March 5 , 1961
THERMODYNAMIC COUPLING OF CHEMICAL REACTIONS
1029
[CONTRIBUTION FROM THE DEPARTMENT OF CHEMISTRY AND CHEMICAL ENGINEERING, STANFORD UNIVERSITY. STANFORD, CALIFORNIA, AND THE DEPARTMENT OF CHEMISTRY, THEUNIVERSITY OF KANSAS, LAWRENCE, KANSAS]
On Thermodynamic Coupling of Chemical Reactions BY FREDERICK 0. KOENIG,FREDERICK H. HORNE’AND DAVID M. MOHILNER RECEIVED AUGUST10, 1960
Any set of net chemical reactions in which thermodynamic coupling occurs can always be transformed by linear combination into a thermodynamically equivalent set in which such coupling does not occur, and conversely. Therefore the physical interest of thermodynamic coupling considered without reference to reaction mechanism is doubtful.
Introduction We consider a closed system of u constituents, Q1-.Q., whose mole numbers, n1--nu,are changing spontaneously at rates accountable for in terms of p linearly independent net chemical reactions, whereby 2 5 p < u, and the chemical equations of the reactions are, in De Donder notation
of the system; we shall denote it by P. Since T and diS/dt are each always positive or zero and the latter case does not concern us here, we have P =
5
(6)
Ajvj>O
j-1
This thermodynamic result admits of the possibility that one or more of the products Ajyj. be negative (at least one must of course be positive). VijQi 0,j ~ . . . p (1 1 In this case it is said that thermodynamic coupling ill occurs in the set of reactions represented by eq. 1 This means that the reactions themselves have and that each member of the set is a coupled rates, vl...wp,which are given as the solutions of the or a coupling reaction, according as its product equations2 Aivj is negative or positive. It is appealing to dn think of the coupling members of the set as driving vijvj = d t‘i i = 1.. , u each coupled member in the direction opposite j- 1 The existence of these solutions in all cases is to that suggested by its affinity. These ideas guaranteed by the linear independence of the re- are all well known, frequently having been set forth by their originators, De Donder and the memactions, i.e., of the p sets of u coefficients each bers of his school.* These authors have moreover v11. .vu1 claimed for thermodynamic coupling a high degree of physical interest on a number of grounds, not (3) the least of which is that such coupling is “an Y I P . . .YO essential feature of living systems.’’6 It is our purpose to call attention to a fundaWe suppose furthermore that the temperature T and the chemical potential pi of each constituent mental feature of thermodynamic coupling hitherto are a t each instant constant throughout the system. apparently overlooked. This feature arises from The reactions represented by eq. 1 are then known the fact, in itself entirely familiar, that the nature from thermodynamics to have affinities, A1.,.Ap, of the system determines the number, p , of the net chemical reactions needed to account for the defined by dn. changes -’ in accord with eq. 2 but leaves the coAi = VijPi,j 1...p (4) dt i=-1 efficients 3 indeterminate to the extent of an We suppose finally that within the system no arbitrary linear combination. In other words, irreversible processes are occurring other than the any set of chemical equations of type 1 derivable aforementioned changes of the mole numbers, by linear combination of the members of a particudni/dt.a The system is then known from thermody- lar set satisfying eq. 2, likewise satisfies eq. 2. namics to have a rate of entropy production, From this infinity of equivalent sets of reactions diS/dt, given by moreover, thermodynamics offers no criterion for a particular choice; this too is well known. What seems to have been overlooked is that as we pass
2
2
.
