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Equilibrium relationships for non-equilibrium chemical dependencies
Chemical Engineering Science 66 (2011) 111–114
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Chemical Engineering Science journal homepage: www.elsevier.com/locate/ces
Note
Equilibrium relationships for non-equilibrium chemical dependencies G.S. Yablonsky a,b,, D. Constales c,1, G.B. Marin d a
Parks College, Department of Chemistry, Saint Louis University, 3450 Lindell Blvd, Saint Louis, MO 63103, USA The Langmuir Research Institute, 106 Crimson Oaks Court, Lake Saint Louis, MO 63367, USA Department of Mathematical Analysis, Ghent University, Galglaan 2, B-9000 Gent, Belgium d Laboratory for Chemical Technology, Ghent University, Krijgslaan 281 (S5), B-9000 Gent, Belgium b c
a r t i c l e in f o
abstract
Article history: Received 24 June 2010 Received in revised form 10 September 2010 Accepted 6 October 2010 Available online 13 October 2010
It is shown that equilibrium constants determine the time-dependent behavior of particular ratios of concentrations for any system of reversible first-order reactions. Indeed, some special ratios actually coincide with the equilibrium constant at any moment in time. This is established for batch reactors, and similar relations hold for steady-state plug-flow reactors, replacing astronomic time by residence time. Such relationships can be termed time invariants of chemical kinetics. & 2010 Elsevier Ltd. All rights reserved.
Keywords: Kinetics Transient response Thermodynamics process Batch Time invariants Detailed equilibrium
1. Introduction In presenting the foundations of physical chemistry, the basic difference between equilibrium chemical thermodynamics and chemical kinetics is always stressed. A typical problem of equilibrium chemical thermodynamics (ECT) is calculating the composition of a chemical mixture that reacts in a closed system for an infinitely long time. ECT does not consider time. In opposite, chemical kinetics is the science of the evolution of chemical composition in time. Some selected results of theoretical chemical kinetics are obtained from thermodynamic principles, especially the principle of detailed equilibrium: (a) the uniqueness and stability of the equilibrium in any closed system, see Zeldovich (1938) and the analysis in Yablonskii et al. (1991); (b) the absence of damped oscillations near the point of detailed equilibrium, see Wei and Prater (1962) and the analysis in Yablonskii et al. (1991);
Corresponding author at: Parks College, Department of Chemistry, Saint Louis University, 3450 Lindell Blvd, Saint Louis, MO 63103, USA. E-mail addresses: [email protected] (G.S. Yablonsky), [email protected] (D. Constales), [email protected] (G.B. Marin). 1 Financial support from BOF/GOA 01GA0405 of Ghent University is gratefully acknowledged.
0009-2509/$ - see front matter & 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.ces.2010.10.014
(c) some limitations on the kinetic relaxation from the given initial conditions, e.g., based on the known set of equilibrium constants that determine the equilibrium composition, one can find a forbidden domain of compositions that is impossible to reach from the given boundary conditions. See Gorban et al. (1982, 2006) and Gorban (1984). However, the present dogma of physical chemistry holds that it is impossible to present an expression for any non-steady-state chemical system based on its description under equilibrium conditions, except for some relations describing the behavior in the linear vicinity of equilibrium. The goal of this short note is to announce that, contrary to this ‘dogma’, we have obtained relations of equilibrium type for some non-steady-state chemical systems. This has been achieved for all linear cases (with a general proof) and some nonlinear ones. In a forthcoming extended paper these relationships will be explained in more detail. Based on our results, equilibrium thermodynamic relationships can be considered not only as a description of the final point of temporal evolution, but as inherent characteristics of the dynamic picture. The following are examples to illustrate our statement; in all of them we analyze traditional models of chemical kinetics based on the mass-action law. The processes described by these models occur in a closed non-steady-state chemical system with perfect mixing (batch reactor) or in an open steady-state chemical system with no radial gradient (plug-flow reactor), whose description is
112
G.S. Yablonsky et al. / Chemical Engineering Science 66 (2011) 111–114
identical to the batch reactor, replacing astronomic time by residence time. In this short communication, we shall analyze a combination of data of different thought experiments, e.g.,
a chemical reactor is primed with a substance A only, it is primed with substance B only. In both cases, we shall monitor both concentrations A and B and pay special attention to the dependencies of ‘‘B produced from A’’ and ‘‘A produced from B’’. We shall use the notation AA(t) for the temporal concentration dependence of substance A, given the initial condition (A,B,y)¼(1,0,y), i.e., only A occurs, with normalized concentration 1. Similarly, BA(t) is the concentration of substance B for the same initial condition; AB(t) that of A when at t ¼0, (A,B,y)¼(0,1,y). We found interesting relationships between these dependencies.
