epl draft Reciprocal Relations Between Kinetic Curves G.S. Yablonsky(a)1, A.N. Gorban(b)2, D. Constales(c)3, V.V. Galvita4 and G.B. Marin(a)4 1 Parks College, Department of Chemistry, Saint Louis University - Saint Louis, MO 63103, USA 2 Department of Mathematics, University of Leicester - Leicester LE1 7RH, UK 3 Department of Mathematical Analysis, Ghent University - Galglaan 2, B-9000 Gent, Belgium 4 Laboratory for Chemical Technology, Ghent University - Krijgslaan 281 (S5), B-9000 Gent, Belgium PACS 05.20.Dd – Kinetic theory PACS 05.70.Ln – Nonequilibrium and irreversible thermodynamics Abstract. - We study coupled irreversible processes. For linear or linearized kinetics with mi- croreversibility,x ˙ = Kx, the kinetic operator K is symmetric in the entropic inner product. This form of Onsager’s reciprocal relations implies that the shift in time, exp(Kt), is also a symmetric operator. This generates the reciprocity relations between the kinetic curves. For example, for the Master equation, if we start the process from the ith pure state and measure the probability pj (t) of the jth state (j =6 i), and, similarly, measure pi(t) for the process, which starts at the jth pure state, then the ratio of these two probabilities pj (t)/pi(t) is constant in time and coincides with the ratio of the equilibrium probabilities. We study similar and more general reciprocal relations between the kinetic curves. The experimental evidence provided as an example is from the reversible water gas shift reaction over iron oxide catalyst. The experimental data are ob- tained using Temporal Analysis of Products (TAP) pulse-response studies. These offer excellent confirmation within the experimental error. Introduction. – the reciprocal relations were derived for nonlinear coupled A bit of history. In 1931, L. Onsager [1,2] gave transport processes between reservoirs coupled at meso- the backgrounds and generalizations to the reciprocal re- scopic contact points. Now, an elegant geometric frame- lations introduced in 19th century by Lord Kelvin and work is elaborated for Onsager’s relations and their gen- H. v. Helmholtz. In his seminal papers, L. Onsager men- eralizations [9]. tioned also the close connection between these relations Onsager’s relations are widely used for extraction of ki- and detailed balancing of elementary processes: at equilib- netic information about reciprocal processes from experi- rium, each elementary transaction should be equilibrated ments and for the validation of such information (see, for by its inverse transaction. This principle of detailed bal- example, [10]): one can measure how process A affects ance was known long before for the Boltzmann equation process B and extract the reciprocal information, how B [3]. A. Einstein used this principle for the linear kinetics affects A. arXiv:1008.1056v2 [cond-mat.stat-mech] 10 Feb 2011 of emission and absorption of radiation [4]. In 1901, R. The reciprocal relations were tested experimentally for Wegscheider published an analysis of detailed balance for many systems. In 1960, D.G. Miller wrote a remark- chemical kinetics [5]. able review on experimental verification of the Onsager The connections between the detailed balancing and reciprocal relations which is often referred to even now Onsager’s reciprocal relations were clarified in detail by [11]. Analyzing many different cases of irreversible phe- N. G. v. Kampen [6]. They were also extended for vari- nomena (thermoelectricity, electrokinetics, isothermal dif- ous types of coordinate transformations which may include fusion, etc), Miller found that these reciprocal relations are time derivatives and integration in time [7]. Recently, [8], valid. However, regarding the chemical reactions, Miller’s point was : “The experimental studies of this phenomenon (a)Long Term Structural Methusalem Funding by the Flemish Gov- ... have been inconclusive, and the question is still open ernment - grant number BOF09/01M00409 from an experimental point.” (b)Support from the University of Leicester and Isaac Newton In- stitute for Mathematical Sciences According to Onsager’s work [1], the fluxes in chemi- (c)Support from BOF/GOA 01GA0405 of Ghent University cal kinetics are time derivatives of the concentrations and p-1 G.S. Yablonsky et al. potentials are expressed through the chemical potentials. 