JEN-YUANCHIEN
2256
coholic ammonia gave the succinamide derivative, with admixture of the ethyl ester V. Compound I (14.4 g.), finely powdered, was allowed to stand in saturated alcoholic ammonia for six days, with occasional shaking. The appearance changed while the heavy crystalline powder became more voluminous. When all heavy material had disappeared, the product was fi1te:ed off and washed with alcohol. It melted at 228-232 After recrystallization from a mixture of pyridine and acetone it melted a t 240". The admixture of the ester was detected only after reduction (see below). A pure preparation was obtained by ammonolysis with concentrated aqueous ammonium hydroxide. Compound I was warmed with 48% ammonium hydroxide in a stoppered bottle at 60 for two days. The product was washed with water and crystallized from pyridine. To remove pyridine of crystallization, the crystals were suspended in warm alcohol; the volume of the material increased considerably. $fter washing with alcohol and drying, the m. p. was 242 The compound was not soluble in alcohol or acetone, sparingly soluble in dioxane or glacial acetic acid, soluble in hot pyridine. 4-Amin04 ' -&carbamylpropionylaminodiphenylsulfone (IX) .-Compound VIII, obtained by alcoholic ammonolysis, was reduced with Raney nickel catalyst and hydrogen in alcoholic suspension. The crude reduction product melted at 135'. Most of it was soluble in hot water. When recrystallized from hot water, containing a little ammonia, colorless fine needles were obtained of m. p. 140". 'The solubility in water at 26" was 0.4%. The water-insoluble material was identified with VI1 by melting point and analysis. 4-Nitro4 '-/3-phenylcarbamylpropionylaminodiphenylsulfone (X).-Compound I (13.5 8 . ) was refluxed with aniline for fourteen hours. After dilution with alcohol,
.
.
[CONTRIBUTION FROM
THE
VOl. 70
10.5 g. of material m. p. 205-210' was isolated. From hot acetone (charcoal) cream crystals of m. p. 225-220" were obtained. It is sparingly soluble in hot alcohol. 4-Amin04 '-~-phenylcarbamylpropionylaminodiphenylsulfone (XI) was prepared from X using the procedure I1 2. 4-Nitro4 '-&cyclohexylcarbamylpropionylaminodiphenylsulfone (XII) .-Compound I (7.29.) was refiuxed with cyclohexylamine for five hours. The yield was 8.7~ g., the m. p. 276'. Recrystallized from :lacia1 acetic acid, the cream needles melted at 275-276 The compound was not soluble in alcohol or acetone, soluble in hot pyridine and hot glacial acetic acid. 4-Amino4 '-/3-cyclohexylcarbamylpropionylaminodiphenylsulfone (XII) was prepared from XI1 using the procedure 11 2.
.
Summary The preparation of succinic acid derivatives of 4-nitro-4'-aminodiphenylsulfone and of 4,4'-diaminodiphenylsulfone has been described. 4Amino-4'-succinimidodiphenylsulfone (11) and 4'amino-4' - /3 - carboethoxypropionylaminodiphenylsulfone (VII) were tested in experimental pneumonia and tuberculosis and found to be active. 4Amino- 4'- /3 - carboxypropionylaminodiphenylsulfone (IV) and 4-amino-4'-/3-carbamylpropionylaminodiphenylsulfone (IX) were also active in experimental pneumococcus infections in mice. RECEIVED' MARCH29, 1948 (7) Original manuscript received October 30, 1947.
DEPARTMENT OF CHEMISTRY A N D CHEMICAL ENGINEERING, STATE UNIVERSITY
IOWA
OF
Kinetic Analysis of Irreversible Consecutive Reactions BY JEN-YUANCHIEN The kinetics of consecutive reactions have been of considerable interest since even simple chemical change may go through a number of intermediate steps. When the rate constants of each step are comparable in magnitude, the integration of the kinetic equations presents considerable difficulties. Especially when reactions of high order in the intermediate steps are involved, the resulting system of non-linear differential equations can only be solved by successive approximations. Hill' has recently outlined a general scheme of such methods, but the amount of labor required would be tremendous because of the fact that the rate constants themselves are the unknown parameters sought. In this paper, certain types of two step irreversible consecutive reactions are considered, where the solution in closed form has been found. These solutions serve as an extension to the summary of formulas compiled by MoelwynHughes.2 (1) T.L. Hill, Tars JOURNAL, $4, 466 (1942). (2) E. A. Mdwyn-Hughes. "Physical Chemistry." Cambridge University Press, Appendix 9, 1940, pp. 633-641.
