DOCSLIB.ORG

A Probabilistic Rate Theory Connecting Kinetics to Thermodynamics Denis Michel

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
A Probabilistic Rate Theory Connecting Kinetics to Thermodynamics Denis Michel A probabilistic rate theory connecting kinetics to thermodynamics Denis Michel To cite this version: Denis Michel. A probabilistic rate theory connecting kinetics to thermodynamics. 2017. hal- 01569569v7 HAL Id: hal-01569569 https://hal.archives-ouvertes.fr/hal-01569569v7 Preprint submitted on 15 Mar 2018 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. A probabilistic rate theory connecting kinetics to thermodynamics∗ Denis Michelz z Universite de Rennes1-IRSET. Campus de Villejean. 35000 Rennes France. Email: [email protected]. Abstract. Kinetics and thermodynamics are largely classical version (without tunneling effects), the empiri- disconnected in current theories because Arrhenius acti- cal Arrhenius equation vation energies (Ea) have strictly no influence on equilib- rium distributions. A first step towards the incorporation − Ea of rate theories in thermodynamics is the identification k = A e kB T (1) of the pre-exponential term of the Arrhenius equation as an entropic quantity. A second step examined here where kB is the Boltzmann constant and T is the tem- is the possible contribution of Ea in equilibrium land- perature, includes (i) a so-called preexponential factor A, scapes. Interestingly, this possibility exists if envisioning containing the elementary frequency and configurational the energetic exponential term of Arrhenius rate con- restrictions, and (ii) a unitless exponential function con- stants as the probability that the energy of the reactant taining an energy barrier called activation energy Ea. is sufficient for the transition. This radically new ap- In the present paper, the exponential factor of Arrhe- proach encompasses Maxwell-Boltzmann distributions nius will be reformulated as the probability to reach a and solves inconsistencies in previous theories, in partic- threshold energy, as initially proposed but incompletely ular on the role of temperature in kinetics and thermo- appraised in [3]. This simple hypothesis will lead to ma- dynamics. These probabilistic rate constants are then jor changes in thermodynamic relationships. reintroduced in dynamic systems to provide them with the two distinct facets of time: the time step and the time arrow. Keywords: Kinetics; equilibrium; heat capacity; en- thalpy; entropy; Gibbs free energy. Notes to readers: The unitless statistical entropy usually written H will be renamed here S to avoid confusion with enthalpy. The letter C will be used for heat capacity but not for concentration. 1 Introduction Rate constants are the drivers of all dynamic systems and are widely used in physical chemistry and modeling stud- ies. By the restrictions they impose on chemical transi- Figure 1. Representation of the energy barrier according to tions, rate constants prevent our organized world from the Arrhenius principle. The barrier is supposed to restrict falling directly into its state of weaker free energy and reaction kinetics but to have no influence on equilibrium ra- z maximal entropy. Yet the profound nature of these con- tios. E is a fixed energy threshold necessary for crossing the stants remains obscure and the main focus of this study barrier in both directions and H are the mean enthalpies of is to clarify their energetic component. After multiple reaction of the particles. attempts of description, its most common expression re- mains the exponential factor of Arrhenius [1, 2]. In its Using the notations of Fig.1, the Arrhenius approach ∗Reference: Michel, D. 2018. A probabilistic rate theory connecting kinetics to thermodynamics. Physica A 503, 26-44 1 reads 2.3 A non-probabilistic exponential law z The exponential term of the Arrhenius law has cer- k = A e−(E −Hi)=kB T (2a) ij ij tain appearences of a probability, but as explained in and for the reverse reaction the following section, as it stands it cannot correspond z −(E −Hj )=kB T to the exponential distribution. The purpose of the kji = Aji e (2b) present study is precisely to examine the theoretical con- In equilibrium the forward and reverse fluxes equalize sequences of the hypothesis that it is a probability of such that sufficient energy for the transition. This simple hypoth- eq eq ni kij = nj kji (2c) esis justified later leads to the novel formulas giving n k A z z i ji ji (E −Hi−E +Hj )=kB T = = K = z E ji e − E − 1+ a nj kij Aij sufficient energy H ( H ) eq (2d) P ( ) = e = e (3a) Aji Aji = e−(Hi−Hj )=kB T = e−∆H=kB T Aij Aij which gives the equilibrium relationship z E disappears. The equilibrium ratio is independent of z ni Aji Hi E the energetic barrier and depends only on the difference ln = ln + 1 − nj Aij Hj Hi of energy between the reactants. These currently admit- eq (3b) ted relationships can be naively questioned. Aji Hi Ea = ln + 1 − 1 + Aij Hj Hi 2 Some concerns and alternative whose structure is completely different from the usual proposal result ni Aji Hj − Hi Gj − Gi 2.1 On the universality of kBT as a mean ln = ln + = (4) n A k T k T energy j eq ij B B 1=kBT is introduced in statistical mechanics as a con- In Eq.