- Email: [email protected]
A kinetic analysis of the exchange of deuterium with hydrides
Chemical Engineering Science, 1963, Vol. 18, pp. 253-258. Pergamon Press Ltd., Oxford.
Printed in Great Britain.
A kinetic analysis of the exchange of deuterium with hydrides R. J. MIKOVSKY and J. WEI Socony Mobil Oil Company, Inc., Research Department,
Paulsboro Laboratory,
New Jersey
(Received 1 December 1962) Abstract-In the exchange reaction between deuterium and hydride molecules, the consecutive single-step m~hanism is characterized by interactions only between the ith deutero-isomer and the (i f 1) and (i - 1) isomers. Rate equations have been set up on this basis and their exact and unique solutions found. This permits the deutero-isomer distribution to be expressed analytically as a function of the time. If the initial state contains only undeuterated hydride or a binomial distribution of isomers, the distribution has been shown to maintain its binomial character to equilibrium.
An application of the analysis to two exemplary cases has been made.
THE relationship
between the mechanism for an isotopic exchange of deuterium with a hydride (hydrocarbon) and the binomial distribution of product isomers is well appreciated. For example,
in the work of DALLINGA et al. [l], KEMBALL [2], KLOOSTERZIEL [3], PASSet al. [4] and WAGNERet al. [5], the binomial distribution has been treated as evidence for the simplest possible exchange mechanism wherein deuterium atoms are inserted into the hydride one at a time. Deviations from such distributions have been looked upon as evidence for a multiple-exchange process (SCHUIT et al. [6], DALLINGA and TERMATEN[7], ROWLINSON et al. 187). Kinetic analyses, such as those of HARRIS [9] and ANDERSON and KEMBALL [IO] have been made but, in the absence of analytical expressions, comparison of observed and calculated deutero-isomer concentrations is usually made on the basis of parameters derived from the data. The purpose of this communication is to show that (I) exact analytical solutions of the rate equations for the single-step exchange mechanism are possible, and (2) that, if the hydride is initially undeuterated or is present as a binomial distribution of isomers, the product isomer distribution will retain its binomial character at every stage in the approach to equilibrium. KINETIC ASPECTS
heterogeneous reaction with saturated hydrocarbons is usually of a positive order in hydrocarbon and of a negative order in deuterium (BOND 1111). The relatively simple statistical nature of the single-step exchange mechanism, however, is free of much of this complexity. Model and rate equations Consider a closed reaction system containing hydride and hydrogen-like molecules. Ifthe hydride contains n hydrogen atoms that are exchangeable and equivalent, there can be (n i- 1) isomeric states whose mole fractions can be represented by PO, p1, PZ, ... Pi, . . . P, where i is the number of deuterium atoms in the isomer. Because the total number of hydride molecules is invariant the
$iopi=n where 71 is a constant, viz, the mole fraction of hydride. The hydrogen-like molecules include the species hydrogen, hydrogen deuteride, and deuterium whose atoms are all considered equivalent; any isotope effect is neglected. The atomic concentration of deuterium may be designated by g E 20, + HD and that of hydrogen as p E 2H, + HD where D2, HD and Hz are the mole fractions of the respective species. Conservation of the total number of hydrogen-like atoms leads to
In general, the kinetics of exchange between deuterium and hydrides can be quite complex. The
u+4=2* 253
(2)
R. J. MIKOVSKYand J WEI
where $ is a constant, viz. the mole fraction of hydrogen-like molecules. As the exchange reaction proceeds, the net result is that deuterium atoms are transferred from the pool of hydrogen-like molecules into the pool of deutero-isomers. However, the total number of deuterium atoms in the system remains constant. This is expressed by 4 +
~ iPi = 6 i=O
where 6 is a constant. If the reaction system consists initially of only undeuterated hydride molecules, the value of 6 equals q(0) or the initial atomic concentration of deuterium. A consecutive single-step exchange reaction can be schematically represented by P, * P, P P, P . . . Pi . . . P P, where the forward direction is effected by reaction with deuterium atoms whose concentration is q and the reverse direction by reaction with hydrogen atoms whose concentration is p. Alternatively,
equation
(2), is
Solutions.for equilibrium state The binomial distribution of deutero-isomers, which has been used (WAGNER et al. [5]) to characterize the consecutive single-step exchange reaction, can easily be shown to follow from equations (4) at equilibrium. Setting the rates equal to zero and successively adding the equations (4) leads to Pi (i + 1)~’ “‘Pi;r=(n-i)q’...