Integration of the Kinetic Equations I. Uni-unimolecular Reaction.-The general solution for a unimolecular reaction chain of any number of steps has been found.8 For the particular case of a two-step reaction ki
A+B+C
ka (1)
starting with a. mole of A, the solution may be expressed in terms of a dimensionless variable 7 and a dimensionless parameter K in the following form, where A, B and C are concentrations of A, B and C, respectively, a t t. A =
C=
\
QT
a0
-A -B
where T
= e-kI1 and
K
= kn/kl
I
11. Uni-bimolecular reaction.-Two separate cases are to be considered, one with a single (3) H.Bateman, Proc. Camb.Phil. Soc., 16.423 (1910).
KINETICANALYSIScw IRREVERSIBLE CONSECUTIVE REACTIONS
June, 1948
starting material and the other with two starting materials.
t = 0: 1
:
ao x1
0
0
x2
X:
2257
The complete solution of the problem is summarized as A = aoT 1
These kinetic equations describe the processes :
!
&l/dt = -klxl thz/dt = kixi kzx: tdxs/dt = 1 / 2 k i ~ :
(2i) (2ii) (2iii)
-
Integration of (2i) gives XI
(3)
= aoe-kll
and then the Riccati equation (2ii) becomes dxz/dt = aokle-klt
- kzx:
y
The kinetic equations are
{ dy/dt d(ao
- x)/dt
- x) - y)(do - y) = -kl(ao
k2(x
x = ao(l
(5)
- aoklkze-kli u = 0
A further change of the independent variable t to r = e-klj yields an equation of the Bessel type4
(99 (9ii)
- e--Llt)
then (9ii) becomes dy/dt =
kz(a0
- aoe--L1*- y)(& - y)
This is also a Riccati equation which in general cannot be solved. However, it is known that if one particular solution y1 is found, the equation can be reduced to linear form by the transformation y = y1 * l / v . Following Bruins’ y1 = a0 is obviously a solution, and Y = do
The solution of (6) is u = aJo(2idG)
0
do-y
Integration of (9i) gives
1 du(t) kzU(t) dt
which transforms the non-linear first order equation (4) in x2 into a linear second order equation in u(t) d2u/dt2
do
0 x-y
a0
ao-x
(4)
On using the transformation zz(t) =
t = 0: t :
- l/v(t)
leads to an equation for o(t)
+ P’iHi1)(2id/KT)
a,
where a,O’are a.rbitraryconstants, i = and Jo,Hi1)are Beasel functions of the first and third kind of order zero respectively. Then, by the recurrence relations of Besscl functions6
- aok& - e-kli) v
z
= kr
where XO = (UO - &)/no The solution of this linear differential equation in o is where 7
=
e+‘,
K
= aokz/kl
and and hence the solutions for xz is, by (5)
r(n,x) =
where the arbitrary constant 0 is determined by the initial condition xz(0) = 0, which gives (7ii)
For xs(t), use is made of (2ii) and (aiii), i. e. v -
( 4 ) Cf.T. v. K&rrn&n and M. A. Biot, “Mathematical Methods in Engineering,” McGmw-Hill Book Co.,Inc., New York, N. Y . , 1940, Chapter XI, p. 66. (5) Cf.G. N. Watson, “A Treatise on the Theory of Bessel Functions,” Cambridge University Press, Second Edition, 1944, pp. 45, 74. (6) Tables of Jo(i+), -iJi(i#), iH!’)(ix) and -Hi‘)&) are given in E. Jahake and F. Emde, “Tables of Functions,” Dover Publicationa. New York, N. Y.,1943, pp. 224-229, 236-243.
I’
fm-l
e-€ dE
(1Oii)
is the incomplete r-functionv8 Evaluation of the arbitrary constant y by the initial condition gives
Hence A =
a07
C=do-D B =ao-A
I‘
-C
’
where T
=
e-klt,
K
= aoka/kl, XO = (a,
-
- &)/do
(7) E. M. Bruins, Rec. Iroo. ckim., 69, 799 (1940). (8) Values of I(x,n) lYn,x)/r(n) have been tabulated by K. Pearson, “Tables of the Incomplete r-function,” His Majesty’s Stationery Office, London, 1922.
