Eyring equation

Eyring equation Eyring (′ī?ri?) equation (physical chemistry) An equation, based on statistical mechanics, which gives the specific reaction rate for a chemical reaction in terms of the heat of activation, entropy of activation, the temperature, and various constants. The Eyring equation also known as Eyring–Polanyi equation in chemical kinetics relates the reaction rate to temperature. It was developed almost simultaneously in 1935 by Henry Eyring, M.G. Evans and Michael Polanyi. This equation follows from the transition state theory and is trivially equivalent to the empirical Arrhenius equation which are both readily derived from statistical thermodynamics in the kinetic theory of gases. The general form of the Eyring equation somewhat resembles the Arrhenius equation: ‡where ΔG is the Gibbs energy of activation. It can be rewritten as: To find the linear form of the Eyring equation: where: , = reaction rate constant , = absolute temperature , = enthalpy of activation , = gas constant , = Boltzmann constant , = Planck's constant , = entropy of activation Everything should be made as simple as A certain chemical reaction is performed at possible, but not simpler. different temperatures and the reaction rate is ― Albert Einstein ，，lnkT1Tdetermined. The plot of versus ‡,,HRgives a straight line with slope from which the enthalpy of activation can be ‡，，lnkh，,SRderived and with intercept from which the entropy of activation is derived. B References , Evans, M.G.; Polanyi M. (1935). "Some applications of the transition state method to the calculation of reaction velocities, especially in solution". Trans. Faraday Soc. 31: 875. doi:10.1039/tf9353100875. , Eyring, H. (1935). "The Activated Complex in Chemical Reactions". J. Chem. Phys. 3: 107. doi:10.1063/1.1749604. , Eyring, H.; Polanyi M. (1931). Z. Phys. Chem. Abt. B 12: 279. , Laidler, K.J.; King M.C. (1983). "The development of Transition-State Theory". J. Phys. Chem. 87: 2657–2664. doi:10.1021/j100238a002. , Polanyi, J.C. (1987). Some concepts in reaction dynamics. Science. 236. pp. 680–690 , 1 Eyring Equation Peter Keusch, University of Regensburg Both the Arrhenius and the Eyring equation describe the temperature dependence of reaction rate. Strictly speaking, the Arrhenius equation can be applied only to the kinetics of gas reactions. The Eyring equation is also used in the study of solution reactions and mixed phase reactions - all places where the simple collision model is not very helpful. The Arrhenius equation is founded on the empirical observation that rates of reactions increase with temperature. The Eyring equation is a theoretical construct, based on transition state model. The bimolecular reaction is considered by 'transition state theory'. According to the transition state model, the reactants are getting over into an unsteady ‡intermediate state (AB ) on the reaction pathway: There is an 'energy barrier' on the pathway between the reactants (A, B) and the product (C). The barrier determines a 'threshold energy' or minimum of energy necessary to permit the reaction to occur. It is called 'activation enthalpy' ('activation energy'). Fig. 1 shows the energy of the molecules along the reaction coordinate which measures the progress of the reaction. Along the flat region at the left, the particles are approaching each other. They possess kinetic energy and their potential energy is constant. The beginning of the rise in the curve signifies that the two molecules have enough energy to have an effect on each other. During the approach, the particles slow down as their kinetic energies furnish the potential energy to climb the curve. If the reacting particles possess sufficient energy they can ascend the left Figure 1: Energy profile side of the 'barrier' all the way up to the summit. E: Potential energy; Reaction coordinate: Attaining of the summit can be interpreted as parameter changing during the course of the follows: The approaching reactant molecules had reaction (as bond length or bond angle); Transition state: Maximum of energy in the sufficient kinetic energy to overcome the mutual path way repulsive forces between the electron clouds of their constituent atoms and thus come very close ‡to each other. An 'activated complex' AB or 'transition state' is formed at the potential energy maximum. The high-energy complex represents an unstable molecular arrangement, in which bonds break and form to generate the product C or to degenerate back to the reactants A and B. Once the energy barrier is surmounted, the reaction proceeds downhill to the product. 