Chemical Engineering - Chemical Reaction Engineering and Applied Chemical Kinetics

Chapter 4

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Notes of Applied Chemical Kinetics course Carlo Cavallotti Chapter 1. Introductory aspects and classification of chemical reactions Chapter 2. Kinetic schemes and reaction mechanisms Chapter 3. Kinetic theory of Gases Chapter 4. Fundamentals of Statistical Thermodynamics and Molecular Quantum Mechanics Chapter 5. Transition State Theory and further developments Chapter 4. Fundamentals of Statistical Thermodynamics and Molecular Quantum Mechanics 4.1 Introduction and molecular kinematics Although kinetic theory of gases allows understanding many fundamental aspects of the gas phase reactivity, the attempt to apply it systematically to estimate the kinetic constant of complex reactions pointed out its fundamental limitations. In particular it has been demonstrated that, to describe adequately a chemical reaction characterized by a significant rearrangement of the molecular structure of reactants in the transition to the configuration of the reaction products, it is necessary to describe explicitly the internal motion of the atoms. Such aspect is not considered in the kinetic theory of gases, which considers molecules as spheres. The explicit treatment of internal molecular motions is a complex problem, as it requires describing the evolution in time of the spatial configuration of each atom involved in the reactive process. Defining N the total number of atoms involved in the reaction, the total number of coordinates of which we have to describe the evolution in time is 3N, i.e. three coordinates (xi, yi, zi) for each atom. From this point of view, the kinetics of a molecular system can be equated to that of an integral system made by N spheres of mass mi. The study of the kinematics of such systems is a problem that is well known in analytical mechanics, and it is usually faced using as reference the center of mass of the multibody system and decomposing the molecular motions in two typologies: external motions, related to translation and rotation in space of the molecular system considered as a whole, and internal molecular motions. As the atoms are linked between them by molecular bonds, internal motions can be described, at a first level of approximation, as oscillations around equilibrium positions. Assuming that the restoring force to the equilibrium position is proportional to the displacement, and that the internal motions are independent among them, the internal molecular motions can be treated as a set of harmonic oscillators. Such treatment of the kinematics of a multibody system is called adiabatic harmonic oscillator approximation, and will be used later in these notes. The limitations due to this approach will be discussed at the end of this chapter, and in chapter 5. Summarizing, the kinematics of a multiatomic system requires the description of the temporal evolution of a set of 3N coordinates of which: − 3 represent the external translational motions of the system referred to its center of mass. − 3 represent the rotation of the molecule with respect to the three axes of inertia. − the remaining degrees of freedom, 3N-6, can be described as oscillations around the equilibrium positions. In the case of a linear molecule, the distribution of the degrees of freedom is different as one of the three principal moments of inertia is null. Thus only 2 external rotational motions, and 3N-5 internal oscillations are allowed. Both the external roto-translational motions and the internal vibrational motions affect the reaction rate, and thus it is necessary to develop a theory that is able to adequately describe them properly in order to develop a consistent kinetic theory. In literature, three approaches exist that differ significantly for the level of complexity. Here we will briefly describe their characteristics: − classical molecular dynamics in a reactive force field: the dynamics of the system is described by integrating the second law of Newton (F=ma) for each atom in a force field V, that is a function of the coordinates of every atom, such that the force that is exerted on each atom is given by F=-∇V. − description of the energy of the system, and of its distribution in the molecular degrees of freedom using statistical thermodynamics, and quantum mechanics. The reactive properties of the system are extrapolated by the static study of the potential energy surface using transition state theory. − quantum molecular dynamics based on the integration of the Schrodinger equation, in its time dependent form. The extrapolation of the kinetic constant for a generic reaction using the first or the third approach by direct integration of the dynamic equations is usually a slow process, and it is extremely heavy from a computational point of view, although recently it is more and more applied to study certain aspects of some chemical processes, such as the dynamics of protein systems. The most used approach in literature, which allows predicting values of kinetic constants that are in excellent agreement with experimental data, is the second one. To systematically apply it, however, it is necessary to know some of the fundamentals of statistical thermodynamics and quantum mechanics that, not being part of the program of the chemical engineering course anymore, will be presented in the next paragraphs. The exposition here reported has been organized so that it allows to have both a deep understanding of the fundamental concepts the kinetic theory that will be presented in the next chapter and to allow to the student to apply them directly to the estimation of molecular properties and kinetic constants of interest. 