2. The process starts at the state “everything is in Ar”, The fluxes (near equilibrium) are linear functions of poten- and we measure the concentration of Aq. The result b tials and the reciprocal relations state that the coefficient is cq(t) (“how much Aq is produced from the initial matrix of these functions is symmetric. It is impossible to Ar”) (the dual experiment). measure these coefficients directly. To find them one has to solve the inverse problem of chemical kinetics. This The results of the dual experiments are connected by the problem is often ill-posed. identity a b Such a difficulty, appearance of ill-posed problems in the cr (t) cq(t) verification of the reciprocal relations, is typical because eq eq , cr ≡ cq these relations connect the kinetic coefficients. Sometimes it is possible to find them directly in separate experiments where c are concentrations and ceq are equilibrium con- but if it is impossible then the inverse problem arises with centrations. all the typical difficulties. The symmetry with respect to the observables-initial In our work we, in particular, demonstrate how it is data exchange gives the general rule for production of the possible to verify the reciprocal relations without the dif- reciprocal relations between kinetic curves. ferentiation of the empiric kinetic curves and solving the Many real processes in chemical engineering and bio- inverse problems, and present the experimental results chemistry include irreversible reactions, i.e. the reactions which demonstrate these relations for one reaction kinetic with a negligible (zero) rate of the reverse reaction. For system. For this purpose, we have to formulate the recip- these processes, the micro-reversibility conditions and the rocal relations directly between the measurable quantities. backgrounds of classical Onsager relations are not appli- These reciprocal relations between kinetic curves use cable directly. Nevertheless, they may be considered as the symmetry of the propagator in the special entropic in- limits of systems with micro-reversibility when some of ner product. A dual experiment is defined for each ideal the rate constants for inverse reactions tend to zero. We kinetic experiment. For this dual experiment, both the ini- introduce the correspondent weak form of detailed bal- tial data and the observables are different (they exchange ance, formulate the necessary and sufficient algebraic con- their positions), but the results of the measurement is es- ditions for this form of detailed balance and formulate the sentially the same function of time. observables-initial data symmetry for these systems. The structure of the paper. We start from the classi- The experimental evidence of the observables-initial cal Onsager relations and reformulate them as conditions data symmetry is presented for the reversible water gas on the kinetic operator K for linear or linearized kinetic shift reaction over iron oxide catalyst. The experimental equationsx ˙ = Kx. This operator should be symmetric data are obtained using Temporal Analysis of Products in the entropic inner product, whereas the matrix L that (TAP) pulse-response studies. These offer excellent con- transforms forces into fluxes should be symmetric in the firmation within experimental error. standard inner product, i.e. L = L . The form of recip- ij ji Two forms of the reciprocal relations: forces, rocal relations with special inner product is well known in fluxes and entropic inner product. – Let us consider chemical and Boltzmann kinetics [13,14]. They are usually linear kinetic equations or kinetic equations linearized near proved directly from the detailed balance conditions. Such an equilibrium xeq (sometimes, it may be convenient to relations are also universal just as the classical relations move the origin to xeq): are. Real functions of symmetric operators are also symmet- x˙ = Kx . (1) ric. In particular, the propagator exp(Kt) is symmetric. Therefore, we can formulate the reciprocal relation be- tween kinetic curves. These relations do not include fluxes In the original form of Onsager’s relations, the vector and time derivatives, hence, they are more robust. We of fluxes J and the vector of thermodynamic forces X are formulate them as the symmetry relations between the connected by a symmetric matrix, J = LX, Lij = Lji. observables and initial data (the observables-initial data The vector X is the gradient of the corresponding thermo- symmetry). dynamic potential: Xi = ∂Φ/∂xi. For isolated systems, A particular case of this symmetry for a network of Φ is the entropy. For other conditions, other thermody- monomolecular chemical reactions or for the Master equa- namic potentials are used. For example, for the constant volume V and temperature T conditions, Φ is F/T and tion, which describe systems with detailed balance, seems − rather unexpected. Let us consider two situations for a for the constant pressure P and temperature conditions, Φ linear reaction network. is G/T , where F is the Helmholtz energy (free energy) and− G is the Gibbs energy (free enthalpy). These free 1. The process starts at the state “everything is in Aq”, entropy functions are also known as the Massieu–Planck and we measure the concentration of Ar. The result potentials [12]. Usually, they are concave. a is cr (t) (“how much Ar is produced from the initial For the finite-dimensional systems, like chemical kinet- Aq”). ics or the Master equation, the dynamics satisfy linear p-2 Reciprocal Relations Between Kinetic Curves (linearized) kinetic equationx ˙ = Kx, where This exchange of the observer and the initial state trans- forms an ideal experiment into another ideal experiment 2 ∂ Φ 2 (we call them dual experiments). The left and the right Kij = Lil i.e. K = L(D Φ)xeq .