CA ðtÞ ¼ 1
1
l1 l2
ðl1 expðl2 tÞl2 expðl1 tÞÞ,
AB ðtÞ ¼
k 1 ðexpðl2 tÞexpðl1 tÞÞ, l1 l2
BB ðtÞ ¼
1 ðl2 ðl1 k2þ Þexpðl2 tÞ þ l1 ðk2þ l2 Þexpðl1 tÞÞ, k2þ ðl1 l2 Þ
CB ðtÞ ¼ 1
1
l1 l2
ððl1 k2þ Þexpðl2 tÞ þ ðk2þ l2 Þexpðl1 tÞÞ:
Again, it turns out that BA and AB are always in fixed proportion: kþ BA ðtÞ ¼ 1 ¼ Keq , AB ðtÞ k1
ð4Þ
and again this is the constant of equilibrium of the A2B reaction. k1þ
k2þ
k3þ
k 1
k 2
k 3
2.3. The cycle of three reversible first-order reactions A $ B $ C $ A In the Laplace domain, defining the symbols
2. Linear cases
s1 ¼ k1þ þ k1 þ k2þ þ k2 þ k3þ þk3 ,
k1þ
2.1. A single, first-order reversible reaction A $ B
s2 ¼ k1þ k2þ þ k2þ k3þ þ k3þ k1þ þ k1þ k2 þ k2þ k3 þ k3þ k1
k 1
þ k 1 k3 þ k2 k1 þk3 k2 ,
By elementary mathematical techniques, for t Z 0, AA ðtÞ ¼
þ þ k 1 þ k1 expððk1 þ k1 ÞtÞ , þ k1 þk1
BA ðtÞ ¼
k1þ ð1expððk1þ k1þ þ k 1
DðsÞ ¼ sðs2 þ s1 s þ s2 Þ, the transformed concentrations AA, BA and AB are given by
,
LAA ðsÞ ¼
þ þ þ þ þ s2 þðk 1 þ k3 þ k2 þk2 Þs þðk1 k3 þ k1 k2 þ k2 k3 Þ , DðsÞ
k ð1expððk1þ þk 1 ÞtÞÞ , AB ðtÞ ¼ 1 k1þ þ k 1
LBA ðsÞ ¼
k1þ s þðk1þ k3þ þ k1þ k 2 þ k2 k3 Þ , DðsÞ
LAB ðsÞ ¼
þ þ þ k 1 s þðk1 k3 þ k1 k2 þ k2 k3 Þ DðsÞ
þk 1 ÞtÞÞ
þ k þ þ k 1 expððk1 þk1 ÞtÞ : BB ðtÞ ¼ 1 þ k1 þ k1
and the ratio of the latter two, by
Comparing BA and AB, it is clear that they are constant in a fixed proportion: kþ BA ðtÞ ¼ 1 ¼ Keq , AB ðtÞ k1
ð1Þ
which is the thermodynamic constant of equilibrium. It is remarkable that this ratio holds at any moment of time t 40, and not merely in the limit t- þ 1. 2.2. Two consecutive first-order reactions, the first being reversible: k1þ
k2þ
A $ B-C k 1
Using, e.g., Laplace domain techniques, the following analytical solutions are obtained: if we define the expressions qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ þ 2 þ þ k1þ þk ðk1þ þk 1 þk2 7 1 þk2 Þ 4k1 k2 ð2Þ l1,2 ¼ 2 and verify that
l1 4 k2þ 4 l2 4 0, l1 4k1þ 4 l2 40, we can write AA ðtÞ ¼
BA ðtÞ ¼
1 ðl1 ðk2þ l2 Þexpðl2 tÞ þ l2 ðl1 k2þ Þexpðl1 tÞÞ, k2þ ðl1 l2 Þ k1þ
l1 l2
ðexpðl2 tÞexpðl1 tÞÞ,
ð3Þ
k þ s þ ðk1þ k3þ þ k1þ k LBA ðsÞ 2 þ k2 k3 Þ ¼ 1 þ k þ k þ k þ Þ LAB ðsÞ k þ k k1 s þðk 2 3 1 2 0 1 3
1
B C 1 B C B1 þC þ þ þ @ A k k þ k k þ k k þsk 1 2 3 1 3 1 2 1þ k 1 k k1þ k2þ k3þ k k 1 2 3
¼
k1þ k 1
¼
k1þ , k 1
where the Onsager relationship k1+ k2+ k3+ ¼k1 k2 k3 was used in the final step. Hence for all s, LBA ðsÞ and LAB ðsÞ are in fixed proportion given by the equilibrium constant (k1+ /k1 ). Since the inverse Laplace transform is linear, the same proportion holds in the time domain: kþ BA ðtÞ ¼ 1 ¼ Keq,1 : AB ðtÞ k1
ð5Þ
Similarly, kþ CB ðtÞ ¼ 2 ¼ Keq,2 , BC ðtÞ k2
kþ AC ðtÞ ¼ 3 ¼ Keq,3 : CA ðtÞ k3
ð6Þ
As an example, we show in Fig. 1 the time dependence of BA/AA, BB/AB and the time-invariant ratio BA/AB, for the case of isomerization of butenes in Wei and Prater (1962, Eq. (129)): 4:623
