RET'ERSIBLE R E I C ' I ' I O S S
E T J O H S ~V.~1111EI.L
In a paper b!, TYalker and Hamblj-' 011 the conversion of amnioniiiiii cj-anate into iirea the!. state that it is capable of cay. inathematical proof t h a t i n a rc:,t-i.sibIc nionoiiiolecular reaction the form of tlie differential equation i 5 the same as in the case of a reactioii rnniiing to the eiid in oiie direction, tlie difference being that tlie eynilibriuni point is used instead of the original active mass. I n the ordinar\- noii:enclatiire instead of the dx d-r foriiiiila = k(-\ - .I-) the formula to be used is - = k I (E -.L- 1. dt dt TYa.lker and Hanibl!. do not shon- the relatioilship betneen k :ind ki hiit merely by integration obtain the equation I
log ~=-const, E
- -1'
Thcv foiiiid that in the case in hand the valiies obtained were not constant and the\- therefore proceeded to calciilate the constaiit 011 the basis of the reactioii being bimolecidar, siiiipl\. reiiiarkiiig neglecting as in tlie previous case tlie slow re\-er;t. action 1 1 ~ taking . tlie value of the eiid poiiit as tlie original actil-e ' /
iiiass n.c find the espressioii
I
X
t
E-X
is iiideecl constant ".
~-
In another paper I have gi\-en a inatlieniatical proof of tlie first statemelit xliicli liolcls stricti!. no matter where tlie poiiit o f ec~iiilibriuinmay be, hut in eiiclea\-oriiig in the same n-a? to confirin the formula for a reverse liiiiolecular reaction I foniid that I could riot do so. In tr!.iiig to discover in what iiiaiiner Tl-allicr and Haiiihl!- arriI-ecl a t their coiicliision the following mode uf treating the subject occurred to me. Xssuiiiiiig that the eyiiatioiis suggested are correct. let iis
find n-hat conditions must he fnlfillecl. niolecnlar reaction the equation is
For a reversible iiioiio-
where kI arid k2 are the reaction velocities in the two directions. But according to the assumption dx
-& = lz,
x)
~
where [ is the equilibrium point a i d k- is some constant so far unknown. lye have therefore
8,( A - . x )
-
Izp = /+-(E - x)
or k,X
--
k , x - k,x
-
Equate coefficients of like pon-er of
-
AT2[
;u
k3x.
and we have
K I A = k J and k , -r k,= k , .
It reniaiiis to be seeii xvliether these equations are consistent ancl for this purpose we may make use of the relation between k I a d bo obtained from the conditions of equilibrium where
,ince at equilibriiun the rate of change is zero. gives h,-\ = (k,- /$>)E
T h i s eqaation
which exactly coincides with the value obtained from the t n o equations of condition. It fo1lon.s therefore that in tliii case our aisumption is completelj. justifiable. I n the simplest case of a reversible bimolecular reaction we ha\ e d-I
d~
k, ( A
~
x)? X'p? ~
=k,(E-Z)2
according to oiir assumption.
Expanding
k,A ?- 2 k , X x - k,x' - k,x? = &E'
-
2h,EX
+ k,x?
\\-hence by equating coefficients of like power of x rE. A? =:k,l',2k,,l = 2k,[ and fz, - k , = k.,.
From the first of these eyiiatioiis of condition I2 2
E
k
-1 =
K.
E' aiidfroiii x2
T h i s can be true only when E = iii h \vhicli case the reaction n-ill riiii to tlie end, and when f - = o in \vhich case tlie reaction n-ill not go on a t all. It ~voiildappear from the paper by TT-alker and Xpplej-aid' that. n-liile the change from aiiiiiioiiiiiin cyanate into urea is coilsidered bimolecular the reT-erseaction is taken as nionomolecular. T h i s assiiiiies that
the second
k,
= -,
n.r
;it = k , (.\
~-
.v)
~
rE.&
:= k ,
( E -x ) '
froiu n-hich by expanding and equating as before the equations of conditioii are k,X' = k E'. ? k , . l -7 k , 2k,E. k , = L' 3 ' From tlie first and third of these equations =: *I2 ; therefore = -4. T h e negative \ - d u e is meaningless ; the positive 1-alue as in the other case means that the reaction runs to tlie end and is not rex-ersible. ,liter I liacl worked out the problem as above, I foiiiid that Ostn-ald* has gone into the matter ver>-fully and for a reverse iiionoinolecular reaction gives tlie equation f 2
H e n-orks out C in teriiis of k I , K 2 , and A) and shows incidentall> \vit+iontlaying stress upon the point that this last espressioii i i eqiial to E. T lie reL-ersible bimolecular equation
ma!. be tlirovii into another form which n-ill show to what e s I
Jour, Cheni. SOC.67, ;j3 j rS96). 2, 2 j I ant1 ff irS9;).
' Lelirhuch 11,
tent JTalker and Hambl !-were wrong in their assumption. Expanding and arranging the powers of x
as may be pro\-ed by putting
and solying the equation R If -2 a then kI - Rtlz= kl( I -- a) and the equation he-
k,
comes dX
F=R]([
x
__ ( I I--a
5
-1
a,) ( z -
x ~
I --a
a )
( 1 - 1
)
----.x 1311 a
Since tlie conditions of the experiineiit necessitate that E shall A be positil-e arid not greater tliaii -1the value E == ___- is the 1 7 1 .
a
only possible one under tlie circumstances. T h i s is the value designated E2. Nevertheless the 1-alue E, is calculable and inlist be taken into acconiit. T h e equation of TTalker and Hambly
dzl
was - = k ( s - E,'. where they regard k a5 a constant n-ithont dt showing its relation to the \ elocity coefficients. T h i s form coincide, with the one above onl: when f I : f 2 ?in which case it is easilj seen they are both equal to that is the reaction is not a re.\ ersible one. Ostn-aldKarrives a t a formula which differs froiii that which I have worked out in that he has omitted tlie factor ( I - a). H e quotes from Gnldberg and TTaager2 bnt in modif? iiig their form has left out tlie factor named. Integrating tlie expression
Using ordinar\- logarithnis we have
T h e following table shows the constant obtaiiied by IValker and Hambly's calculation and that worked out according to the preceding formula
45 72
4.4 6.j
i
18.j 16.4
0.00528 O.OOj49
Lehrhuch 11,2, 2j6. Jour, prakt. Cherri. [ 2 ] 19, S I (1S79 I.
0.0003j
0
j
000389