2
-
P
The product T
is called the power of irreversibility
(1) National Science Foundation Codperative Graduate Fellow. (2) We shail take the v i j to have the dimensions of mass and t o be measured in moles. The rates v j then have the dimensions of reciprocal time. (3) The system is to be thought of as subject to any further conditions compatible with those just explicitly laid down. For example, we could require: (i) that the system be homogeneous, or alternatively, heterogeneous with some particular number of phases and (ii) that volume and temperature of the system be particular functions of time
V = V ( t ) ,T
T(t)
(44
and so forth. Conditions of type 4a are sometimes called “constraints.” All such further conditions are however irrelevant to our ensuing discussion,
from set to set, although the sum
Ajvj
remains
j-1
invariant, the number of the products Ajvj having a given sign, and so the proportion of coupled to (4) T. De Donder, “L’A5nit6,” Paris, Gauthier-Villars (1927); Bull. Acad. Roy. Bclg. Cl. Sci., 23, 770 (1937); T. De Donder and P. Van Rysselberghe, “Thermodynamic Theory of A5nity;’ Stanford University Press, 1936; P. Van Rysselberghe, BUZZ. Acad. Roy. Bclg. CI. Sci.. 22,1330 (1936); 23,416 (1937); J . Phrs. Chcm., 41,787 (1937): I. Prigogine and R. Defay, “Thermodynamique Chimique,” 2 vols., Liege, editions Deoser, 1944, 1946; sec. ed. (1950); I. Prigogine and R. Defay, “Chemical Thermodynamics,” translated by D. H. Everett, Longmans Green, New York, N . Y.,1954; I. Prigogine, “Thermodynamics of Irreversible Processes,” C. C. Thomas, Springfield, Ill., 1956. (5) I. Prigopine and R. Defay, ref. 4, P. 42.
1030
F. 0. KOENIG,F. H . HORNEAND D. M. MOHILNER
coupling reactions, in general vary.6 We shall show that it is always possible by linear combination to transform any set of reactions with thermodynamic coupling (one or more Ajuj < 0 ) into a set without (all Ajvj > 0), and c~nversely.~Thus the presence or absence of thermodynamic coupling in a given system undergoing chemical change will be seen to depend not upon the physico-chemical nature of the change but upon the choice of the chemical reactions taken to describe it. This conclusion plainly compels us to revise downward the aforementioned estimate of the physical interest of thermodynamic coupling. Two Reactions.--We shall treat first the case p = 2. We consider a pair of reactions with the chemical equations
2
Vl> V2>
0
Ai < 0 A,> 0
(15.1) (15.2)
Equation 13.1and ineq. 15.1 imply AI’ < 0
(16)
Hence for the desired ineq. 10.1 to hold, it is necessary and sufficient that VI’
0 P = AlVl Ap*> 0
(8.1) (8.2) (8.3)
+
We shall show that it is always possible to find a real number, a , such that the equivalent pair of reactions with the chemical equations
f: uilQi
(9.1)
= 0
i=l (vi1
(14.1) (14.2)
This choice, in conjunction with ineq. 8.1 and 8.2, implies
c
2
i=l
0 0
a> ui1Qi =
i =1
0
Vol. 83
+
aYi2)Qi
=
0
(9.2)
VZI
>0
(19)
Hence for the desired ineq. 10.2 to hold, it is necessary and sufficient that A*’> 0
(20)
But this condition, by virtue of eq. 13.2 and ineq. 15.1 and 15.2, is seen to be equivalent to
A > IAIl CY
Ineq. 8 3 14.1, 14.2, 15.1 and 15.2imply
Hence ineqs. 18 and 21 can hold simultaneously and constitute necessary and sufficient conditions for the desired ineq. 10.1 and 10.2. Thus CY will produce the desired transformation if and only if it lies in the range
is free of coupling; this means
>
AI’v~’ 0 Az’~2’> 0 P = Ai‘al‘ A ~ ‘ v ? ‘ >0
+
(10.1) (10.2) (10.3)
where the primes refer to the new pair of reactions. Proof: Equation 2 requires V i l V 1 + Vi2V2 = Y i l V , ’ ( V i 2 + olYil)ii’2’, i = 1 . . .u (11) whence by solving simultaneously the two equations corresponding, respectively, to any two values of i, we find that the rates transform according to
+
Vlf
= Dl
-
’ = z’2
ol‘1’2
t12
(12.1) (12.2)
From eq. 4, 7.1, 7.2, 9.1 and 9.2 we find that the affinities transform according to A,’ = Ai Ai’ = A2
+ aAi
(13.1) (13.2)
It is readily verified that these equations of transformation leave P invariant, as required by eq. 8.3 and 10.3. We now suppose, as we evidently may without loss of generality, that eq. 7.1 and 7.2 are written in the directions which make (6) If we consider the sets having a given reaction, say the k’th, in common, we see from eq. 4 t h a t the affinity A k is invariant and from eq. 2 t h a t the rate uk must in general vary from set to set. This variation can, as we shall see, change the sign of % and so of A k u k . (7) T. De Donder, ref. 4 (1937), and I. Prigogine and R. Defay, ref. 4 , (19441,p. 77, have considered transformations of the kind involved in our discussion but stopped short of our result.