3:724
3:371
10:344
1:000
5:616
cis-2-butene $ 1-butene $ trans-2-butene $ cis-2-butene:
This behavior is typical of the examples given here: if the system is started from the initial values (A,B,C) ¼(1,0,0), the ratio BA/AA
G.S. Yablonsky et al. / Chemical Engineering Science 66 (2011) 111–114
directly to the time domain:
1 B_A/A_A
BA ðtÞ ¼ KA-...-B : AB ðtÞ
B_B/A_B
0.8 ratio of concentrations B/A
113
B_A/A_B
(B/A) starting from (A,B)=(0,1)
ð10Þ
3. Nonlinear cases 3.1. Nonlinear reversible reaction (forward second order, backward
0.6
k1þ
first order) 2A $ B
B starting from (1,0) divided by A starting from (0,1)
k 1
0.4
In accordance to the mass conservation law, the balance is 0.2
AðtÞ þ 2BðtÞ ¼ 1:
(B/A) starting from (A,B)=(1,0)
0 0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
time Fig. 1. Time dependence of BA/AA, BB/AB and the time-invariant ratio BA/AB, for the case of isomerization of butenes in Wei and Prater (1962, Eq. (129)).
grows at first from 0, eventually reaching the equilibrium value (note that for these parameters BA/AA slightly overshoots the limit). Similarly, when started from (0,1,0), the corresponding BB/AB initially decreases from þ 1 and eventually reaches same limit, viz the equilibrium ratio. But surprisingly the combination BA/AB is constantly equal to the equilibrium value, for all times t 4 0.
In order for the A and B trajectories to reach the same equilibrium, we choose to start them from (1,0) and (0,1/2) respectively. The nonlinear differential equation is dAðtÞ ¼ 2k1þ A2 ðtÞ þ k 1 ð1AðtÞÞ, dt which can be solved analytically as sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! kþ kþ 1 þ 8 1 þ 1 þ tanh tk1 8 1 þ1 2 k1 k1 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! , AA ðtÞ ¼ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ þ k1 k1 kþ 1 þ 8 þ1 þ 4 þ 1 tanh tk1 8 1 þ 1 2 k1 k1 k1 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! kþ 1 þ tk1 8 1 þ 1 1 2 k1 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! , BA ðtÞ ¼ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ þ k k kþ 1 þ 8 1 þ 1 þ 4 1 þ 1 tanh tk1 8 1 þ 1 2 k1 k1 k1 kþ
2 k1 tanh
2.4. The cycle of four reversible first-order reactions k1þ
k2þ
k3þ
k4þ
k 1
k 2
k 3
k 4
A$B$C $D$A Although the expressions become more involved, it is straightforward to verify that in this case the fixed equilibrium proportions also hold, given the Onsager relationship k1+ k2+ k3+ k4+ ¼k1 k2 k3 k4 : kþ BA ðtÞ ¼ 1 ¼ Keq,1 , AB ðtÞ k1
ð7Þ
ð8Þ
with a similar relationship for DB/BD.
kþ BA ¼ 1 ¼ Keq,1 AA AB k1 we see that the denominator involves the A concentrations of both trajectories, AA and AB. Only this can ensure a ratio that equals the equilibrium constant at every time t 4 0.