JEN-YUANCHIEN
2258
For the particular case when a0 = do,it is necessary to return to (1Oi) and in this case o =
ex7
[f(- ~ i ( - - Y l+)
Vol. 70
Thus the complete solution12is
+ y']
~i(-a)j
a G
where
211-
c where
= 1
I
is the exponential integralIeand a is some positive number.1° :Evaluation of y' by the initial condition gives =
+
e-6
e-rr
+
Ki - - ~ i ( - K i )
and
J =
e-K(r-1)
+dit,
- JI
*
(17)
= kr/di
K
-E~(K))
Ke-"(Ei(m)
(12')
E~(--K)J
III. Bi-unimolecular Reaction.-One of the two possible cases of consecutive bi-unimolecular reactions is ki kz 2A+B+C t =
0:
0 y
ucl-2xx-y
:
t
(111)
0
a0
The kinetic equations are
t
- 2%) --
d(ao
-ki(ao
dt d y / d t = ki(x
Integration of (13i) gives uo
- 2~ = ~
- 2~)'
- y)
(13i) (13ii)
+
/ ( l aakit)
(14)
Integration of (Mi) gives
Then (13ii) becomes
2+
a0 1 + U<
(19)
5 (1 + aoklt)2 - k2Z:
(20)
xi =
kly = --d kikz aokit 2 1
+
(1,5)
Then = ki
z&
This linear equation has the solution
dt
This is also a Riccati equationla and inspection shows that The indefinite integral in the above expression for
xa = m
y can be evaluated by a change of variable. Let 'r =
1
then
+ a&,
and
=
K
cyao
t
is a particular solution, where a is a constant so chosen as to satisfy the equation. Thus
kz/aokl
a =
2 K
(1
+ d m ) , = kz/kl K
Let the general solution be %a =
where the exponential integral"
- 2kaX1~
dv/dt
- Ke-*r
Ei(KT)
+
y'eK(1
u =
- l)]
Evaluation of the constant y' by the initial condition y(0) = 0 gives y =
5 2
[I
-
ka
(21)
The solution to this equation is
And therefore ')' = :2 [I
xa + l / v ( t )
then the equation for er will be
(?-K(r-l)
-
Ke-rr(&(Kl)
- .&(K))]
(16)
(9) "Tables of Sine Cosine and Exponential Integrals," Federal Works Agency, New York, 1940; or E. Jahnke and F. Emde, "Tables of Functions," loc. c k , pp. 6-9, (10) Here the indefinite integral is changed to a definite integral with a lower limit a by changing the additive constant y' to avoid the difficulty of infinite Ei(0). (11) In Jabnke and Emde, roc. d t . , this function is tabulated under the notation iZl:x).
7w+1[-
&T-P
+ .]
where T = 1 4-aoklf and p = 2 a ~ 1 = 4-2;. (12) The solution of this type of reaction bas been previously reported by A. A. Bdandin and L. S.Lecbenson, Corn#. rend. U R S S . . SB, 22 (1943), but apparently they gave a wrong sign on J for both B and C. (13) The equation (20) can also be solved by the following transformation xt
=
1 du kiic dt
1
-P
+ uoklt = e*
which leads to a second order linear equation in effidents:
dau -dsa
-ds d'-
KU
= 0,
K
I(
with constant EO-
= kz/kl
KINETIC ANALYSIS OF IRREVERSIBLECONSECUTIVE REACTIONS
June, 1948
Integration of ( M i ) by using (1%) gives
A = a0/7 7~e-r
D-do
l-xo 1-(T)(hoK)KdoK[r(K+
C = & - D B = I/l(ao - A )
2
where Thus the complete solution14is
2A
(b) t = ,O: 1
:
7
= 1
it1
-+ B
a0
uo-2x
B
kz +D + C
0
& do-y
x - y
The kinetic equations are d(a0
- 2x)/dt
(IVb)
0 y
- kl(ao - 2x)
=
( d y / d t = kz(x - y ) ( & - Y)
(25i) (25ii)
Integrating (25i) gives a0
- 2x
=
an
~
1
+ aokd
Then (25ii) becomes
As in the case Ilb, y1 = do is obviously a solution of this Riccati equation, so y = do
2259
- l/o(t)
'leads to a 1inea.r equation in v
The solution for v is
(14) The solution of this problem has been considered by J. Hirniak, Acta Physdco-chimica U R S S , 14, 613 (1941). He arrived a t essentially the same result as (24) but through an entirely different and lengthy derivatlon. However, his b ditTers from of this paper, which should be identical, because he overlooked the fact that four molecules not two of A are consumed to give one molecule of C.