(4), the pre-exponential terms A disappear stant called β corresponding to a Lagrangian multiplier from the latter form containing Gibbs free energies G. or a distribution parameter, and finally connected to This modification results from the identification of the temperature by analogy with classical thermodynamics. pre-exponential factor as an entropic parameter, which This introduction of kBT is elegant and perfectly rational can then be incorporated in the Gibbs free energies. This in this context but could be too restrictive for an exten- pertinent operation will also be used here and applied to sion to more complex systems. kBT is concretely useful Eq.(3b), but let us first explain the entropic meaning of here to adimension the exponent of the Arrhenius expo- the pre-exponential factor. This factor can be defined nential and is usually interpreted as the mean particle as A = τ −1Ω−1 where τ −1 is the universal thermal fre- energy. This interpretation holds in the kinetic theory quency (Section 6.2.2), giving their time−1 unit to the of ideal gases but is much less clear for complex chem- rate constants, and Ω−1 is the reciprocal of the number ical systems in which the mean reactional energies are of possible configurations of the reactant, one of which precisely different for each type of particle. allowing the reaction to proceed, as simply illustrated in Fig.2. As a consequence, ln(Aji=Aij) can be identi- 2.2 Insensitivity of dynamic equilibria to fied, after elimination of the identical frequencies, as the difference of statistical entropies activation energies The accepted conclusion that the presence and the height Aji Ωij ln = ln = Sij − Sji of energy barriers do not participate to equilibrium ra- Aij Ωji tios between the reactants, can appear not completely intuitive because one could expect that increasing the Note the inversion of the suffixes between the ratios height of the barrier exponentially amplifies the differ- of A and Ω. In addition, if a given reactant is involved ential capacity of the reactants to jump over, including in several reactions, one can assume that it is not nec- in equilibrium. By contrast, in the potential surface of essary to precise the nature of the product in reactional the transition state theory [4], the activated complexes entropy, because it is expected to be the same for all reside on mountain crests but their energy levels do not the reactions starting from this reactant. Indeed, even if influence the populational distributions. each reaction requires a specific reactional configuration, 2 this configuration belongs the same set of total configura- Large Ea do not invert but accentuate the differences de- tions Ω of this reactant. Converting configurational into termined in both cases by the relative energy levels. This thermodynamic entropy through the Boltzmann formula point results from a mathematical interpretation of the S = S=kB, allows to detail Eq.(4) step-by-step exponential factor. n A H − H ln i = ln j + j i 3 A new probabilistic interpreta- n A k T j eq i B tion of the exponential factor Ω H − H = ln i + j i Ω k T j B 3.1 The exponential distribution S − S H − H = i j + j i kB kBT The exponential component of the Arrhenius law and of Hj − Hi − T (Sj − Si) the Boltzmann distribution, can be simultaneously de- = kBT rived from the exponential distribution, which is basi- (Hj − TSj) − (Hi − TSi) cally mathematical. According to this law, the proba- = bility that a particle from an homogeneous population kBT with an average number of energy quanta per particle Gj − Gi = hEi, reaches a fixed threshold number Ez, is kBT Such a fusion of entropy and enthalpy into free en- Ez z − hEi ergy will be retained here for whole systems but not P (E > E ) = e (5) for individual reactants, because molecular entropy and Since Ez is a temperature-independent threshold and enthalpy play different roles in reactions, which are de- hEi is temperature dependent, temperature remains at scribed by different probabilities (Fig.2). the denominator of the exponent, but an important dif- ference with current rate equations is the absence of kBT .
