Non-isothermal Activation Kinetics
Arango-Restrepo Andrés, Rubi J. Miguel
1 Secció de Física Estadística i Interdisciplinària, Departament de Física de la Matèria Condensada,
Facultat de Física, Universitat de Barcelona, Martí i Franquès 1
08028 Barcelona, Avd. Diagonal 647, 08028 Barcelona, Spain
E-mail: aarangor@unal.edu.co2 Escuela de Química, Facultad de Ciencias, Universidad Nacional de Colombia
Calle 59A No 63-20, Bloque 21, Núcleo El Volador, Medellín, Colombia
E-mail: mrubi@ub.edu
Received:
Received: 30 March 2017; revised: 03 May 2017; accepted: 06 May 2017; published online: 30 September 2017
DOI: 10.12921/cmst.2017.0000022
Abstract:
We analyze the activation kinetics of a system immersed in a non-isothermal bath. Using mesoscopic non-equilibrium thermodynamics, we show that activation is not only driven by the affinity but also by the temperature gradient. Both thermodynamic forces play a role in the kinetics. The presence of a thermal gradient makes the detailed balance
principle not fulfilled. We show that although the law of mass action holds locally, in terms of the local temperature, it is in general not valid globally, when the local values of the activation rate and the fugacity difference are replaced by their corresponding spatial averages. We analyze numerically the deviations of that global law from the actual activation kinetics as a function of the temperature gradient and the activation energy. Our analysis shows how to control the reaction rate by means of a temperature gradient.
Key words:
kinetics, law of mass action, non-equilibrium thermodynamics, non-isothermal process
References:
[1] C. R. Hickenboth, J.S. Moore, S.R. White, N.R. Sottos, J. Baudry, and S.R. Wilson, Biasing reaction pathways with mechanical force, Nature, 446 , 423-427 (2007).
[2] B.M. Rosen, V. Percec, A reaction to stress, Nature, 446, 381-382 (2007).
[3] M.M. Caruso, D.A. Davis, Q. Shen, S.A. Odom, N.R. Sottos, S.R. White and J.S. Moore, Mechanically-Induced Chemical Changes in Polymeric Materials, Chem. Rev., 109, 5755–5798 (2009).
[4] E.M. Lupton, C. Bräuchle, and I. Frank, Understanding Mechanically Induced Chemical Reactions, NIC Symposium 2006 , edited by G. Münster, D. Wolf and M. Kremer, NIC Series, 32, 57-64 (2006).
[5] I. Tinoco and C. Bustamante, The effect of force on thermodynamics and kinetics of single molecule reactions, Biophys. Chem., 101–102, 513–533 (2002).
[6] J.M. Rubi, D. Bedeaux, and S. Kjelstrup, Unifying Thermodynamic and Kinetic Descriptions of Single-Molecule Processes: RNA Unfolding under Tension, J. Phys. Chem B, 111, 9598–9602 (2007).
[7] H. Kramers., Brownian motion in a field of force and the diffusion model of chemical reactions, Physica, 7, 284-304 (1940).
[8] S. Glasstone, K. J. Laidler and H. Eyring, The Theory of Rate Processes: The Kinetics of Chemical Reactions, Viscosity, Diffusion and Electrochemical Phenomena ( McGraw-Hill, New York, 1941).
[9] P. Hänggi, P. Talkner, and M. Borkevec, Reaction-rate theory: fifty years after Kramers, Rev. Mod. Phys., 62, 251-341 (1990).
[10] T.L. Hill, Free Energy Transduction and Biochemical Cycle Kinetics (Dover, New York, 1989).
[11] H. Dekker, Nonequilibrium thermodynamics of nonisothermal activation: escape over mesoscopic barriers. I. The stochastic process, Physica A, 173, 381-410 (1991).
[12] H. Dekker, Nonequilibrium thermodynamics of nonisothermal activation: escape over mesoscopic barriers. II. The escape rate, Physica A, 173, 411-444 (1991).
[13] D. Reguera and J.M. Rubi, Homogeneous nucleation in inhomogeneous media. I. Nucleation in a temperature gradient, J. Chem. Phys, 119, 9877-9887 (2003.