and this condition can always be met. We note incidentally that since ineq. 14.1 and 14.2 imply v 1 , / ~ 2 > 0, ineq. 23 implies cy > 0. For example, the synthesis of urea in liver, when written 2XH3 COz = (iYH2)zCO t- HzO
+
s CaH12Os +
0 2
= CO2
+ H20
has been citeds as a case of thermodynamic coupling, with A1 = -11 kcal., A2 = 115 kcal., V I = 1.4 X lo-* m i n - ’ , up = 6.0 X 10-8min.-1, P = 6.7 X 10-6kcal. min.-’
But this set can be transformed into one in which Al’vl’ and AZ’V~’ are both positive: ineq. 23 yields 10.4 > a > 0.24; We may therefore choose CY = 1,which gives
+ H2O 6 C6H12O6 + 2NH3 = (NH2)2CO + 2H20 2”3
+ COz
(KH2)pCO
1
with A,’ = -11 kcal., A,‘ Inin.-], = 6.0X ti2‘
=
104 kcal., Inin.-’,
P
VI’
=
-4.6
= 6.7 X
X rnin.-’
(8) T. De Donder and P. Van Rysselberghe. ref. 4, (1937); I Prigogine and R. Defay, ref. 4, (1944), p. 74; see also H. Borsook and G. Keighley, Proc. N u l l . A c a d . S c i . U . S., 19, 626, 720 (1933).
March 5 , 1961
THERMODYNAMIC COUPLING
The converse of the proposition just proved and illustrated is also true. We consider a pair of reactions with the chemical equations
5
vil’@
=0
(24.1)
ViP’Qi
=0
(24.2)
OF CHEMICAL
REACTIONS
1031
set of real numbers, al’..czp-l,such that the equivalent set of reactions with the chemical equations
2
0 , j = 1 . .. p - 1
vijQi =
(32.1)
i-1
i l l
(32.2)
c
i=l
i=l
is free of couplingg; this means
We suppose this pair to be free of thermodynamic coupling; this means A l ‘ ~ l ‘ >0
(25.1) (25.2) (25.3)
As’vZ’> 0
+
P = Al’vl’ Az’vz’> 0 It is then always possible to find a real number, a‘, such that, of the equivalent pair of reactions with the chemical equations
2 I
=0
vil’Qi
(26.1)
i=l C
+
(viz’
i= 1
j=l
a’vi1’)Qi
=0
P
=
Aj’~j’>
0
P = Alvl + AZVZ> 0 where the unprimed letters refer to the new pair of reactions. The proof of this proposition is so readily constructed by inverting step by step that given of the previous one, that we shall omit it, and state only the necessary and sufficient conditions on a’ to which it leads. We suppose eq. 24.1 and 24.2 to be written in the directions which make VI’
(28.1) (28.2)
0
We then find that a‘ will produce the desired transformation if and only if
0
(33.2)
j=l
Proof: By considerations parallel to those leading to eq. 12.1 and 12.2, we find (34.1) ~ j = ’ ~j - a j ~ p , j= 1 . . . p -v 1 =
Up’
(34.2)
vp
and by considerations parallel to those leading to eq. 13.1 and 13.2, wefind Aj’ = A j , j = 1 . .. p - 1 (35.1)
(26.2)
the first member is coupled to the second; this means
(33.1)
P
A,‘ =
Aivi
0 , j = 1 . .. p
Aj’vj’>
+
A p
P--l
(35.2)
ajAj
j=l
It readily verified that these equations of transformation leave P invariant. We now suppose that eq. 30 are written in the directions which make Uj> 0 , j = 1... p (36) This choice, in conjunction with ineq. 31.1 and 31.2, implies A j < 0,j= 1 . . . T Aj> 0,j = 7 1 . .. p
(37.1) (37.2)
+
Equation 35.1 and ineq. 37.1 imply Ai‘< 0,j = l . . . ~
(38)
Hence for the first 7 members of the desired set of inequalities (33.1) to hold, it is necessary and sufficient that
0 , j =
7
+ 1..