Load more

Recommended publications
  • Patnaik-Goldfarb-2016.Pdf
    CONTINUOUS ACTIVATION ENERGY REPRESENTATION OF THE ARRHENIUS EQUATION FOR THE PYROLYSIS OF CELLULOSIC MATERIALS: FEED CORN STOVER AND COCOA SHELL BIOMASS * *,** ABHISHEK S. PATNAIK and JILLIAN L. GOLDFARB *Division of Materials Science and Engineering, Boston University, 15 St. Mary’s St., Brookline, MA 02446 **Department of Mechanical Engineering, Boston University, 110 Cummington Mall, Boston, MA 02215 ✉Corresponding author: Jillian L. Goldfarb, [email protected] Received January 22, 2015 Kinetics of lignocellulosic biomass pyrolysis – a pathway for conversion to renewable fuels/chemicals – is transient; discreet changes in reaction rate occur as biomass composition changes over time. There are regimes where activation energy computed via first order Arrhenius function yields a negative value due to a decreasing mass loss rate; this behavior is often neglected in the literature where analyses focus solely on the positive regimes. To probe this behavior feed corn stover and cocoa shells were pyrolyzed at 10 K/min. The activation energies calculated for regimes with positive apparent activation energy for feed corn stover were between 15.3 to 63.2 kJ/mol and for cocoa shell from 39.9 to 89.4 kJ/mol. The regimes with a positive slope (a “negative” activation energy) correlate with evolved concentration of CH4 and C2H2. Given the endothermic nature of pyrolysis, the process is not spontaneous, but the “negative” activation energies represent a decreased devolatilization rate corresponding to the transport of gases from the sample surface. Keywords: Arrhenius equation, biomass pyrolysis, evolved compounds, activation energy INTRODUCTION Fossil fuels comprise the majority of the total energy supply in the world today.1 One of the most critical areas to shift our dependence from fossil to renewable fuels is in energy for transportation, which accounts for well over half of the oil consumed in the United States.
    [Show full text]
  • Temperature Dependence of Reaction Rates
    Module 6 : Reaction Kinetics and Dynamics Lecture 29 : Temperature Dependence of Reaction Rates Objectives In this Lecture you will learn to do the following Give examples of temperature dependence of reaction rate constants (k). Define the activated complex. Outline Arrhenius theory of temperature dependence of k. Rationalize the temperature dependence of molecular speeds and molecular energies through appropriate distribution functions. Summarize other theories for temperature dependence of k. 29.1 Introduction Temperature dependence of physical and chemical parameters is of great interests to chemists and predicting correct temperature dependence is a test as well as a challenge for framing suitable theories. In thermodynamics, temperature dependence of heat capacities and enthalpies was used in calculating equilibrium constants at different temperatures. The departure of real gas behaviour from the ideal gas behaviour is expressed through the temperature dependent viral coefficients. In chemical kinetics, it was observed that for many reactions, increasing the temperature by 10oC, doubled the rates. In rate processes, the temperature dependence is quite striking. In this lecture, we will consider preliminary attempts at explaining this temperature dependence and take up the detailed explanations in later lectures. A knowledge of the distribution of molecular speeds at a given temperature and the population of energy levels is essential in understanding the temperature dependent rate processes and these aspects will also be outlined here. The most common analysis of temperature dependence of reaction rates over a small temperature range of a few tens of degrees Celsius has been through the Arrhenius equation given below. k = A e - Ea / RT (29.1) Where A is the pre exponential factor (commonly referred to as the frequency factor) and Ea is the energy of activation.