4.2 Statistical thermodynamics: historical introduction and microcanonical systems Statistical thermodynamics comes from the brilliant insights of Ludwig Boltzmann, who, at the beginning of his scientific career (1866), set himself to give a purely analytical and completely general formulation of the second law of thermodynamics. For this purpose Boltzmann relied on the intuitions of Maxwell, who pointed out that the impossibility of violating the second law of thermodynamics is due to its statistical nature. The demonstration of Maxwell of the violability of the second law of thermodynamics was based on the assumption that a being (later known as Maxwell’s demon) may exist with such knowledge of the properties of a macroscopic system that he may be able to separate the fastest particles from the slowest ones in a reversible way, which is in violation of the laws of thermodynamics. The impossibility of the existence of a being with such knowledge is due to the difficulty of knowing exactly the properties of a macroscopic thermodynamic system, the number of components of which is usually in the order of magnitude of the Avogadro number (1024), i.e. extraordinarily elevated. On these bases Boltzmann set himself to give a statistical formulation of the second principle, and in particular to find an analytical expression for the state function introduced in some formulations of the second principle of thermodynamics, known as entropy. The approach of Boltzmann is based initially on the study of the properties of an isolated thermodynamical system. That is a portion of space defined by a closed surface, called control surface, which cannot be crossed neither by energy nor by mass. In statistical thermodynamics, isolated systems are known as microcanonical systems. If it is assumed that in the system N particles are present, each one with a particular energy εi, the total energy of the system is given by: € E = ε i i = 1 N ∑ (4.1) For a system with such characteristics, Boltzmann introduces the concept of microstate, defined as a particular state of the system, characterized by a well defined microscopic energetic configuration, where by energetic configuration it is intended the distribution of the energy among the particles that form the microstate. The concept of microstate has been introduced to underline that, for a fixed total energy of a microstate, there is more than one way to distribute the energy among its particles. Since this aspect is of fundamental importance, it is good to clarify it with an example. Consider a microcanonical system formed by 3 molecules, which can assume energies ε1, ε2, or ε3 such that ε2=2ε1 and ε3=3ε1. Be 5ε1 the total energy E of the system. To count the microstates it is necessary to consider two possibilities, depending on whether the molecules are distinguishable between them, and thus we can assign to each one a different symbol (e.g. A, B, or C), or undistinguishable, thus the identification symbol is the same for all (e.g. A, A, A). The energy can be distributed in different ways, as summarized in table 4.1 Table 4.1 Possible microstates for a microcanonical system formed by three particles that can have energies ε1, ε2, or ε3 such that ε2=2ε1 and ε3=3ε1. Distinguishable Undistinguishable Microstate A B C ETOT A A A ETOT 1 ε3 ε1 ε1 5ε1 ε3 ε1 ε1 5ε1 2 ε1 ε3 ε1 5ε1 ε1 ε3 ε1 5ε1 3 ε1 ε1 ε3 5ε1 ε1 ε1 ε3 5ε1 4 ε2 ε2 ε1 5ε1 ε2 ε2 ε1 5ε1 5 ε2 ε1 ε2 5ε1 ε2 ε1 ε2 5ε1 6 ε1 ε2 ε2 5ε1 ε1 ε2 ε2 5ε1 The table shows that, for the same energy level, 6 different microstates can exist for a system where the particles are distinguishable. However the situation is different for undistinguishable particles, as it is easy to verify that in such case microstates 1, 2, and 3, and microstates 4, 5, and 6 are identical between them. Thus for undistinguishable particles the total number of microstates is 2. The analysis of microstates reported in Table 4.1 allows introducing the concept of degree of degeneration of a microstate, which we will see later in these notes. It can be noted that, for distinguishable particles, states 1, 2, and 3 have the same way of distribution of the energy (a molecule with energy ε3 and two with energy ε1), but they differ for the way this is assigned to different molecules. In such case it is said that there is only one microstate, where one molecule has energy ε3 and two have energy ε1, that in the case of distinguishable particles has degree of degeneration 3, while in the case of distinguishable particles it has degree of degeneration 1. The same holds for microstates 4, 5, and 6. The total number of microstates of a system with energy E is a quantity that strongly characterized it and for this reason it has been defined by Boltzmann as Ω(E). It is now possible to introduce the definition of probability of a generic microstate i, defined as the ratio between the number of microstates of type i and the total number of microstates. It can be expressed mathematically as: € ω i = N i E ( ) Ω E ( ) (4.2) in the example above, the microstate where one molecule has energy ε3 and two molecules have energy ε1 has a probability of 