. p
+
=
AjVj>
0
Vj‘>O,j
=
7 +
1
I
.
.
%> aj,j =
7
+ 1..
UP
(41)
p
- 1
(42)
. p
-
1
(431
(9) In the language of linear transformation theory this proposition m a y be stated as follows: it is always possible t o transform the set (30) with coupling into a set without, by a matrix of the form
(31.3)
We shall show that it is always possible to find a
1
This condition, by virtue of eq. 34.1 and ineq. 36, is seen to be equivalent to
(31.1) (31.2)
j=l
-
Hence for members no. 1 to p - 1 inclusive of the desired set of inequalities 33.1 to hold, it is necessary and sufficient that
P
P
(40)
Equation 35.1 and ineq. 37.2 imply
i=l
Ajvj>O,j=
= l...T
VP
The General Case.-The general case of p reactions, with 2 5 p < u,is resolvable by a treatment modeled upon that just accorded the case p = 2. We consider p reactions with the chemical equations
2
aj>
0
o . . .
0 O . . . @
1
1032
F. 0.KOENIG,F. H. HORNEAND D. M. MOHILNER
Equation 34.2 and ineq. 36 imply (44)
0
VP’>
Hence for the p’th member of the desired set of inequalities 33.1 to hold, it is necessary and sufficient that Ap’> 0
vj
-
c
= I...T
+-->uj,j
IAil
UP
c .= - --Bj
ai>’ UP
Ai
T
c=
f
l . . . p
-1
P
~
(P
Accepting this result for the present, we note that, owing to ineq. 31.3 and 36 0
We suppose this set to be free of thermodynamic coupling; this means
(49)
(56.1)
2 Aj’vj‘> 0
(56.2)
P
=
j-1
It is then always possible to find a set of real numbers, a1’. . CY'^-^, such that, of the equivalent set of reactions with the chemical equations = 0, j = 1 . . . p
Vij’Qi
T+1...P-1
(50)
(51.1)
and this condition can always be met. It remains only to show that ineq. (46.1) and (46.2) constitute a sufficient but not necessary condition for ineq. 45. By virtue of ineq. 37.1 and 37.2 and a few obvious rearrangements, we transcribe ineq. 46.1 and 46.2 to the single statement VP
(vi
-
crj~p)>
(57.1)
j= 1
the first 7 members are coupled to the remaining p - 7 ; this means Ajvj
It follows from ineq. 49 that ineq. 40 and 46.1 can hold simultaneously and from ineq. 50 that ineq. 43 and 46.2 can hold simultaneously. Thus the aj will produce the desired transformation if they lie in the ranges
- 1- A j
-1
i-1
and furthermore, owing to ineq. 37.2
C
Aj’vj’> 0, j = 1.. . p
i-1
&>o,j=
(55)
(48)
whence
L
= 0,j = 1...p
vi,’Q,
i=l
(46.2)
(47)
- 1)VP
c>
2
(46.1)
where Cis defined by 4l
are not a necessary one follows from the obvious fact that ineq. 53 does not imply ineq. 52. The converse of the proposition just proved is also true. We consider a set of reactions with the chemical equations
(45110
We shall show later that for ineq. 45 to hold, the following conditions on the aj are sufficient though not necessary
Vol. 83
0 , j = 1.. . p
-1
(52)
From this we deduce by summation
E
2
= T
1.. .T
+l...p
(58.1) (58.2)
0
(58.3)
Ajvj>
j= I
where the unprimed letters refer to the new set of reactions. The proof of this proposition is so readily constructed from that of the preceding one that we shall omit it and state only the sufficient conditions on the aJ’(j= 1.. . p - 1) to which it leads. We suppose eq. 55 to be written in the directions which make Oj’ < 0 , j = 1.. .T vi’> 0, j = T 1.. . p