    [Show full text]
  • Activation Energy (Ea)
    Activation Energy (Ea) Ea value indicates that how temp. changes during processing or storage affect the k value of the reaction. The higher Ea value of the reaction, the more sensitive for the reaction to temp. changes during storage or processing. Ea value is specific for each chemical, microbial and enzymatic reaction. 1 √ Ea cannot be directly measured. √ Ea is calculated from Arrhenius equation. This equation (described by Svante Arrhenius in 1889) gives the relationship between k and temp. of processing or storage. Therefore, we need k and temp. values to determine Ea 2 Arrhenious equation –Ea/ RT k = ko e k: Reaction rate constant (for any reaction order) ko: frequency factor (same unit as k) Ea: Activation energy of the reaction (cal/mole of J/mole) R: Gas constant (1.987 cal/(mole K) or 8.314 J/(mole K) T: Temperature (K) 3 Take ln of both sides – Ea 1 ln k = (—— ——) + ln ko R T ↕ ↕ ↕ ↕ y = a x + b Find the equivalence of this equ. on log10 4 To determine Ea value graphically √ First identify the quality factor of concern and then determine k values at least at three different temp., preferably at five different processing or storage temp. √ Then, plot k values vs 1/T values. Using aritmetic graph paper: Take ln of k values and reciprocal of temp. values in Kelvin and then plot ln k vs 1/T. Slope will be equal to –Ea/R. Using semi-log graph paper: Plot original k values vs 1/T values. Slope will be equal to –Ea/2.303R.
    [Show full text]
  • Eyring Activation Energy Analysis of Acetic Anhydride Hydrolysis in Acetonitrile Cosolvent Systems Nathan Mitchell East Tennessee State University
    East Tennessee State University Digital Commons @ East Tennessee State University Electronic Theses and Dissertations Student Works 5-2018 Eyring Activation Energy Analysis of Acetic Anhydride Hydrolysis in Acetonitrile Cosolvent Systems Nathan Mitchell East Tennessee State University Follow this and additional works at: https://dc.etsu.edu/etd Part of the Analytical Chemistry Commons Recommended Citation Mitchell, Nathan, "Eyring Activation Energy Analysis of Acetic Anhydride Hydrolysis in Acetonitrile Cosolvent Systems" (2018). Electronic Theses and Dissertations. Paper 3430. https://dc.etsu.edu/etd/3430 This Thesis - Open Access is brought to you for free and open access by the Student Works at Digital Commons @ East Tennessee State University. It has been accepted for inclusion in Electronic Theses and Dissertations by an authorized administrator of Digital Commons @ East Tennessee State University. For more information, please contact [email protected]. Eyring Activation Energy Analysis of Acetic Anhydride Hydrolysis in Acetonitrile Cosolvent Systems ________________________ A thesis presented to the faculty of the Department of Chemistry East Tennessee State University In partial fulfillment of the requirements for the degree Master of Science in Chemistry ______________________ by Nathan Mitchell May 2018 _____________________ Dr. Dane Scott, Chair Dr. Greg Bishop Dr. Marina Roginskaya Keywords: Thermodynamic Analysis, Hydrolysis, Linear Solvent Energy Relationships, Cosolvent Systems, Acetonitrile ABSTRACT Eyring Activation Energy Analysis of Acetic Anhydride Hydrolysis in Acetonitrile Cosolvent Systems by Nathan Mitchell Acetic anhydride hydrolysis in water is considered a standard reaction for investigating activation energy parameters using cosolvents. Hydrolysis in water/acetonitrile cosolvent is monitored by measuring pH vs. time at temperatures from 15.0 to 40.0 °C and mole fraction of water from 1 to 0.750.
    [Show full text]
  • The Arrhenius Equation
    CHEM 2880 - Kinetics Determining Rate Order As mentioned earlier, the rate order of a reaction can only be determined experimentally. A number of methods can be used to determine both the reaction order and the rate constant. 1. Integration method - the concentration (of reagent or product) is measured at various time intervals, and the integrated rate law equations are plotted. The order is determined by the plot with the best linear fit (most consistent k value). 2. Isolation method - if more than one reactant is involved, the reaction is run several times, each time changing the initial concentration of one of the reagents. Any corresponding change in the rate is due to the reagent that changed concentration and the corresponding order can be determined. Once the order wrt to each reagent has been determined the overall rate can be calculated. 3. Reagents in excess - this method can be used to reduce the order of a reaction (as mentioned earlier), and make complicated rate laws easier to analyze. 40 CHEM 2880 - Kinetics 4. Differential method - for an nth-order reaction, the rate law is: taking logarithms of both sides yields: Thus a plot of log v vs log[A] will be linear with a slope of n and an intercept of log k. This method often involves measuring the initial rate of reaction at several different initial concentrations. This has the advantages of: avoiding any complications caused by build up of products; and measuring rate at a time when the concentration is known most accurately. From: Physical Chemistry for the Chemical and Biological Sciences, R.