[14] D. Reguera and J.M. Rubi, Homogeneous nucleation in inhomogeneous media. II. Nucleation in a shear flow, J. Chem. Phys. 119, 9888-9893 (2003).
[15] I. Santamaria-Holek, A. Gadomski and J.M. Rubi, Controlling protein crystal growth rate by means of temperature, J. Phys. Cond. Matt., 23, 235101 (2011).
[16] D. Reguera, J.M. Rubi and J.M.G. Vilar, The Mesoscopic Dynamics of Thermodynamic Systems, J. Phys. Chem. B, 109, 21502-21515 (2005).
[17] Vilar, J. M. G., and Rubi, J. M., Proc. Nat. Acad. Sci., Thermodynamics “beyond” local equilibrium, 98, 11081-11084 (2001).
[18] I. Pagonabarraga, A. Perez-Madrid and J.M. Rubi, Fluctuating hydrodynamics approach to chemical reactions, Physica A, 237, 205-219 (1997).
[19] S.R. de Groot and P. Mazur, Non-Equilibrium Thermodynamics (Dover, New York, 1984).
[20] S. Kjelstrup, J.M. Rubi, D. Bedeaux, Energy dissipation in slipping biological pumps, Phys. Chem. Chem. Phys., 7, 4009-4018 (2005).
[21] G. Gomila and J.M. Rubi, Relation for the nonequilibrium population of the interface states: Effects on the bias dependence of the ideality factor, J. Appl. Phys., 81, 2674-2681 (1997).
[22] J.M. Rubi and S. Kjelstrup, Mesoscopic nonequilibrium thermodynamics gives the same thermodynamic basis to Butler-Volmer and Nernst equations. J. Phys. Chem. B, 107, 13471-13477 (2003)
[23] J.M. Rubi, D. Bedeaux, S. Kjelstrup, and I. Pagonabarraga, Chemical Cycle Kinetics: Removing the Limitation of Linearity of a Non-equilibrium Thermodynamic Description, Int. J. Thermophys., 34, 1214–1228 (2013).
[24] A. Perez-Madrid, D. Reguera, and J.M. Rubi, Origin of the violation of the fluctuation-dissipation theorem in systems with activated dynamics, Physica A, 329, 357-364 (2003).
[25] J.W. Dufty and J.M. Rubi, Generalized Onsager symmetry, Phys. Rev. A, 36, 222-225 (1987).
[26] J. Ross, and P. Mazur, Some Deductions from a Formal Statistical Mechanical Theory of Chemical Kinetics, J. Chem. Phys., 35, 19-28 (1961).
[27] J.M. Rubi and A. Perez-Madrid, Far-from equilibrium kinetic processes, J. Non-Equilib. Thermodyn., 40, 275-281 (2015).
[28] D. Bedeaux, I. Pagonabarraga, J.M. Ortiz de Zárate, J.V. Sengers, and S.Kjelstrup, Mesoscopic non-equilibrium thermodynamics of non-isothermal reaction-diffusion, Phys. Chem. Chem. Phys., 12, 12780–12793 (2010).
[29] K. Maruta, T. Kataoka, N.I. Kim, S. Minae, and R. Fursenko, Characteristics of combustion in a narrow channel with a temperature gradient, Proc. Combust. Inst., 30 2429-2436 (2005).
[30] Y. Mao, T. Ynag and P.S. Cremer, A microfluidic device with a linear temperature gradient for parallel and combinatorial measurement, J. Am. Chem. Soc., 124, 4432-4435 (2002).
[31] J.Y. Cheng, C.J. Hsieh, Y.C. Chuang and J.R. Hsieh, Performing microchannel temperature cycling reactions using reciprocating reagent shuttling along a radial temperature gradient, Analyst., 130, 931-940 (2005).
[32] C. Gota, K. Okabe, T. Funatsu, Y. Harada, S. Uchiyama, Hydrophilic Fluorescent Nanogel Thermometer for Intracellular Thermometry, J. Am. Chem. Soc. 131, 2766–2767 (2009).
[33] S. H, Kim, et al. Micro-Raman thermometry for measuring the temperature distribution inside the microchannel of a polymerase chain reaction chip, J. Micromech. Microeng., 16, 526–530 (2006).
[34] P. Low, B.Kim, N. Takama, & C.Bergaud, High-Spatial-Resolution Surface-Temperature Mapping Using Fluorescent Thermometry, Small 4, 908–914 (2008).