(59.1) (59.2)
+
We define C’by C’
P
=i
(P
(60)
- I)%’
and then find that the aj’ will produce the desired transformation if
,
< %, j
=
I . . .T
(61.1)
VP
By virtue of eq. 47, 34.1, 34.2 and 35.1 this becomes (54) j=l
But the left side of this inequality is seen by eq. 33.2 to be equal to Apt. Hence ineq. 46.1 and 46.2 are a sufficient condition for ineq. 45. That they
-
P =
ai‘
(531
- -
O,j
(IO) Our treatment of the general case, which up to this point has been completely parallel to that of the case p 2, will now cease to be so, in that we shall for simplicity be content with a condition on the ai, namely, ineq. 46.1 and 48.2, which will be seen t o be sufficient for ineq. 45 but not necessary, whereas for the analogous ineq. 20 we readily found a condition on a,namely, ineq. 21, which is necessary and sufficient.
p
-1
(61.2)
a condition which can always be met. Conclusion.-It therefore appears that, in any case in which we know the values of dni/dt and wi (i 7 1. . .a), we can confer or revoke thermodynamic coupling by properly choosing a set of net chemical reactions out of an infinity of permitted sets, which is to say, by a stroke of the pen. It follows that without a further criterion for choice out of this infinity, thermodynamic coupling is neither an essential feature of any special class of systems dead or alive, nor an explanation of anything. Thermodynamics, as already said, supplies no such criterion. If any exists, the only possible source would seem to be the theory of reaction mechanism. Thus without reference to the latter,
March 5, 1961
BENZENE CHEMISORPTION
further concern with thermodynamic coupling would seem to be useless. Of what use it might be in conjunction with reaction mechanism remains an open question.
[CONTRIBUTION FROM THE CHEMICAL
ON
1033
NICKEL CATALYST
Acknowledgments.-The
authors take pleasure
in thanking Professors William J. Argersinger, Jr., and Richard J. Bearman, of the University of Kansas, for their helpful discussions.
LABORATORY OF NORTHWESTERN UNIVERSITY, EVANSTON, ILLINOIS]
The Mechanism of Benzene Chemisorption on a Supported Nickel Catalyst BY J. A. SILVENTAND P. W. SELWOOD RECEIVED AUGUST15, 1960 A further study of benzene chemisorption on a nickel-kieselguhr catalyst by the low frequency ax. permeameter magnetic method has shown, as previously reported, that the process takes place by the formation of approximately six bonds, up t o a temperature of about 120'. This is assumed to mean flat, non-dissociative adsorption. A t higher adsorption temperatures the number of bonds increases rapidly until a t 200" substantially complete dissociation of the benzene molecule is achieved. These conclusions are supported by analysis of the saturated hydrocarbons obtained by flushing the adsorbed benzene with hydrogen and trapping the products. The conclusions are also confirmed by measuring the volume of hydrogen denied access t o the surface by the presence of pre-adsorbed benzene, admitted to the catalyst a t progressively higher temperatures.