    [Show full text]
  • Continuous Activation Energy Representation of the Arrhenius Equation for the Pyrolysis of Cellulosic Materials: Feed Corn Stover and Cocoa Shell Biomass
    CELLULOSE CHEMISTRY AND TECHNOLOGY CONTINUOUS ACTIVATION ENERGY REPRESENTATION OF THE ARRHENIUS EQUATION FOR THE PYROLYSIS OF CELLULOSIC MATERIALS: FEED CORN STOVER AND COCOA SHELL BIOMASS * *,** ABHISHEK S. PATNAIK and JILLIAN L. GOLDFARB *Division of Materials Science and Engineering, Boston University, 15 St. Mary’s St., Brookline, MA 02446 ** Department of Mechanical Engineering, Boston University, 110 Cummington Mall, Boston, MA 02215 ✉Corresponding author: Jillian L. Goldfarb, [email protected] Received January 22, 2015 Kinetics of lignocellulosic biomass pyrolysis – a pathway for conversion to renewable fuels/chemicals – is transient; discreet changes in reaction rate occur as biomass composition changes over time. There are regimes where activation energy computed via first order Arrhenius function yields a negative value due to a decreasing mass loss rate; this behavior is often neglected in the literature where analyses focus solely on the positive regimes. To probe this behavior feed corn stover and cocoa shells were pyrolyzed at 10K/min. The activation energies calculated for regimes with positive apparent activation energy for feed corn stover were between 15.3 to 63.2 kJ/mol and for cocoa shell from 39.9to 89.4 kJ/mol. The regimes with a positive slope (a “negative” activation energy) correlate with evolved concentration of CH 4 and C 2H2. Given the endothermic nature of pyrolysis, the process is not spontaneous, but the “negative” activation energies represent a decreased devolatilization rate corresponding to the transport of gases from the sample surface. Keywords : Arrhenius equation, biomass pyrolysis, evolved compounds, activation energy INTRODUCTION Fossil fuels comprise the majority of the total There are multiple methods used to analyze energy supply in the world today.
    [Show full text]
  • Activation Energies and Temperature Dependencies of the Rates of Crystallization and Melting of Polymers
    polymers Review Activation Energies and Temperature Dependencies of the Rates of Crystallization and Melting of Polymers Sergey Vyazovkin Department of Chemistry, University of Alabama at Birmingham, 901 S. 14th Street, Birmingham, AL 35294, USA; [email protected] Received: 20 April 2020; Accepted: 5 May 2020; Published: 7 May 2020 Abstract: The objective of this review paper is to survey the phase transition kinetics with a focus on the temperature dependence of the rates of crystallization and melting, as well as on the activation energies of these processes obtained via the Arrhenius kinetic treatment, including the treatment by isoconversional methods. The literature is analyzed to track the development of the basic models and their underlying concepts. The review presents both theoretical and practical considerations regarding the kinetic analysis of crystallization and melting. Both processes are demonstrated to be kinetically complex, and this is revealed in the form of nonlinear Arrhenius plots and/or the variation of the activation energy with temperature. Principles which aid one to understand and interpret such results are discussed. An emphasis is also put on identifying proper computational methods and experimental data that can lead to meaningful kinetic interpretation. Keywords: activation energy; Arrhenius equation; crystallization; differential scanning calorimetry (DSC); melting; multi-step kinetics 1. Introduction Evidently, it is a fact of nature that the rate of a broad variety of processes responds exponentially to temperature changes. For chemical reactions, this fact is reflected in a number of empirical equations [1], including the one proposed by Arrhenius [2]. Among those, the Arrhenius equation has thrived because it was backed up by the theories of the activated complex and transition state [3,4].