2 Principles of the transition state theory: (1) There is a thermodynamic equilibrium between the transition state and the state of reactants at the top of the energy barrier. (2) The rate of chemical reaction is proportional to the concentration of the particles in the high-energy transition state. ‡The change in the concentration of the complex AB over time can be described by the following equation: ‡Due to the equilibrium between the 'activated complex' AB and the reactants A and B, the ‡components k ? [A] ? [B] and k ? [AB ] cancel out. Thus the rate of the direct reaction is 1-1‡proportional to the concentration of AB : k is given by statistical mechanics: 2 -23-1k = Boltzmann's constant [1.381?10 J ? K] B T = absolute temperature in degrees Kelvin (K) -34h = Plank constant [6.626?10 J ? s] -12-1k is called 'universal constant for a transition state' (~ 6 ? 10 sec at room temperature). 2‡Additionally, [AB ] can be derived from the pseudo equilibrium between the transition state ‡molecule AB and the reactant molecules by application of the mass action law: ‡K = thermodynamic equilibrium constant Due to the equilibrium that will be reached rapidly, the reactants and the activated complex decrease at the same rate. Therefore, considering both equation (5) and (6), equation (4) becomes: Comparing the derived rate law (1) with expression (7) yields for the rate constant k of the overall reaction Additionally, thermodynamics gives a further description of the equilibrium constant: 3 ‡Furthermore G is given by , R = Universal Gas Constant = 8.3145 J/mol K ‡-1G = free activation enthalpy [kJ ? mol] ,‡-1-1S = activation entropy [J ? mol ? K] ,‡-1H = activation enthalpy [kJ ? mol] ,‡ H is the difference between the enthalpy of , the transition state and the sum of the enthalpies of the reactants in the ground state. It is called activation enthalpy (Fig. 2). S is for the entropy, the extent of randomness or disorder in a system. The difference between the entropy of the transition state and the sum of the entropies of the ‡reactants is called activation entropy S. ,‡G is the Gibb's free energy change. ,‡G is equal to the According to equation (10) ,‡change in enthalpy H minus the product of ,‡ T and the change in entropy S of temperature,‡Figure 2: Enthalpie of activation the chemical system. G may be considered to ,‡be the driving force of a chemical reaction. G ,‡‡determines the spontaneity of the reaction. G< 0 => reaction is spontaneous; G = 0 => ,,‡system at equilibrium, no net change occurs; G > 0 => reaction is not spontaneous , Combining equation (9) with expression (10) and solving for lnk yields: The Eyring equation is found by substituting equation (11) into equation (8): A plot of ln(k/T) versus 1/T produces a straight line with the familiar form y = -mx + b (Fig. ‡‡3), where x = 1/T; y = ln(k/T); m = -H / R; b = y (x = 0). H can be calculated from ,, ‡the slope m of this line: H = -m ? R . , From the y-intercept 4 ‡S can be determined and thus the calculation ,‡G for the appropriate reaction of , temperatures according to equation (10) is allowed. ‡ Figure 3: Determination of H , A comparison between the Arrhenius equation ‡‡and the Eyring equation (13) shows, that lnA and S on the one hand and E and H on ,,a the other hand are analogous quantities. The two energies are therefore frequently used interchangeably in the literature to define the activation barrier of a reaction. The activation ‡energy E is related to the activation enthalpy H as follows ,a ‡‡low values of E and H => fast rate; high values of E and H =>slow rate. The typical ,,aa‡values of E and H lie between 20 and 150 [kJ / mol]. ,a The study of the temperature dependence supplies the above all mechanistically important ‡‡values lnA or S, equivalent in their mechanistical significance. lnA- and S-values are ,, sensible sensors. They give informations about the degree of order in the transition state. (1) ‡low values of lnA correspond to large negative values of S (unfavorable). The activated , complex in the transition state has a more ordered or more rigid structure than the reactants in the ground state. This is generally the case if translational, rotational, and vibrational degrees of freedom become 'frozen' on the route from the initial to the transition state. The reaction rate is slow. (2) high values of lnA correspond to positive values (less negative values) of ‡S (favorable). A positive value for entropy of activation indicates that the transition state is , highly disordered compared to the ground state. Degrees of freedom are liberated in going from the ground state to the transition state, which, in turn, increase the rate of the reaction. References: Chemical Kinetics Kinetics: Characterization of Transition States Rate Law and Stoichiometry Convex Arrhenius plots and their interpretation 5