[35] F. Vetrone, et al., Temperature Sensing Using Fluorescent Nanothermometers, ACS Nano 6, 3254–3258 (2010).
[36] S. Wang, S. Westcott, & W. Chen, Nanoparticle Luminescence Thermometry, J. Phys. Chem. B 106, 11203–11209 (2002).
We analyze the activation kinetics of a system immersed in a non-isothermal bath. Using mesoscopic non-equilibrium thermodynamics, we show that activation is not only driven by the affinity but also by the temperature gradient. Both thermodynamic forces play a role in the kinetics. The presence of a thermal gradient makes the detailed balance
principle not fulfilled. We show that although the law of mass action holds locally, in terms of the local temperature, it is in general not valid globally, when the local values of the activation rate and the fugacity difference are replaced by their corresponding spatial averages. We analyze numerically the deviations of that global law from the actual activation kinetics as a function of the temperature gradient and the activation energy. Our analysis shows how to control the reaction rate by means of a temperature gradient.
Key words:
kinetics, law of mass action, non-equilibrium thermodynamics, non-isothermal process
References:
[1] C. R. Hickenboth, J.S. Moore, S.R. White, N.R. Sottos, J. Baudry, and S.R. Wilson, Biasing reaction pathways with mechanical force, Nature, 446 , 423-427 (2007).
[2] B.M. Rosen, V. Percec, A reaction to stress, Nature, 446, 381-382 (2007).
[3] M.M. Caruso, D.A. Davis, Q. Shen, S.A. Odom, N.R. Sottos, S.R. White and J.S. Moore, Mechanically-Induced Chemical Changes in Polymeric Materials, Chem. Rev., 109, 5755–5798 (2009).
[4] E.M. Lupton, C. Bräuchle, and I. Frank, Understanding Mechanically Induced Chemical Reactions, NIC Symposium 2006 , edited by G. Münster, D. Wolf and M. Kremer, NIC Series, 32, 57-64 (2006).
[5] I. Tinoco and C. Bustamante, The effect of force on thermodynamics and kinetics of single molecule reactions, Biophys. Chem., 101–102, 513–533 (2002).
[6] J.M. Rubi, D. Bedeaux, and S. Kjelstrup, Unifying Thermodynamic and Kinetic Descriptions of Single-Molecule Processes: RNA Unfolding under Tension, J. Phys. Chem B, 111, 9598–9602 (2007).
[7] H. Kramers., Brownian motion in a field of force and the diffusion model of chemical reactions, Physica, 7, 284-304 (1940).
[8] S. Glasstone, K. J. Laidler and H. Eyring, The Theory of Rate Processes: The Kinetics of Chemical Reactions, Viscosity, Diffusion and Electrochemical Phenomena ( McGraw-Hill, New York, 1941).
[9] P. Hänggi, P. Talkner, and M. Borkevec, Reaction-rate theory: fifty years after Kramers, Rev. Mod. Phys., 62, 251-341 (1990).
[10] T.L. Hill, Free Energy Transduction and Biochemical Cycle Kinetics (Dover, New York, 1989).
[11] H. Dekker, Nonequilibrium thermodynamics of nonisothermal activation: escape over mesoscopic barriers. I. The stochastic process, Physica A, 173, 381-410 (1991).
[12] H. Dekker, Nonequilibrium thermodynamics of nonisothermal activation: escape over mesoscopic barriers. II. The escape rate, Physica A, 173, 411-444 (1991).
[13] D. Reguera and J.M. Rubi, Homogeneous nucleation in inhomogeneous media. I. Nucleation in a temperature gradient, J. Chem. Phys, 119, 9877-9887 (2003.
[14] D. Reguera and J.M. Rubi, Homogeneous nucleation in inhomogeneous media. II. Nucleation in a shear flow, J. Chem. Phys. 119, 9888-9893 (2003).
[15] I. Santamaria-Holek, A. Gadomski and J.M. Rubi, Controlling protein crystal growth rate by means of temperature, J. Phys. Cond. Matt., 23, 235101 (2011).
[16] D. Reguera, J.M. Rubi and J.M.G. Vilar, The Mesoscopic Dynamics of Thermodynamic Systems, J. Phys. Chem. B, 109, 21502-21515 (2005).