    [Show full text]
  • From Arrhenius to Vacancy Formation: Xtra Notes We Began with the General Arrhenius Relation
    From Arrhenius to Vacancy Formation: Xtra Notes We began with the general Arrhenius relation. Many phenomena depend not just on energy but on the energy of some process relative to thermal energy. When the process is “thermally activated” the Arrhenius relationship can describe its rate. Rate (how often the Activation energy for process occurs in time) process to occur -(Ea/RT) K = A * e Average thermal Pre-exponential (also kinetic energy called “frequency factor”) Some things: 1) if the Ea is per mole use the gas constant R (8.314 J/mol*K), if it’s per atom use the Boltzmann constant KB (8.62x10-5 eV/K) 2) The pre-exponential factor A is related to how the process is attempted (we learn about it more later when we do reaction rates) 3) Ea is the energy required to “activate” the process. It’s also known as the barrier height for a standard reaction coordinate vs. Energy plot. 4) The units of K and A must be the same, as with Ea and RT (or KBT). 1 From Arrhenius to Vacancy Formation: Xtra Notes K=A*e -(Ea/RT) ln(K)=ln(A)-Ea/RT Take the log of the Arrhenius equation and we get a nice linear relationship between ln(K) and 1/T, like this: Some things: Y-intercept = lnA 1) This is sometimes called an “Arrhenius plot,” showing the inverse relation between reaction rates and temperature 2) The negative slope gives the activation energy, Ea: slope = -Ea/R 3) Extrapolation of the Arrhenius plot slope=-Ea/R back to the y-intercept gives ln(A) ln(k) 4) This plot shows how activation energy and temperature affect the sensitivity of the reaction rate 1/Temperature 2 From Arrhenius to Vacancy Formation: Xtra Notes Next we related a thermally activated process to vacancy formation.
    [Show full text]
  • Arrheniusan Activation Energy of Separation for Different Parameters Regulating the Process
    Physicochem. Probl. Miner. Process., 54(4), 2018, 1152-1158 Physicochemical Problems of Mineral Processing ISSN 1643-1049 http://www.journalssystem.com/ppmp © Wroclaw University of Science and Technology Received May 14, 2018; reviewed; accepted June 21, 2018 Arrheniusan activation energy of separation for different parameters regulating the process Jan Drzymala Wroclaw University of Science and Technology, Faculty of Geoengineering, Mining and Geology, Wybrzeze Wyspianskiego 27, 50-370 Wroclaw, Poland Corresponding author: [email protected] (Jan Drzymala) Abstract: The Arrhenius model, that relates the activation energy with the kinetic constant and process temperature, was applied for flotation as a separation process, and next was extended to other incentive parameters such as the frother concentration, NaCl content and hydrophobicity. It was shown that determination of the activation energy caused by other incentive parameters (i.e. particle size, surface potential) was also possible. The units of the activation energy depend on the type of the separation process and incentive parameter. For contact angle regulating flotation the activation energy unit is mJ/m2, while for the frother concentration is J. It is known that instead in joules, the activation energy can also be expressed in J/mol and in kT or RT units, where k is the Boltzmann constant, R gas constant and T is absolute temperature in kelvins. Even though different formulas of the specific Gibbs potential were used for calculation of activation energy caused by various incentive parameters, there was generally a good agreement between the extend of changes of the first order kinetic constants of the process and activation energy value.
    [Show full text]
  • The Basic Theorem of Temperature-Dependent Processes
    Review The Basic Theorem of Temperature-Dependent Processes Valentin N. Sapunov 1,* , Eugene A. Saveljev 1, Mikhail S. Voronov 1, Markus Valtiner 2,* and Wolfgang Linert 2,* 1 Department of Chemical Technology of Basic Organic and Petrochemical Synthesis, Mendeleev University of Chemical Technology of Russia, Miusskaya sq. 9, 125047 Moscow, Russia; [email protected] (E.A.S.); [email protected] (M.S.V.) 2 Department of Applied Physics, Vienna University of Technology, Wiedner Hauptstrasse 8-10/134-AIP, 1040 Vienna, Austria * Correspondence: [email protected] (V.N.S.); [email protected] (M.V.); [email protected] (W.L.) Abstract: The basic theorem of isokinetic relationships is formulated as “if there exists a linear correlation “structure∼properties” at two temperatures, the point of their intersection will be a common point for the same correlation at other temperatures, until the Arrhenius law is violated”. The theorem is valid in various regions of thermally activated processes, in which only one parameter changes. A detailed examination of the consequences of this theorem showed that it is easy to formulate a number of empirical regularities known as the “kinetic compensation effect”, the well- known formula of the Meyer–Neldel rule, or the so-called concept of “multi-excitation entropy”. In a series of similar processes, we examined the effect of different variable parameters of the process on the free energy of activation, and we discuss possible applications. Keywords: isokinetic relationship; Meyer–Neldel; multi-excitation entropy; free energy of activation Citation: Sapunov, V.N.; Saveljev, E.A.; Voronov, M.S.; Valtiner, M.; Linert, W.