Pb -_=_-; Pf p; nq’
where the primes denote the equilibrium concentrations which are realized as the time t + co. Reduction of these ratios to a common denominator gives
Pd_ -p,:
pf _ n P’ n. --J 9 “‘p:, i 0 O(
p’ n-i -q ...; )
(nI1)($) (6)
+= Pi+qF?Pi+l+p where The rate of reaction may be characterized by a specific rate constant k which bears a reciprocaltime dimension. If a catalyst is used, this rate constant also contains the catalyst concentration, e.g. min-l g-r. The rate equations are homogeneous in this k, as defined, and may be written
PA_1 np’ p:,=qY-
;
n
0 i
is the binomial
coefficient.
mation of these ratios can be equated expansion, thus
The sum-
to a binomial
iio;= p’ +1”=1
Lo 4’
n
p:
1
As a consequence, tPo=
-qnP,+pP, p,:=n
~
Pi = q(n - i + + pti +
l)Pi_1 - [q(n - i) + pilPi
(4)
41
[Pf +
II
4’
I
Insertion of this result into the ratios (6) gives the final distribution of deutero-isomers
lIpi+
i&n =qP,_,-pnP,
Pi = +j
The closed nature of the system also allows the rate of change of the deuterium content of the hydrogen-like molecules to be related to the specific rate constant. The relation, with the use of 254
(--&)i(&)“-i
(8)
A kinetic analysis of the exchange of deuterium with hydrides
Solutions for transient states The binomial distribution of deutero-isomers that obtains at equilibrium implies that the solutions to the equations (4) may be of binomial character at any time t. Thus, Pi(t) =
0r
[y(t)]‘[l
7C
-
y(t)]“-’
Integration fixes q at
x exp[ -(nn + 2$)kf] +
(9)
The solution to equation (12) may now be given.
i+2k~y=kq(l)=k([qO-(~~)] x
WAGNER et ai. [5] have used this expression by treating y(t) as the fractional number of hydride bonds that have been exchanged at the time t.
x exp[-(nn
Insertion of equation (9) into the general rate equation (4) yields
$ iJi= q(n -i+l)
+ 2$)kt] + (-$&))
and, as a consequence i r 1 $-l(l
(
_ yy-‘+l
+
)
r(t) =
+ p(i + 1) i jr 1 ++l(l ( >
_ yy-i-l
_
- Cq(n - ij + pi]
x exp[-(nn
This expression reduces to
x [i(l - 7) - (n -
ihlCdl- 7)- ~71 (10)
Another expression for the time rate of change of the ith deutero-isomer can be had by differentiation of equation (9). This gives pi=
01
y-(1
- y)“+-’
(11)
Comparison of this equation with equation (10) gives a relation from which y may be derived, viz.
i = 4dl
- Y)- PYI
Because p + q = 2$, we have
;‘+ 2kJ/y = kq
(12)
The atomic concentration of deuterium, q, is also a function of time and must be specified before an explicit form for y can be given. By combining equations (2), (3) and (5), we may write 4 = 2k$6 - (2$ + nnjkq
(14)
Initially, if the hydride is completely undeuterated, then y(O) = 0 and q(0) = 6. Thus equation (14) simplifies t 0 (1 - expC-(nn
)(
x [i(l - y) - (n - i)y]i
+ 2$)kt] +
+ 2WtJ]
(15)
Equations (9), (14) and (15) thus give a completely explicit solution to the equations (4). If the hydride molecules are initially present in a binomial distribution then this distribution may be characterized by a y(0) > 0 and will retain its binomial character to equilibrium. Furthermore, for a set of differential equations such as (4) and (5) which possess given initial values of P,(O), q(O), and p(O), there exists a unique solution (LEFSCHETZ[12]). Therefore equation (9), wherein y(t) is described by equation (14), is the one and only solution. APPLICATION KEMBALL [13] has shown that the exchange between deuterium and ammonia proceeds via a consecutive single-step exchange mechanism over a
255
R. J.
and J. WEI
MIKOVSKY
)-
)--
)NH, 0
l-
,-
Time.