[17] Vilar, J. M. G., and Rubi, J. M., Proc. Nat. Acad. Sci., Thermodynamics “beyond” local equilibrium, 98, 11081-11084 (2001).
[18] I. Pagonabarraga, A. Perez-Madrid and J.M. Rubi, Fluctuating hydrodynamics approach to chemical reactions, Physica A, 237, 205-219 (1997).
[19] S.R. de Groot and P. Mazur, Non-Equilibrium Thermodynamics (Dover, New York, 1984).
[20] S. Kjelstrup, J.M. Rubi, D. Bedeaux, Energy dissipation in slipping biological pumps, Phys. Chem. Chem. Phys., 7, 4009-4018 (2005).
[21] G. Gomila and J.M. Rubi, Relation for the nonequilibrium population of the interface states: Effects on the bias dependence of the ideality factor, J. Appl. Phys., 81, 2674-2681 (1997).
[22] J.M. Rubi and S. Kjelstrup, Mesoscopic nonequilibrium thermodynamics gives the same thermodynamic basis to Butler-Volmer and Nernst equations. J. Phys. Chem. B, 107, 13471-13477 (2003)
[23] J.M. Rubi, D. Bedeaux, S. Kjelstrup, and I. Pagonabarraga, Chemical Cycle Kinetics: Removing the Limitation of Linearity of a Non-equilibrium Thermodynamic Description, Int. J. Thermophys., 34, 1214–1228 (2013).
[24] A. Perez-Madrid, D. Reguera, and J.M. Rubi, Origin of the violation of the fluctuation-dissipation theorem in systems with activated dynamics, Physica A, 329, 357-364 (2003).
[25] J.W. Dufty and J.M. Rubi, Generalized Onsager symmetry, Phys. Rev. A, 36, 222-225 (1987).
[26] J. Ross, and P. Mazur, Some Deductions from a Formal Statistical Mechanical Theory of Chemical Kinetics, J. Chem. Phys., 35, 19-28 (1961).
[27] J.M. Rubi and A. Perez-Madrid, Far-from equilibrium kinetic processes, J. Non-Equilib. Thermodyn., 40, 275-281 (2015).
[28] D. Bedeaux, I. Pagonabarraga, J.M. Ortiz de Zárate, J.V. Sengers, and S.Kjelstrup, Mesoscopic non-equilibrium thermodynamics of non-isothermal reaction-diffusion, Phys. Chem. Chem. Phys., 12, 12780–12793 (2010).
[29] K. Maruta, T. Kataoka, N.I. Kim, S. Minae, and R. Fursenko, Characteristics of combustion in a narrow channel with a temperature gradient, Proc. Combust. Inst., 30 2429-2436 (2005).
[30] Y. Mao, T. Ynag and P.S. Cremer, A microfluidic device with a linear temperature gradient for parallel and combinatorial measurement, J. Am. Chem. Soc., 124, 4432-4435 (2002).
[31] J.Y. Cheng, C.J. Hsieh, Y.C. Chuang and J.R. Hsieh, Performing microchannel temperature cycling reactions using reciprocating reagent shuttling along a radial temperature gradient, Analyst., 130, 931-940 (2005).
[32] C. Gota, K. Okabe, T. Funatsu, Y. Harada, S. Uchiyama, Hydrophilic Fluorescent Nanogel Thermometer for Intracellular Thermometry, J. Am. Chem. Soc. 131, 2766–2767 (2009).
[33] S. H, Kim, et al. Micro-Raman thermometry for measuring the temperature distribution inside the microchannel of a polymerase chain reaction chip, J. Micromech. Microeng., 16, 526–530 (2006).
[34] P. Low, B.Kim, N. Takama, & C.Bergaud, High-Spatial-Resolution Surface-Temperature Mapping Using Fluorescent Thermometry, Small 4, 908–914 (2008).
[35] F. Vetrone, et al., Temperature Sensing Using Fluorescent Nanothermometers, ACS Nano 6, 3254–3258 (2010).
[36] S. Wang, S. Westcott, & W. Chen, Nanoparticle Luminescence Thermometry, J. Phys. Chem. B 106, 11203–11209 (2002).