2.5. Sketch of proof for general systems of first-order reactions We use the terminology and results of Roelant et al. (2010). Let A and B be substances such that a path from either to the other exists. In view of Onsager relations, then there must exist a reversible path from A to B. The denominators of AB and BA in the Laplace domain are the same, but their numerators differ. For AB, the numerator consists of contributions from all forests where A is the root of a tree and B is in that tree. Let Z, X, Y denote other nodes as in the denominator of the left-hand term in (9): Z-A-X-B’Y kA-X kX-B ¼ ¼ KA--B : Z-A’X’B’Y kX-A kB-X
1 þ tk 2 1
The remarkable proportion in this case differs slightly from the linear examples:
and similarly for CB/BC, DC/CD and AD/DA. Furthermore kþ kþ CA ðtÞ ¼ 1 2 ¼ Keq,1 Keq,2 , AC ðtÞ k1 k2
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! kþ 8 1 þ 1 k1 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! : AB ðtÞ ¼ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ k kþ 1 þ 8 1 þ 1 þ tanh tk1 8 1 þ 1 2 k1 k1 2tanh
ð9Þ
Reversing the X-A and B-X arrows amounts to multiplying the forest’s term by ðkA-X kX-B Þ=ðkX-A kB-X Þ. In view of the Onsager relations, this value does not depend on X, but only on A and B; in fact, it is the equilibrium constant KA-...-B . If several X or Y or Z occur, the reasoning is the same. Consequently, every term in the numerator of AB is in that proportion to the corresponding term in the numerator of BA, and the fixed proportion for all s translates
3.2. Nonlinear reversible reaction (forward and backward second k1þ
order) 2A $ 2B k 1
The mass conservation law is AðtÞ þ BðtÞ ¼ 1, which offers no difficulties for the initial values, (1,0) and (0,1). The differential equation dAðtÞ 2 ¼ 2k1þ A2 ðtÞ þ 2k 1 ð1AðtÞÞ dt can be solved analytically as AA ðtÞ ¼
1 sffiffiffiffiffiffiffi , qffiffiffiffiffiffiffiffiffiffiffiffi k1þ þ tanh 2t k 1þ k 1 1 k 1
114
G.S. Yablonsky et al. / Chemical Engineering Science 66 (2011) 111–114
sffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffi k1þ tanh 2t k1þ k 1 k 1 sffiffiffiffiffiffiffi , BA ðtÞ ¼ qffiffiffiffiffiffiffiffiffiffiffiffi k1þ þ tanh 2t k k 1þ 1 1 k 1 sffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffi k 1 k1þ k þ tanh 2t 1 k1 sffiffiffiffiffiffiffi , AB ðtÞ ¼ qffiffiffiffiffiffiffiffiffiffiffiffi k þ 1 k tanh 2t k 1þ 1 1 k1þ
1 sffiffiffiffiffiffiffi BB ðtÞ ¼ : qffiffiffiffiffiffiffiffiffiffiffiffi k1 þ k 1þ tanh 2t k 1 1 k1þ Again eliminating time, the similar proportion for this case is kþ BA BB ¼ 1 ¼ Keq,1 , AA AB k1 where both numerator and denominator have undergone a duplication in A and B trajectories, still producing the equilibrium constant at all times t 40.
4. Conclusions We are going to describe these results in detail in a full-length paper. Then the similar approach will be applied to different systems, i.e., CSTR reactors, to reaction–diffusion TAP systems, etc. Presenting simply the result of this paper, it is a surprising relationship between A from B and B from A. An ‘‘ABBA rule’’ for short, which reminds the authors of their youth. They speak strangely but I understand—ABBA, ‘‘Eagle’’ (1978). References Gorban, A.N., 1984. Equilibrium Encircling: Equations of Chemical Kinetics and their Thermodynamic Analysis. Novosibirsk, Nauka 226pp. (in Russian). Gorban, A.N., Yablonskii, G.S., Bykov, V.I., 1982. Path to equilibrium. Int. Chem. Eng. 22 (2), 368–375. Gorban, A.N., Kaganovich, B.M., Filippov, S.P., Keiko, A.V., Shamansky, V.A., Shirkalin, I.A., 2006. Thermodynamic Equilibria and Extreme Analysis of Attainability Regions and Partial Equilibrium. Springer, Berlin, Heidelberg, New York 282pp. Roelant, R., Constales, D., Van Keer, R., Marin, G.B., 2010. Identifiability of rate coefficients in linear reaction networks from isothermal transient experimental data. Chem. Eng. Sci. 65 (7), 2333–2343. Wei, J., Prater, C.D., 1962. Structure and analysis of complex reaction systems. Adv. Catal. 13 (203), 202–392. Yablonskii, G.S., Bykov, V.I., Gorban, A.N., Elokhin, V.I., 1991. Kinetic models of catalytic reactions. In: Compton, R.G. (Ed.), Comprehensive Chemical Kinetics, vol. 32. Elsevier, Amsterdam, pp. 392. Zeldovich, Ya.B., 1938. Proof of the uniqueness of the solution of mass-action law equations. Zh. Fiz. Khim. 11 (5), 685–687 (in Russian).
