despite of their apparent stationariness

Let t h e following simple type of consecutive reactions be considered: ko k

S ~ ~ - - + ~ r o d u c t s k'o

where S is a rezctant which decomposes through an intermediate X to form products Then

and

d[X)/dt= ko [ S J - ( k t o t k ) [ X )

where t is the time after the initiation of the reaction and k's are rate constants given in the reaction scheme.. By accurate integration these equations give

( S

I=-

/ ( k 0 t k ' o t k ) ~ - 4 k o k

IS"

- - - { ( k t o + k + al)eUlt

-

( k t o t k+ *)eUzt

)

and these in turn give

In the above equations [ S ) o means the initial value of I S ) and

C-t

sign for 0 1 , and -sign for 0 , ) If BODENSIEIN'S approximation is made:

d [ X j / d t = k o [ S > - ( k t o + k ) [ X 1 = 0

then

[ X I = ko[Sl/(kto+ k )

In any case the total reaction rate

v

is given by

v=

k [ X 1

In general 01,u2<0 and

1

u,

I

>

1

a1

1,

so that for large values of t

eazt <eO1t

and (22) reduces to

If the quantity [ S l + ( X ) i s denoted by Q, (24 can be transformed to

[ X ) =koQ/(kto+ko+ k + 01)

and so

V = { k ko/(kto+ ko+ k + a l ) } Q (2 5)

If the decomposition rate is measured by the amount of products iormed, Q appears as the total concen- tration of the reactant molecules that remain a t time t Then, 1251 means that the reaction will appear as first order and stationary after a sufficient time elapse.

In BODENSIEIN'S approximation, rate equation is derived fro.= 1231 in the form:

v = ( k ko/(kfo+ko+ k j ) Q 1261

I n general, (25) and (26) are not identical unless a1 can be neglected zn (25) The last condition is satisfied,

for example, if kto> ko, k This is the case when X is unstable and formed only with difficulty

Acknowledgment The auther wishes to express his sincere appreciation to Prof 0 SHIMAMURA,

Univ Tokyo, for his constant encouragement and many valuable discussions

References

(1) BODENSIEIN, M : Z Physik Chem 8 5 , 329(1913) Jafian, Pure C h m Sec (Nzhon Kagaku Zasshi), (2) M ~ r s u ~ o r o , S , TSUBOI, Y : J Chem Soc 7 7 , 1763 (1956)