min
Fig. 1. Relative production of deutero-ammonias. variety of metals including nickel, platinum, tungsten, silver and copper. He has been able to numerically integrate rate equations similar to equations (4) in order to describe data on the rate of production of the deutero-isomers. The data of Fig. 1 were obtained by KIMBALLusing tilms of the metals mentioned; they show the relative rates of production of the isomers after an adjustment is made for the different activities of the metals. The solid lines of Fig. 1 are calculated curves based on the present analysis. The values of the various parameters were obtained from the experimental conditions and were as follows: n = 3, II = O-5, 6 = 2$ = l-0, and k = 0.0133 min-‘. The value of k is one-third that determined by KEMBALLin keeping with the respective definitions of the specific rate constant. As can be seen, the theoretical curves are in excellent agreement with the data and are actually coincidental with the curves obtained by KEMBALLby numerical integration. The equilibrium values are consistent with those expected for randomly deuterated isomers :
0 Dataof
%h%BALL
-
Calculated.
P;,= 21.6, Pi = 43.2, Pi = 28.8 and Pj = 6.4 per cent. As a second example of the applications of the equations developed herein, the data of KEMBALL [2] on the deuteration of neopentane serve very well. The data points of Fig. 2 show the changes occurring with time for undeuterated t-butyl ions and the first four deuterated species. The t-butyl ion was used as an indication for the reaction due to the difficulty of obtaining parent peaks with the mass spectrometer. The solid lines are the calculated values utilizing equations (9) and (15) and empirical values for the parameters. The parameters were n = 9, TC= O-0682,6 = 2+ = l-364 and k = O-00266 rnin-I. The values of IZ,n and 6 are 8 of the experimental values to adjust for the measurement of t-butyl-ion concentration rather than neopentane concentration; the value of k is & of the experimental value determined by KEMBALL,again in keeping with the respective definitions of the rate constants. Although numerical integration would have been
256
A kinetic analysis of the exchange of deuterium with hydrides DISCUSSION
k/s-::_, C, H7D:
C, H, D:
0
IO
20
Time,
30
CaH, ‘A+ 40
50
min
Fig. 2. Exchange of neo-pentane with deuterium. 3 Data of KEMBALL - Calculated.
alternative for describing the data of Fig. 2, the present analysis eliminates the tediousness of such a procedure in addition to yielding exact expressions : this consideration becomes increasingly important as the number of deutero-isomers rises. In addition, numerical integration must be considered an approximation and cannot serve as proof of the invariance of the binomial character. an
The derivation of analytical expressions for the concentration of the deutero-isomers has consequences in addition to the proof of the invariance of the binomial character. The time derivatives of the concentrations are given by equation (11) and are equal to zero when y’-’ = 0, y = i/n and $J= 0. The last condition is the equilibrium condition and indicates that the deutero-isomer concentrations approach equilibrium values asymptotically, i.e. as t + co and 9 * 0. The first condition signifies that when y = 0 (when undeuterated hydride is the reactant initially), the deutero-isomers concentrations with i > 1 leave the origin with zero slope. The slopes of P,,(t) and PI(t) are fixed at finite values by equation (11). The intermediate condition sets the value of y for a stationary point at i/n. Thus the maxima in the curves of Figs. 1 and 2, for example, occur at predeterminable values of y. This condition also permits linding the isomers whose concentrations do not pass through maxima but rise from the origin through a point of inflection and approach equilibrium asymptotically, viz. those isomers whose i/n > y’, where y’ is the equilibrium value established by the experimental conditions. The parameter y(t), because it implicitly contains the time and rate constant, can be considered a normalizing parameter against which the relative amounts ot’ deutero-isomers may be compared. Thus the relative production of the isomers in terms of y(t) becomes a criterion for the single-step exchange mechanism.