Introduction In an earlier investigation by the low frequency a.c. permeameter method,' it was shown that benzene chemisorbed on silica-supported nickel is taken up primarily by the formation of six bonds. This suggests associative adsorption with the plane of the ring parallel to the nickel surface, although various other interpretations are not excluded by the magnetic data alone. The maximum temperature of adsorption, in the earlier work, was 150'. In the present work the observations are extended to 200°, a t which temperature extensive dissociative adsorption may be expected, and the conclusions reached by the magnetic method are confirmed through the use of several other lines of investigation. Experimental
Results
Magnetization-volume isotherms for benzene on nickel are shown in Fig. 1. It previously has been shown2 that the isotherms for hydrogen have a slope which is somewhat dependent on temperature even a t temperatures well above those at which appreciable physical adsorption may take place. This effect seems to be related to the range of nickel particle diameters present in any catalyst sample. Whatever may be the reason for this behavior it is clearly necessary to compare the isot h e m slope of any adsorbate with that of hydrogen at the same temperature, if accurate determination of the number of surface bonds per molecule adsorbed is required. Isotherms for hydrogen are, therefore, shown in Fig. 2. It will be noted that the slopes for benzene rise much more rapidly with Magnetization-volume isotherms were obtained as pre- increasing temperature of adsorption than do those viously described.' All measurements were made on Uni- of hydrogen. versal Oil Products nickel-kieselguhr containing 52.8y0 Table I shows the effect of pre-adsorbed benzene of nickel. The reduction and evacuation of samples, together with purification of reagents have all been described. on the volume of hydrogen which may be adsorbed Subsequent to several runs it was necessary to determine by the nickel a t -78' up to a pressure of 1 atm. the products obtainable by flowing hydrogen over the ad- The data are presented as (1) temperature at which sorbent-adsorbate system. The hydrogenated products were selectively collected in cold traps for analysis in a mass benzene was admitted to the surface, ( 2 ) volume (s.c.) of benzene vapor adsorbed per g. of nickel, spectrometer or, for some samples, by vapor phase chromatography. Conditions of adsorption and hydrogenation (3) volume of hydrogen (s.c.) adsorbed per g. of are given i n connection with the data presented below. nickel at - 7 8 O , (4) volume of hydrogen (s.c.) In one series of experiments it was desired to estimate the area of nickel occupied by adsorbed benzene or its dissocia- denied access to the nickel and ( 5 ) molecules of tion products. This was possible because benzene is not hydrogen denied access to the nickel per molecule The procedure was of benzene vapor adsorbed. hydrogenated over nickel a t - 7 8 O . l t o determine the pressure-volume isotherm for hydrogen Another result, which we believe to be of signifiThe hydrogen then was deover the catalyst a t -78'. cance, is that in every case represented in Table I sorbed by evacuation a t 360', following which a measured quantity of benzene was admitted t o the surface. The the slope of the magnetization-volume isotherm for sample then was cooled to -78' and another pressurehydrogen on a surface covered in part by benzene volume isotherm for hydrogen was obtained. The benzene, or its dissociation products, was thus shown to deny access was the same as that on a bare nickel surface. This was true in spite of the fact that the total of hydrogen to part of the nickel surface. The extent to which this denial of access occurred was, of course, a meas- volume of hydrogen which could be taken up by the ure of the dissociative process which had occurred during nickel was diminished substantially by the presence chemisorption of the benzene. These experiments were of pre-adsorbed benzene. performed over a considerable range of benzene adsorption temperatures. It is well known that adsorbed hydrocarbon In one experiment the benzene was adsorbed a t a lower molecules and their dissociation products are often temperature, then heated, then cooled again to determine quite difficult to remove by hydrogenation in situ. whether dissociative adsorption occurs primarily during the In spite of this enough product was obtained to adsorption step or whether it may occur in a pre-adsorbed intact molecule if the temperature is raised. confirm the conclusions reached on the basis of the (1) P.W.Selwood, THISJOVRNAL, 79, 4637 (1957).
(2)
P. W.Selwood, ibid.. 78, 3893
(1956).