Contents lists available at ScienceDirect
Chemical Engineering Science journal homepage: www.elsevier.com/locate/ces
Note
Equilibrium relationships for non-equilibrium chemical dependencies G.S. Yablonsky a,b,, D. Constales c,1, G.B. Marin d a
Parks College, Department of Chemistry, Saint Louis University, 3450 Lindell Blvd, Saint Louis, MO 63103, USA The Langmuir Research Institute, 106 Crimson Oaks Court, Lake Saint Louis, MO 63367, USA Department of Mathematical Analysis, Ghent University, Galglaan 2, B-9000 Gent, Belgium d Laboratory for Chemical Technology, Ghent University, Krijgslaan 281 (S5), B-9000 Gent, Belgium b c
a r t i c l e in f o
abstract
Article history: Received 24 June 2010 Received in revised form 10 September 2010 Accepted 6 October 2010 Available online 13 October 2010
It is shown that equilibrium constants determine the time-dependent behavior of particular ratios of concentrations for any system of reversible first-order reactions. Indeed, some special ratios actually coincide with the equilibrium constant at any moment in time. This is established for batch reactors, and similar relations hold for steady-state plug-flow reactors, replacing astronomic time by residence time. Such relationships can be termed time invariants of chemical kinetics. & 2010 Elsevier Ltd. All rights reserved.
Keywords: Kinetics Transient response Thermodynamics process Batch Time invariants Detailed equilibrium
1. Introduction In presenting the foundations of physical chemistry, the basic difference between equilibrium chemical thermodynamics and chemical kinetics is always stressed. A typical problem of equilibrium chemical thermodynamics (ECT) is calculating the composition of a chemical mixture that reacts in a closed system for an infinitely long time. ECT does not consider time. In opposite, chemical kinetics is the science of the evolution of chemical composition in time. Some selected results of theoretical chemical kinetics are obtained from thermodynamic principles, especially the principle of detailed equilibrium: (a) the uniqueness and stability of the equilibrium in any closed system, see Zeldovich (1938) and the analysis in Yablonskii et al. (1991); (b) the absence of damped oscillations near the point of detailed equilibrium, see Wei and Prater (1962) and the analysis in Yablonskii et al. (1991);
Corresponding author at: Parks College, Department of Chemistry, Saint Louis University, 3450 Lindell Blvd, Saint Louis, MO 63103, USA. E-mail addresses: [email protected] (G.S. Yablonsky), [email protected] (D. Constales), [email protected] (G.B. Marin). 1 Financial support from BOF/GOA 01GA0405 of Ghent University is gratefully acknowledged.
0009-2509/$ - see front matter & 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.ces.2010.10.014
(c) some limitations on the kinetic relaxation from the given initial conditions, e.g., based on the known set of equilibrium constants that determine the equilibrium composition, one can find a forbidden domain of compositions that is impossible to reach from the given boundary conditions. See Gorban et al. (1982, 2006) and Gorban (1984). However, the present dogma of physical chemistry holds that it is impossible to present an expression for any non-steady-state chemical system based on its description under equilibrium conditions, except for some relations describing the behavior in the linear vicinity of equilibrium. The goal of this short note is to announce that, contrary to this ‘dogma’, we have obtained relations of equilibrium type for some non-steady-state chemical systems. This has been achieved for all linear cases (with a general proof) and some nonlinear ones. In a forthcoming extended paper these relationships will be explained in more detail. Based on our results, equilibrium thermodynamic relationships can be considered not only as a description of the final point of temporal evolution, but as inherent characteristics of the dynamic picture. The following are examples to illustrate our statement; in all of them we analyze traditional models of chemical kinetics based on the mass-action law. The processes described by these models occur in a closed non-steady-state chemical system with perfect mixing (batch reactor) or in an open steady-state chemical system with no radial gradient (plug-flow reactor), whose description is
112
G.S. Yablonsky et al. / Chemical Engineering Science 66 (2011) 111–114
identical to the batch reactor, replacing astronomic time by residence time. In this short communication, we shall analyze a combination of data of different thought experiments, e.g.,
a chemical reactor is primed with a substance A only, it is primed with substance B only. In both cases, we shall monitor both concentrations A and B and pay special attention to the dependencies of ‘‘B produced from A’’ and ‘‘A produced from B’’. We shall use the notation AA(t) for the temporal concentration dependence of substance A, given the initial condition (A,B,y)¼(1,0,y), i.e., only A occurs, with normalized concentration 1. Similarly, BA(t) is the concentration of substance B for the same initial condition; AB(t) that of A when at t ¼0, (A,B,y)¼(0,1,y). We found interesting relationships between these dependencies.