REFJIRENCE~ STUARTA. A., SMITP. J. and MACKORE. L. Z. Elekfrochem. 1957 61 1019. 111 DALLINGAG., VERRLJN PI KIMBALLC., Trans. Faraday Sot. 1954 50 1344. r31 KLOOSTERP~LH., Chemisorption (Edited by GARNERW. E.) p. 76. Academic Press, New York, 1956. A. B. and Buaww R. L., Jr., J. Phys. Chem. 1960 82 6281. [41 PASSG., LITTLEWOOD D. P., J. Chem. Phys. 1952 20 338. [51 WAGNERC. D., WILSONJ. D., OTVOSJ. W. and STEVENSON L. L., Chemisorption (Edited by GARNERW. E.) [61 SCHUITG. C. A., Denoan N. H., DORGEL~G. J. H. and VANRELJEN p. 39. Academic Press, New York 1956. DALLINGAG. and TERMAT~N G., Rec. Trav. Chim. 1960 79 737. ROWLIN~~NH. C., BURWIU R. L., Jr. and TUXWORTH R. H., J. Phys. Chem. 1955 59 225. G. M., Dans. Faraday Sot. 195147 716. ANDBR~~NJ. R. and -ALL C., Proc. Roy. Sot. 1954 A226 472. BOND G. C., Quart. Rev. Chem. Sot. 1954 8 279. LEFSHETZ S., Differential Equations: Geometric Theory p. 29 ff. Interscience, New York 1957. KEMBALL C., Proc. Roy. Sot. 1952 A214 413.
257
R. J. MIKOVSKYand J. WEI
R&sum&Lkhange entre le deutkrium et des molecules d’hydrures selon un mkcanisme conskcutif simple est caractkrid par des interactions entre le deutero-isomere i et les isomeres i + 1 et i - 1. On a etabli les equations cinetiques dont on a trouve les solutions exactes et uniques, ce qui permet d’exprimer de facon analytique la distribution des deutero-isomeres en fonction du temps. Lorsqu’a l’etat initial il n’y a que de l’hydrure non deutkre ou une distribution binomiale d’isomeres, la distribution garde son caractere binomial jusqu’a l’kquilibre, cette methode d’analyse a et6 appliquke il deux exemples.
258
Printed in Great Britain.
A kinetic analysis of the exchange of deuterium with hydrides R. J. MIKOVSKY and J. WEI Socony Mobil Oil Company, Inc., Research Department,
Paulsboro Laboratory,
New Jersey
(Received 1 December 1962) Abstract-In the exchange reaction between deuterium and hydride molecules, the consecutive single-step m~hanism is characterized by interactions only between the ith deutero-isomer and the (i f 1) and (i - 1) isomers. Rate equations have been set up on this basis and their exact and unique solutions found. This permits the deutero-isomer distribution to be expressed analytically as a function of the time. If the initial state contains only undeuterated hydride or a binomial distribution of isomers, the distribution has been shown to maintain its binomial character to equilibrium.
An application of the analysis to two exemplary cases has been made.