1,&7)
-r(K+
1,h~K)) '
-
+ aoklt,
(29)
C K
= k2/2k1,
Xe
= (a0
- 2do)jao
I
Discussion All the solutions obtained for the two step irreversible consecutive reactions considered above have been expressed in terms of a dimensionless variable 7 and some dimensionless parameters defined for each type of reaction in question. The general behavior of these solutions is shown in Figs. 1 and 2 for hypothetical reactions with K = 0.5. For the reaction type IIb and Ivb, equal initial concentrations for A and D are assumed. In these plots, 1/1 aok1.f is chosen as abscissa. This particular choice appears to be most suitable to show the different characters of the reaction mechanism. It is obvious that if the h t step is bimolecular, the decay curve for A will be linear. A unimolecular step will result in a change of the direction of curvature in the formation curve of both B and C, while no inflection point will be found on the curve if the steps involved are bimolecular. Therefore, such a plot will give directly an insight into the reaction mechanism. A linear time scale is also attached on the top of each curve. Experimentally the aim is always the evaluation of the rate constants of each step and thus an assignment of possible reaction mechanism. In view of the difl6culties involved in a simultaneous evaluation of the two rate constants, it appears desirable to follow the decay of the initial compound A. This will give the rate constant k1 immediately. The evaluation of K can then be made, by comparing the experimental formation curve of either the product C or the intermediate B, or the decay curve of the other starting compound D, to
+
JEN-YUANCHBN
2260 kit.
0
0.4 0.60.81.0
0.2
2
3 45
10
m
1.0
an empirical unimolecular equation with an apparent induction period. For example, for a0 = ,2 and K = 0.5, the theoretical curve and an empirical equation like
vi 0.8
8
3
*-
VOl. 70
-
C’ = 1
0.6
(30)
agree very well from 15 to 90% completion of the reaction as shown in Table I. This case corresponds to an apparent unimolecular rate constant of approximately half the true value of kz. As K decreases the agreement will be better, and finally when aok1-i 03
0.4
8
0 0.2
0 1.o
0.8
Fig. 1.-Uni-uni-
0
e-0.Z58(r-l.j0)
0.6 0.4 0.2 0 [I k i t ] - ’ . and uni-bi-molecular reactions, a. = 1, k = 0.5.
+
aokit. 0.4 0.6 0.81.0
0.2
2 3 4 5 10m
Therefore
1.0
Ea 3
3 c
0.8
which is the unimolecular formation of C from B as would be expected.
0.6 0.4
TABLE I COMPARISON OF EQUATIONS (17)AND (30) FOR a. = 2 AND
8 0.2 0 1 .o
Fig. 2.-Bi-uni-
ii
0.6
0.8
+
0.4
0.2
0
[I ~ o k i t 1 - 1 . and bi-hi-molecular reactions, a0 = 2, k = 0.5.
a set of computed curves. It might be mentioned here that for the special case of reaction type I, Swain16has devised a graphical method of evaluating kl and kz from a single experimental C). curve of (a0 - A Experimental studies on the kinetics of consecutive reactions have been handicapped by their complexity. The present analysis shows definitely that a separate determination of the kinetic behavior of the initial step leads to an essential simplication of the problem. With the value of a0 chosen as 1 or 2, depending on the reaction mechanism of the first step of the reaction, the graphs shown in Figs. 1and 2, with different values of K can be constructed and can then be used to analyze all experimental data in any given case. This procedure amounts to plotting all experimental values of concentrations relative to the concentration of the initial compound A and on a time scale of (1 aoklt)-l. A similar procedure has been used for the study of concurrent consecutive reactions.lB It is interesting to note here that for the reaction type 111, the formation curve of C as computed from equation (17) could well be represented by
+
+
(15) C. 0. Swain, TISIS JOURNAL, 66, 1696 (1944). (IS) C. Potter and R. R. McLaughlin, Can. J . Research, B25, 405 (1947).
I
C
1.0 1.4 1.8 2.2 2.6 3.0 3.5 4.0 5.0 6.0 8.0 10.0 20.0
0 0.029 .092 .166 .242 .314 .398 .473 .593 .682 .794 .a55 .943
= 0.5
C’
0.073 ,162 .243 ,316 .398
.468 .588 .680 .807 .a84 .991
A%
-21
-3.0 f0.4
+0.6
*o.o -1.1 -0.8 -0.3 $1.6
+3.4 +5.1
In conclusion, it is hoped that the solutions of irreversible consecutive reactions here presented will be useful for the study of many organic reactions. Some of the results appear to be particularly simple to apply. The only limitation will lie in the reversibility of the reactions. For such cases the kinetic equations have not been integrated a t the present moment. Acknowledgment.-The author is indebted to Professor R. E. Langer of the Mathematics Department, University of Wisconsin, for valuable suggestions and to Professor G. Glockler for a discussion of the topic involved.