CA ðtÞ ¼ 1
1
l1 l2
ðl1 expðl2 tÞl2 expðl1 tÞÞ,
AB ðtÞ ¼
k 1 ðexpðl2 tÞexpðl1 tÞÞ, l1 l2
BB ðtÞ ¼
1 ðl2 ðl1 k2þ Þexpðl2 tÞ þ l1 ðk2þ l2 Þexpðl1 tÞÞ, k2þ ðl1 l2 Þ
CB ðtÞ ¼ 1
1
l1 l2
ððl1 k2þ Þexpðl2 tÞ þ ðk2þ l2 Þexpðl1 tÞÞ:
Again, it turns out that BA and AB are always in fixed proportion: kþ BA ðtÞ ¼ 1 ¼ Keq , AB ðtÞ k1
ð4Þ
and again this is the constant of equilibrium of the A2B reaction. k1þ
k2þ
k3þ
k 1
k 2
k 3
2.3. The cycle of three reversible first-order reactions A $ B $ C $ A In the Laplace domain, defining the symbols
2. Linear cases
s1 ¼ k1þ þ k1 þ k2þ þ k2 þ k3þ þk3 ,
k1þ
2.1. A single, first-order reversible reaction A $ B
s2 ¼ k1þ k2þ þ k2þ k3þ þ k3þ k1þ þ k1þ k2 þ k2þ k3 þ k3þ k1
k 1
þ k 1 k3 þ k2 k1 þk3 k2 ,
By elementary mathematical techniques, for t Z 0, AA ðtÞ ¼
þ þ k 1 þ k1 expððk1 þ k1 ÞtÞ , þ k1 þk1
BA ðtÞ ¼
k1þ ð1expððk1þ k1þ þ k 1
DðsÞ ¼ sðs2 þ s1 s þ s2 Þ, the transformed concentrations AA, BA and AB are given by
,
LAA ðsÞ ¼
þ þ þ þ þ s2 þðk 1 þ k3 þ k2 þk2 Þs þðk1 k3 þ k1 k2 þ k2 k3 Þ , DðsÞ
k ð1expððk1þ þk 1 ÞtÞÞ , AB ðtÞ ¼ 1 k1þ þ k 1
LBA ðsÞ ¼
k1þ s þðk1þ k3þ þ k1þ k 2 þ k2 k3 Þ , DðsÞ
LAB ðsÞ ¼
þ þ þ k 1 s þðk1 k3 þ k1 k2 þ k2 k3 Þ DðsÞ
þk 1 ÞtÞÞ
þ k þ þ k 1 expððk1 þk1 ÞtÞ : BB ðtÞ ¼ 1 þ k1 þ k1
and the ratio of the latter two, by
Comparing BA and AB, it is clear that they are constant in a fixed proportion: kþ BA ðtÞ ¼ 1 ¼ Keq , AB ðtÞ k1
ð1Þ
which is the thermodynamic constant of equilibrium. It is remarkable that this ratio holds at any moment of time t 40, and not merely in the limit t- þ 1. 2.2. Two consecutive first-order reactions, the first being reversible: k1þ
k2þ
A $ B-C k 1
Using, e.g., Laplace domain techniques, the following analytical solutions are obtained: if we define the expressions qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ þ 2 þ þ k1þ þk ðk1þ þk 1 þk2 7 1 þk2 Þ 4k1 k2 ð2Þ l1,2 ¼ 2 and verify that
l1 4 k2þ 4 l2 4 0, l1 4k1þ 4 l2 40, we can write AA ðtÞ ¼
BA ðtÞ ¼
1 ðl1 ðk2þ l2 Þexpðl2 tÞ þ l2 ðl1 k2þ Þexpðl1 tÞÞ, k2þ ðl1 l2 Þ k1þ
l1 l2
ðexpðl2 tÞexpðl1 tÞÞ,
ð3Þ
k þ s þ ðk1þ k3þ þ k1þ k LBA ðsÞ 2 þ k2 k3 Þ ¼ 1 þ k þ k þ k þ Þ LAB ðsÞ k þ k k1 s þðk 2 3 1 2 0 1 3
1
B C 1 B C B1 þC þ þ þ @ A k k þ k k þ k k þsk 1 2 3 1 3 1 2 1þ k 1 k k1þ k2þ k3þ k k 1 2 3
¼
k1þ k 1
¼
k1þ , k 1
where the Onsager relationship k1+ k2+ k3+ ¼k1 k2 k3 was used in the final step. Hence for all s, LBA ðsÞ and LAB ðsÞ are in fixed proportion given by the equilibrium constant (k1+ /k1 ). Since the inverse Laplace transform is linear, the same proportion holds in the time domain: kþ BA ðtÞ ¼ 1 ¼ Keq,1 : AB ðtÞ k1
ð5Þ
Similarly, kþ CB ðtÞ ¼ 2 ¼ Keq,2 , BC ðtÞ k2
kþ AC ðtÞ ¼ 3 ¼ Keq,3 : CA ðtÞ k3
ð6Þ
As an example, we show in Fig. 1 the time dependence of BA/AA, BB/AB and the time-invariant ratio BA/AB, for the case of isomerization of butenes in Wei and Prater (1962, Eq. (129)): 4:623