THE relationship
between the mechanism for an isotopic exchange of deuterium with a hydride (hydrocarbon) and the binomial distribution of product isomers is well appreciated. For example,
in the work of DALLINGA et al. [l], KEMBALL [2], KLOOSTERZIEL [3], PASSet al. [4] and WAGNERet al. [5], the binomial distribution has been treated as evidence for the simplest possible exchange mechanism wherein deuterium atoms are inserted into the hydride one at a time. Deviations from such distributions have been looked upon as evidence for a multiple-exchange process (SCHUIT et al. [6], DALLINGA and TERMATEN[7], ROWLINSON et al. 187). Kinetic analyses, such as those of HARRIS [9] and ANDERSON and KEMBALL [IO] have been made but, in the absence of analytical expressions, comparison of observed and calculated deutero-isomer concentrations is usually made on the basis of parameters derived from the data. The purpose of this communication is to show that (I) exact analytical solutions of the rate equations for the single-step exchange mechanism are possible, and (2) that, if the hydride is initially undeuterated or is present as a binomial distribution of isomers, the product isomer distribution will retain its binomial character at every stage in the approach to equilibrium. KINETIC ASPECTS
heterogeneous reaction with saturated hydrocarbons is usually of a positive order in hydrocarbon and of a negative order in deuterium (BOND 1111). The relatively simple statistical nature of the single-step exchange mechanism, however, is free of much of this complexity. Model and rate equations Consider a closed reaction system containing hydride and hydrogen-like molecules. Ifthe hydride contains n hydrogen atoms that are exchangeable and equivalent, there can be (n i- 1) isomeric states whose mole fractions can be represented by PO, p1, PZ, ... Pi, . . . P, where i is the number of deuterium atoms in the isomer. Because the total number of hydride molecules is invariant the
$iopi=n where 71 is a constant, viz, the mole fraction of hydride. The hydrogen-like molecules include the species hydrogen, hydrogen deuteride, and deuterium whose atoms are all considered equivalent; any isotope effect is neglected. The atomic concentration of deuterium may be designated by g E 20, + HD and that of hydrogen as p E 2H, + HD where D2, HD and Hz are the mole fractions of the respective species. Conservation of the total number of hydrogen-like atoms leads to
In general, the kinetics of exchange between deuterium and hydrides can be quite complex. The
u+4=2* 253
(2)
R. J. MIKOVSKYand J WEI
where $ is a constant, viz. the mole fraction of hydrogen-like molecules. As the exchange reaction proceeds, the net result is that deuterium atoms are transferred from the pool of hydrogen-like molecules into the pool of deutero-isomers. However, the total number of deuterium atoms in the system remains constant. This is expressed by 4 +
~ iPi = 6 i=O
where 6 is a constant. If the reaction system consists initially of only undeuterated hydride molecules, the value of 6 equals q(0) or the initial atomic concentration of deuterium. A consecutive single-step exchange reaction can be schematically represented by P, * P, P P, P . . . Pi . . . P P, where the forward direction is effected by reaction with deuterium atoms whose concentration is q and the reverse direction by reaction with hydrogen atoms whose concentration is p. Alternatively,
equation
(2), is
Solutions.for equilibrium state The binomial distribution of deutero-isomers, which has been used (WAGNER et al. [5]) to characterize the consecutive single-step exchange reaction, can easily be shown to follow from equations (4) at equilibrium. Setting the rates equal to zero and successively adding the equations (4) leads to Pi (i + 1)~’ “‘Pi;r=(n-i)q’...
Pb -_=_-; Pf p; nq’
where the primes denote the equilibrium concentrations which are realized as the time t + co. Reduction of these ratios to a common denominator gives
Pd_ -p,:
pf _ n P’ n. --J 9 “‘p:, i 0 O(
p’ n-i -q ...; )
(nI1)($) (6)
+= Pi+qF?Pi+l+p where The rate of reaction may be characterized by a specific rate constant k which bears a reciprocaltime dimension. If a catalyst is used, this rate constant also contains the catalyst concentration, e.g. min-l g-r. The rate equations are homogeneous in this k, as defined, and may be written
PA_1 np’ p:,=qY-
;
n
0 i
is the binomial
coefficient.
mation of these ratios can be equated expansion, thus
The sum-
to a binomial
iio;= p’ +1”=1
Lo 4’
n
p:
1
As a consequence, tPo=
-qnP,+pP, p,:=n
~
Pi = q(n - i + + pti +
l)Pi_1 - [q(n - i) + pilPi
(4)
41
[Pf +
II
4’
I
Insertion of this result into the ratios (6) gives the final distribution of deutero-isomers
lIpi+
i&n =qP,_,-pnP,
Pi = +j
The closed nature of the system also allows the rate of change of the deuterium content of the hydrogen-like molecules to be related to the specific rate constant. The relation, with the use of 254
(--&)i(&)“-i
(8)
A kinetic analysis of the exchange of deuterium with hydrides
Solutions for transient states The binomial distribution of deutero-isomers that obtains at equilibrium implies that the solutions to the equations (4) may be of binomial character at any time t. Thus, Pi(t) =
0r
[y(t)]‘[l
7C
-
y(t)]“-’
Integration fixes q at
x exp[ -(nn + 2$)kf] +
(9)
The solution to equation (12) may now be given.