0.5, both for distinguishable and undistinguishable particles. For a system of undistinguishable particles (which is the typical characteristic of molecules), as the degree of degeneration of a microstate is 1, 4.2 becomes: € ω i = 1 Ω E ( ) (4.3) Based on these definitions, Boltzmann introduces two fundamental postulates, on which all the statistical thermodynamics holds. First postulate. The observed value, averaged in time, of one of the properties of a thermodynamic system at the equilibrium is equal to the mean value of such quantity. In particular, if a variable Yi is associated to a microstate i, the observed mean value of that variable is given by: € Y = ω i Y i i = 1 Ω ( E ) ∑ (4.3) Thus, the characteristic averages of the system can be expressed through the average over the microstates of the correspondent statistical ensemble. This postulate, also known as ergodic theorem, gives a link between the values of the thermodynamic properties calculated with statistical methods, and those observed through experimental measurements. The second postulate is associated to the first. Second postulate. In a thermodynamic system at the equilibrium all the accessible states have the same probability. The meaning of the second postulate is that, even if a microstate is difficult to reach because of a particular microscopic energetic configuration, its probability of existence is however independent from possible kinetic limitations that may affect its accessibility. It is now possible to introduce the definition of entropy of a microcanonical system, formulated by Boltzmann based on the introduction of the concept of microstate: S = kblnΩ(E) (4.4) According to expression 4.4, entropy is directly proportional to the natural logarithm of the total number of microstates of a system. The proportionality constant, that has the value of 1.38x10-23 J/mol/K, is called Boltzmann constant. The choice of a logarithmic dependence on the total number of states can be motivated on the basis of the property of entropy of being an additive function. In fact, given a system formed by its parts A and B, its total entropy is given by the sum of the entropy of A and of that B. Similarly, the number of microstates of the global system is given by the product of the number of microstates A and B (it is left to the student the simple demonstration of this property). It is easy to demonstrate that 4.4 satisfies the additive property: € S tot = S A + S B = k B ln Ω E A ( ) + k B ln Ω E A ( ) = k B ln Ω E A ( ) Ω E B ( ) = k B ln Ω E ( ) (4.5) Relation 4.4 meets the ambition of Boltzmann of discovering an analytical definition of entropy, and it is one of the most important equations of physics. Starting from that, in the next chapters we will derive the fundamental relations of statistical thermodynamics that will accompany us in the study of chemical kinetics. 4.3 Macrocanonical systems. Microcanonical systems are often used in thermodynamics because of their simplicity in order to study fundamental theoretical aspects. However, they can difficultly be used to study to real systems, which usually interacts in a significantly with the environment, either with the exchange of energy or of matter. In particular, a system that exchanges energy with the environment, but not matter, is defined in thermodynamics as a closed system, and in statistical thermodynamics as canonical, or macrocanonical, system. A macrocanonical system can be represented as a portion of space defined by a control surface, a portion of which is in contact with a thermal bath, that is maintained at constant temperature and that exchanges energy continuously with the system. In a macrocanonical system energy is not constant in time, but it statistically oscillates around a mean value following a trend similar to that reported in Figure 4.1. The extension of the concepts introduced for a microcanonical system to a macrocanonical one requires the redefinition of the probability of being in a particular energetic state with total energy Er. For this purpose consider a system formed by the sum of a macrocanonical system (S) and the environment (A). If we define as E the total energy of such system, the energy of the environment is Ea = E – Er. Considering the system sum of S+A, the probability of the system S of having energy Er is equal to the probability of the system S+A of having an energy that is available to be distributed among its degrees of freedom E = Er, being energy Er the energy of the system S. Thus the probability of a system with energy Er can be defined as a function of the ratio between the total number of microstates that are accessible to system A when the available energy is E – Er, and those of the system S+A with total energy E, as: ω S E R ( ) = ω A E − E R ( ) = Ω A E − E R ( ) Ω S + A E ( ) (4.6) It is also known that the total number of microstates for a molecular system (a rigorous demonstration exists for a monoatomic ideal gas and for a set of oscillators) is proportional to the energy E elevated to a constant n, which is the total number of particles in the system (i.e. a very high number): € Ω E ( ) = CE n (4.7) where C is an arbitrary multiplicative constant. E t Figure 4.1 Temporal oscillations of the energy of a macrocanonical system. Introducing 4.7 into 4.6, we get: € ω S E R ( ) = Ω A E − E R ( ) Ω A + S E ( ) = C 1 E − E R ( ) n C 2 E ( ) n = C 1 − E R E % & ' ( ) * n (4.8) remembering that if Er