3:724
3:371
10:344
1:000
5:616
cis-2-butene $ 1-butene $ trans-2-butene $ cis-2-butene:
This behavior is typical of the examples given here: if the system is started from the initial values (A,B,C) ¼(1,0,0), the ratio BA/AA
G.S. Yablonsky et al. / Chemical Engineering Science 66 (2011) 111–114
directly to the time domain:
1 B_A/A_A
BA ðtÞ ¼ KA-...-B : AB ðtÞ
B_B/A_B
0.8 ratio of concentrations B/A
113
B_A/A_B
(B/A) starting from (A,B)=(0,1)
ð10Þ
3. Nonlinear cases 3.1. Nonlinear reversible reaction (forward second order, backward
0.6
k1þ
first order) 2A $ B
B starting from (1,0) divided by A starting from (0,1)
k 1
0.4
In accordance to the mass conservation law, the balance is 0.2
AðtÞ þ 2BðtÞ ¼ 1:
(B/A) starting from (A,B)=(1,0)
0 0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
time Fig. 1. Time dependence of BA/AA, BB/AB and the time-invariant ratio BA/AB, for the case of isomerization of butenes in Wei and Prater (1962, Eq. (129)).
grows at first from 0, eventually reaching the equilibrium value (note that for these parameters BA/AA slightly overshoots the limit). Similarly, when started from (0,1,0), the corresponding BB/AB initially decreases from þ 1 and eventually reaches same limit, viz the equilibrium ratio. But surprisingly the combination BA/AB is constantly equal to the equilibrium value, for all times t 4 0.
In order for the A and B trajectories to reach the same equilibrium, we choose to start them from (1,0) and (0,1/2) respectively. The nonlinear differential equation is dAðtÞ ¼ 2k1þ A2 ðtÞ þ k 1 ð1AðtÞÞ, dt which can be solved analytically as sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! kþ kþ 1 þ 8 1 þ 1 þ tanh tk1 8 1 þ1 2 k1 k1 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! , AA ðtÞ ¼ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ þ k1 k1 kþ 1 þ 8 þ1 þ 4 þ 1 tanh tk1 8 1 þ 1 2 k1 k1 k1 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! kþ 1 þ tk1 8 1 þ 1 1 2 k1 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! , BA ðtÞ ¼ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ þ k k kþ 1 þ 8 1 þ 1 þ 4 1 þ 1 tanh tk1 8 1 þ 1 2 k1 k1 k1 kþ
2 k1 tanh
2.4. The cycle of four reversible first-order reactions k1þ
k2þ
k3þ
k4þ
k 1
k 2
k 3
k 4
A$B$C $D$A Although the expressions become more involved, it is straightforward to verify that in this case the fixed equilibrium proportions also hold, given the Onsager relationship k1+ k2+ k3+ k4+ ¼k1 k2 k3 k4 : kþ BA ðtÞ ¼ 1 ¼ Keq,1 , AB ðtÞ k1
ð7Þ
ð8Þ
with a similar relationship for DB/BD.
kþ BA ¼ 1 ¼ Keq,1 AA AB k1 we see that the denominator involves the A concentrations of both trajectories, AA and AB. Only this can ensure a ratio that equals the equilibrium constant at every time t 4 0.