i+2k~y=kq(l)=k([qO-(~~)] x
WAGNER et ai. [5] have used this expression by treating y(t) as the fractional number of hydride bonds that have been exchanged at the time t.
x exp[-(nn
Insertion of equation (9) into the general rate equation (4) yields
$ iJi= q(n -i+l)
+ 2$)kt] + (-$&))
and, as a consequence i r 1 $-l(l
(
_ yy-‘+l
+
)
r(t) =
+ p(i + 1) i jr 1 ++l(l ( >
_ yy-i-l
_
- Cq(n - ij + pi]
x exp[-(nn
This expression reduces to
x [i(l - 7) - (n -
ihlCdl- 7)- ~71 (10)
Another expression for the time rate of change of the ith deutero-isomer can be had by differentiation of equation (9). This gives pi=
01
y-(1
- y)“+-’
(11)
Comparison of this equation with equation (10) gives a relation from which y may be derived, viz.
i = 4dl
- Y)- PYI
Because p + q = 2$, we have
;‘+ 2kJ/y = kq
(12)
The atomic concentration of deuterium, q, is also a function of time and must be specified before an explicit form for y can be given. By combining equations (2), (3) and (5), we may write 4 = 2k$6 - (2$ + nnjkq
(14)
Initially, if the hydride is completely undeuterated, then y(O) = 0 and q(0) = 6. Thus equation (14) simplifies t 0 (1 - expC-(nn
)(
x [i(l - y) - (n - i)y]i
+ 2$)kt] +
+ 2WtJ]
(15)
Equations (9), (14) and (15) thus give a completely explicit solution to the equations (4). If the hydride molecules are initially present in a binomial distribution then this distribution may be characterized by a y(0) > 0 and will retain its binomial character to equilibrium. Furthermore, for a set of differential equations such as (4) and (5) which possess given initial values of P,(O), q(O), and p(O), there exists a unique solution (LEFSCHETZ[12]). Therefore equation (9), wherein y(t) is described by equation (14), is the one and only solution. APPLICATION KEMBALL [13] has shown that the exchange between deuterium and ammonia proceeds via a consecutive single-step exchange mechanism over a
255
R. J.
and J. WEI
MIKOVSKY
)-
)--
)NH, 0
l-
,-
Time.
min
Fig. 1. Relative production of deutero-ammonias. variety of metals including nickel, platinum, tungsten, silver and copper. He has been able to numerically integrate rate equations similar to equations (4) in order to describe data on the rate of production of the deutero-isomers. The data of Fig. 1 were obtained by KIMBALLusing tilms of the metals mentioned; they show the relative rates of production of the isomers after an adjustment is made for the different activities of the metals. The solid lines of Fig. 1 are calculated curves based on the present analysis. The values of the various parameters were obtained from the experimental conditions and were as follows: n = 3, II = O-5, 6 = 2$ = l-0, and k = 0.0133 min-‘. The value of k is one-third that determined by KEMBALLin keeping with the respective definitions of the specific rate constant. As can be seen, the theoretical curves are in excellent agreement with the data and are actually coincidental with the curves obtained by KEMBALLby numerical integration. The equilibrium values are consistent with those expected for randomly deuterated isomers :
0 Dataof
%h%BALL
-
Calculated.
P;,= 21.6, Pi = 43.2, Pi = 28.8 and Pj = 6.4 per cent. As a second example of the applications of the equations developed herein, the data of KEMBALL [2] on the deuteration of neopentane serve very well. The data points of Fig. 2 show the changes occurring with time for undeuterated t-butyl ions and the first four deuterated species. The t-butyl ion was used as an indication for the reaction due to the difficulty of obtaining parent peaks with the mass spectrometer. The solid lines are the calculated values utilizing equations (9) and (15) and empirical values for the parameters. The parameters were n = 9, TC= O-0682,6 = 2+ = l-364 and k = O-00266 rnin-I. The values of IZ,n and 6 are 8 of the experimental values to adjust for the measurement of t-butyl-ion concentration rather than neopentane concentration; the value of k is & of the experimental value determined by KEMBALL,again in keeping with the respective definitions of the rate constants. Although numerical integration would have been
256
A kinetic analysis of the exchange of deuterium with hydrides DISCUSSION
k/s-::_, C, H7D:
C, H, D:
0
IO
20
Time,
30
CaH, ‘A+ 40
50
min
Fig. 2. Exchange of neo-pentane with deuterium. 3 Data of KEMBALL - Calculated.