    [Show full text]
  • The Development of the Arrhenius Equation
    The Development of the Arrhenius Equation Keith J. Laidler Onawa-Carleton Institute for Research and Graduate Studies in Chemistry, University of Onawa, Onawa, Ontario. Canada KIN 684 Todav the influence of temperature on the rate of chemical k = AFn (1 +GO) (4) reactions is almost always interpreted in terms of what is now known as the Arrhenius equation. According to this, a rate In terms of the absolute temperature T, the equation can he constant h is the product of a pre- exponentid ("frequency") written as factor A and an exponential term k = A'FT(l + G'T) (5) k = A~-EIRT (1) In 1862 Berthelot (9) presented the equation where R is the gas constant and E is the activation energy. The Ink=A'+DT or k = AeDT (6) apparent activation energy E.,, is now defined (I) in terms of this equation as A ronsi(lrrable niimlwr of workers bupportrd thls equation, which was taken seriously at least until 19011 (101. In 1881 Warder (11, 12) prop,wd thr reli~tlonshlp (a k)(b T) = e Often A and E can he treated as temperature independent. + - (7) For the analvsis of more precise rate-temperature data, par- where a, b, and c are constants, and later Mellor (4) pointed ticularly those covering a wide temperature range, it is usual out that if one multiplies out this formula, expands the factor to allow A to he proportional to T raised to a power m, so that in T and accepts only the first term, the result is the equation hecomes k = a' + b'T2 (8) k = A'T~~~-E/RT (3) Equation (7) received some support from results obtained by where now A' is temperature independent.
    [Show full text]
  • Application of Arrhenius Equation and Plackett-Burman Design to Ascorbic Acid Syrup Development
    Naresuan University Journal 2004; 12(2) : 1-12 1 Application of Arrhenius Equation and Plackett-Burman Design to Ascorbic Acid Syrup Development Srisagul Sungthongjeen Department of Pharmaceutical Technology, Faculty of Pharmaceutical Sciences, Naresuan University, Phitsanulok 65000, Thailand. Corresponding author. E-mail address: [email protected] (S. Sungthongjeen) Received 19 April 2004; accepted 29 June 2004 Abstract Stability testing and drug-excipients compatibility of pharmaceutical dosage forms are necessary to perform during the early stages of product development. The aim of this study was to apply Arrhenius equation and Plackett-Burman design for stability testing and drug-excipients compatibility of formulations, respectively. The stability of ascorbic acid in syrup was examined using an accelerated stability testing according to the Arrhenius equation. The formulations were stored at room temperature and over the temperature range of 40-70°C. A linear regression line was obtained from Arrhenius plot of the reaction rate (k) against reciprocal of degree kelvin. The heat of activation was found to be 17.96 kcal/mole. Plackett-Burman experimental design was used to investigate the compatibility of ascorbic acid with various syrup excipients. It was found that glycerin (5%v/v) exerted a significant stabilizing effect on ascorbic acid, whereas sugar cane syrup (33.3%v/v) had destabilizing effect. The other excipients appeared to be no significant effect on ascorbic acid stability. The study demonstrated that Arrhenius equation would be useful for the prediction of product stability and the stabilizing or destabilizing effects of excipients could be identified by Plackett-Burman design. Keywords: Arrhenius equation, Plackett-Burman design, Ascorbic acid syrup Introduction Stability testing of pharmaceutical dosage forms usually begins during the early stages of product development, the main purpose is to establish a product shelf life.
    [Show full text]