2.5. Sketch of proof for general systems of first-order reactions We use the terminology and results of Roelant et al. (2010). Let A and B be substances such that a path from either to the other exists. In view of Onsager relations, then there must exist a reversible path from A to B. The denominators of AB and BA in the Laplace domain are the same, but their numerators differ. For AB, the numerator consists of contributions from all forests where A is the root of a tree and B is in that tree. Let Z, X, Y denote other nodes as in the denominator of the left-hand term in (9): Z-A-X-B’Y kA-X kX-B ¼ ¼ KA--B : Z-A’X’B’Y kX-A kB-X
1 þ tk 2 1
The remarkable proportion in this case differs slightly from the linear examples:
and similarly for CB/BC, DC/CD and AD/DA. Furthermore kþ kþ CA ðtÞ ¼ 1 2 ¼ Keq,1 Keq,2 , AC ðtÞ k1 k2
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! kþ 8 1 þ 1 k1 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! : AB ðtÞ ¼ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ k kþ 1 þ 8 1 þ 1 þ tanh tk1 8 1 þ 1 2 k1 k1 2tanh
ð9Þ
Reversing the X-A and B-X arrows amounts to multiplying the forest’s term by ðkA-X kX-B Þ=ðkX-A kB-X Þ. In view of the Onsager relations, this value does not depend on X, but only on A and B; in fact, it is the equilibrium constant KA-...-B . If several X or Y or Z occur, the reasoning is the same. Consequently, every term in the numerator of AB is in that proportion to the corresponding term in the numerator of BA, and the fixed proportion for all s translates
3.2. Nonlinear reversible reaction (forward and backward second k1þ
order) 2A $ 2B k 1
The mass conservation law is AðtÞ þ BðtÞ ¼ 1, which offers no difficulties for the initial values, (1,0) and (0,1). The differential equation dAðtÞ 2 ¼ 2k1þ A2 ðtÞ þ 2k 1 ð1AðtÞÞ dt can be solved analytically as AA ðtÞ ¼
1 sffiffiffiffiffiffiffi , qffiffiffiffiffiffiffiffiffiffiffiffi k1þ þ tanh 2t k 1þ k 1 1 k 1
114
G.S. Yablonsky et al. / Chemical Engineering Science 66 (2011) 111–114
sffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffi k1þ tanh 2t k1þ k 1 k 1 sffiffiffiffiffiffiffi , BA ðtÞ ¼ qffiffiffiffiffiffiffiffiffiffiffiffi k1þ þ tanh 2t k k 1þ 1 1 k 1 sffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffi k 1 k1þ k þ tanh 2t 1 k1 sffiffiffiffiffiffiffi , AB ðtÞ ¼ qffiffiffiffiffiffiffiffiffiffiffiffi k þ 1 k tanh 2t k 1þ 1 1 k1þ
1 sffiffiffiffiffiffiffi BB ðtÞ ¼ : qffiffiffiffiffiffiffiffiffiffiffiffi k1 þ k 1þ tanh 2t k 1 1 k1þ Again eliminating time, the similar proportion for this case is kþ BA BB ¼ 1 ¼ Keq,1 , AA AB k1 where both numerator and denominator have undergone a duplication in A and B trajectories, still producing the equilibrium constant at all times t 40.
4. Conclusions We are going to describe these results in detail in a full-length paper. Then the similar approach will be applied to different systems, i.e., CSTR reactors, to reaction–diffusion TAP systems, etc. Presenting simply the result of this paper, it is a surprising relationship between A from B and B from A. An ‘‘ABBA rule’’ for short, which reminds the authors of their youth. They speak strangely but I understand—ABBA, ‘‘Eagle’’ (1978). References Gorban, A.N., 1984. Equilibrium Encircling: Equations of Chemical Kinetics and their Thermodynamic Analysis. Novosibirsk, Nauka 226pp. (in Russian). Gorban, A.N., Yablonskii, G.S., Bykov, V.I., 1982. Path to equilibrium. Int. Chem. Eng. 22 (2), 368–375. Gorban, A.N., Kaganovich, B.M., Filippov, S.P., Keiko, A.V., Shamansky, V.A., Shirkalin, I.A., 2006. Thermodynamic Equilibria and Extreme Analysis of Attainability Regions and Partial Equilibrium. Springer, Berlin, Heidelberg, New York 282pp. Roelant, R., Constales, D., Van Keer, R., Marin, G.B., 2010. Identifiability of rate coefficients in linear reaction networks from isothermal transient experimental data. Chem. Eng. Sci. 65 (7), 2333–2343. Wei, J., Prater, C.D., 1962. Structure and analysis of complex reaction systems. Adv. Catal. 13 (203), 202–392. Yablonskii, G.S., Bykov, V.I., Gorban, A.N., Elokhin, V.I., 1991. Kinetic models of catalytic reactions. In: Compton, R.G. (Ed.), Comprehensive Chemical Kinetics, vol. 32. Elsevier, Amsterdam, pp. 392. Zeldovich, Ya.B., 1938. Proof of the uniqueness of the solution of mass-action law equations. Zh. Fiz. Khim. 11 (5), 685–687 (in Russian).