alternative for describing the data of Fig. 2, the present analysis eliminates the tediousness of such a procedure in addition to yielding exact expressions : this consideration becomes increasingly important as the number of deutero-isomers rises. In addition, numerical integration must be considered an approximation and cannot serve as proof of the invariance of the binomial character. an
The derivation of analytical expressions for the concentration of the deutero-isomers has consequences in addition to the proof of the invariance of the binomial character. The time derivatives of the concentrations are given by equation (11) and are equal to zero when y’-’ = 0, y = i/n and $J= 0. The last condition is the equilibrium condition and indicates that the deutero-isomer concentrations approach equilibrium values asymptotically, i.e. as t + co and 9 * 0. The first condition signifies that when y = 0 (when undeuterated hydride is the reactant initially), the deutero-isomers concentrations with i > 1 leave the origin with zero slope. The slopes of P,,(t) and PI(t) are fixed at finite values by equation (11). The intermediate condition sets the value of y for a stationary point at i/n. Thus the maxima in the curves of Figs. 1 and 2, for example, occur at predeterminable values of y. This condition also permits linding the isomers whose concentrations do not pass through maxima but rise from the origin through a point of inflection and approach equilibrium asymptotically, viz. those isomers whose i/n > y’, where y’ is the equilibrium value established by the experimental conditions. The parameter y(t), because it implicitly contains the time and rate constant, can be considered a normalizing parameter against which the relative amounts ot’ deutero-isomers may be compared. Thus the relative production of the isomers in terms of y(t) becomes a criterion for the single-step exchange mechanism.
REFJIRENCE~ STUARTA. A., SMITP. J. and MACKORE. L. Z. Elekfrochem. 1957 61 1019. 111 DALLINGAG., VERRLJN PI KIMBALLC., Trans. Faraday Sot. 1954 50 1344. r31 KLOOSTERP~LH., Chemisorption (Edited by GARNERW. E.) p. 76. Academic Press, New York, 1956. A. B. and Buaww R. L., Jr., J. Phys. Chem. 1960 82 6281. [41 PASSG., LITTLEWOOD D. P., J. Chem. Phys. 1952 20 338. [51 WAGNERC. D., WILSONJ. D., OTVOSJ. W. and STEVENSON L. L., Chemisorption (Edited by GARNERW. E.) [61 SCHUITG. C. A., Denoan N. H., DORGEL~G. J. H. and VANRELJEN p. 39. Academic Press, New York 1956. DALLINGAG. and TERMAT~N G., Rec. Trav. Chim. 1960 79 737. ROWLIN~~NH. C., BURWIU R. L., Jr. and TUXWORTH R. H., J. Phys. Chem. 1955 59 225. G. M., Dans. Faraday Sot. 195147 716. ANDBR~~NJ. R. and -ALL C., Proc. Roy. Sot. 1954 A226 472. BOND G. C., Quart. Rev. Chem. Sot. 1954 8 279. LEFSHETZ S., Differential Equations: Geometric Theory p. 29 ff. Interscience, New York 1957. KEMBALL C., Proc. Roy. Sot. 1952 A214 413.
257
R. J. MIKOVSKYand J. WEI
R&sum&Lkhange entre le deutkrium et des molecules d’hydrures selon un mkcanisme conskcutif simple est caractkrid par des interactions entre le deutero-isomere i et les isomeres i + 1 et i - 1. On a etabli les equations cinetiques dont on a trouve les solutions exactes et uniques, ce qui permet d’exprimer de facon analytique la distribution des deutero-isomeres en fonction du temps. Lorsqu’a l’etat initial il n’y a que de l’hydrure non deutkre ou une distribution binomiale d’isomeres, la distribution garde son caractere binomial jusqu’a l’kquilibre, cette methode d’analyse a et6 appliquke il deux exemples.
258