summary 1. Solutions in closed form for some two step irreversible consecutive reactions are presented, with the results expressed in terms of a dimension-
June, 1948
9,lO-DIoxo-13-CYANOMETHYL-5,8,9,10, 13,14-HEXAHYDROPHENA"THRENE
less variable and dimensionless parameters. Unibi-, bi-uni- and. bi-bi-molecular reactions are the cases considered. 2. Experimental evaluation of the rate constants is discussed. A separate determination of the rate of decay of the initial substance is desir-
2261
able for the interpretation of the mechanism and the evaluation of rate constants. 3. Practical applications of the results are limited only by the possible reversibility of the reaction steps involved, IOWACITY, IOWA
[CONTRIBUTION FROM THE MARIONEDWARDS PARK
RECEIVED DECEMBER 9, 1947
LABORATORY OF BRYNMAWRCOLLEGE]
T h e Synthesis of Ring Systems Related to Morphine. I. 9,10-Dioxo-13cyanomethy1-5,8,9,10,13,14-hexahydrophenanthrene BY MARSHALL GATESAND WILLIAM F. NEWHALL A synthesis of the ring system (I) present in morphine and its close relatives has not yet been achieved although a number of interesting at-
/\
111
1
C,H~/-CN
/\ II
II \/\/I
I I
1
I1
II
\/
I
tempts have been reported.' We have been engaged for some time in an attempt to synthesize derivatives of such a ring system which might be compared with certain degradation products of the morphine alkaloids. Such a synthesis would offer a rigorous solution to the question of the point of attachment of the ethanamine side chain of morphine. The appearance in recent months of several publicationsla*c-d bearing on this general problem has prompted us to offer our results for publication. We have developed a convenient synthesis for 9,lO - dioxo - 113 - cyanomethyl - 5,8,9,10,13,14hexahydrophenanthrene (11) which appears to be flexible enough to allow the introduction of substituents into rings I and I11 by suitable choice of starting materials. The starting point for the preparation of this substance i s the 4(carbethoxycyanomethyl)-1,2naphthoquinone of Sachs and Craveri,* which is available from ammonium 1,2-naphthoquinone-4sulfonate by an improved procedure in 91% yield. This material is reduced, hydrolyzed and decarboxylated in one step giving 4-cyanomethyl-1,2(1) See, for example, (a) Holmes' recent [Tms JOURNAL, 69, 2000 (1947)] extension of (b) Fieser and Holmes' [ibid.. 60, 2548 (1938)) work; also the recent papers of (c) Homing, ibid., 69, 2929 (1947); (d) Newman, ibid., 69, 942 (1947), (e) Grewe, Bel., 76, 1072, 1076 (1948); (1) G h a h and Robinson, J . Chctn. SOC.,606 (1944); (E) Qsnguly, S&nu owd Culture, 0, 819 (1941); (h) Koelach, Tsxs J O ~ N A L ,07, 669 (1946). and others. (4) Sacha and Csavmii, &I., SO, 8684 (1906).
naphthohydroquinone in 91% yield. Dichromate oxidation of this hydroquinone in glacial acetic acid affords 4-cyanomethyl-l,2-naphthoquinone in 83% yield. 0
I
I
/I
II1
\/\/ I CHo
I
CN
The quinone I11 condenses readily with butadiene in acetic acid to give the diketone 11. Excellent quality adduct is readily obtained in 56% yieldn4 Its formulation as I1 appears to be required by the fact that its azine IV, easily prepared by condensation with o-phenylenediamine, yields 1,2,3,4-dibenzophenazine (9,lO-phenanthrenequinone azine) (V) on distillation with zinc dust. Experiments are in progress on the reduction of I1 by a variety of methods. We hope to effect a (3) The position of the cyanomethyl group in this substance has been conclusively demonstrated by Miss Elizabeth R. Carmichael, working in this laboratory on another problem, by hydrolyzing and decarboxylating the azine of this substance to 4-methyl-l,2-naphtho61, 417 (1939)l phenazine [Fieaer and Bradsher, Tar8 JOURNAL, which waa compared with an authentic aample kindly furnished us by Profusor Louia F. Fiesex of Huvprd University. (4) Compare the work of Fiescr and Bradaher, THIS JOURNAL, 61, 417 (1939). in which 2,3-dimethylbutadiene was shown t o add rlowly to 4-bcniyl. end 4-dionrbethorymeCh~1~ 1,?-aaphthoquio-MI.
