Molecular kinetics

Rate equations are quantitative models of the time course of chemical reactions. Although rate equations are based on macroscopic observations, they reflect processes that occur at the molecular scale. This chapter reviews some of the important models that link these two scales. These models are especially useful because they constrain the mathematical form of rate equations and they provide a conceptual basis for thinking about the reactions. Because water is so important in geochemical systems, this chapter focuses on models for reaction rates in the aqueous phase. 

The idea that molecules must collide to form a cluster correctly predicts that the reaction rate increases with concentration (or pressure) because increasing the number of molecules per unit volume will result in more collisions. This is the reason that rate equations stipulate that rates increase with increasing concentration, pressure, or activity. 

Transition-state theory, invented by Eyring (1935) and Evans and Polanyi (1935), is a statistical mechanics model that quantitatively accounts for the three factors that control the reaction process. Eyring used the term activated complex and Evans and Polanyi used the term transition state , to describe the cluster of molecules that is transitional between the reactants and products. These terms continue to be used interchangeably. 

Transition-state theory arose from statistical mechanics and describes the distribution of energy among molecules using partition functions. This means that a firm grasp of the theoretical imderpinnings of transition-state theory requires an understanding of statistical mechanics. Houston (2001) describes many of the concepts discussed here from a statistical mechanics point of view and Fueno (1999) explains partition functions. However, the quasi-equilibrium model presented here is a simpler approach that illustrates 

The quasi-equilibrium model considers a simple reaction where atom A reacts with molecule BC and displaces atom C to produce molecule AB. 

The specific problems discussed in this book require the use of fundamental concepts and equations from various fields like kinetic theory of gases, kinetics of chemical reactions, thermodynamics and mass transfer. This chapter presents some basic relationships relevant to these problems. From the very beginning, the studies of gas-phase radiochemistry of heavy metallic elements have been largely motivated by the quest for new man-made chemical elements. It necessitated experimentation with very short-lived nuclides on one-atom-at-a-time basis. We will pay much attention to this direction of research. Accordingly, we will consider microscopic pictures (at the atomic and molecular level) of the processes underlying the experimental methods and concrete techniques, and follow individual histories of the molecules. 

References are given in the present chapter only when the approach or treatment may be less known. 

Most of the formulae below give the average values of quantities. For integer quantities, like concentration or frequency of collisions, in the case of poor statistics, the possible uncertainties in data evaluation are discussed when dealing with specific applied problems in Chapter 6. 

The average number of molecules per volume unit, the concentration, is  

The number of molecules per cubic centimeter at standard temperature and pressure (STP), the Loschmidt number, is 

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Kinetic theory A mathematical explanation of the behavior of gases on the assumption that gases consist of molecules in ceaseless motion in space. The molecular kinetic energy depends on the temperature of the gas. 

Fig. 8-4. Effect of temperature on atomic (or molecular) kinetic energy distribution.

A plot of the Maxwell distribution for the same gas at several different temperatures shows that the average speed increases as the temperature is raised (Fig 4.27). We knew that already (Section 4.9) but the curves also show that the spread of speeds widens as the temperature increases. At low temperatures, most molecules of a gas have speeds close to the average speed. At high temperatures, a high proportion have speeds widely different from their average speed. Because the kinetic energy of a molecule in a gas is proportional to the square of its speed, the distribution of molecular kinetic energies follows the same trends. 

Internal energy is stored as molecular kinetic and potential energy. The equipartition theorem can be used to estimate the translational and rotational contributions to the internal energy of an ideal gas. 

At a given temperature, all gases have the same molecular kinetic energy distribution. 

Determine the average molecular kinetic energy and molar kinetic energy of gaseous sulfiar hexafluoride, SFfi, at 150 °C. 

The energies generated by forces among ideal gas molecules are negligible compared with molecular kinetic energies. 

A gas will obey the ideal gas equation whenever it meets the conditions that define the ideal gas. Molecular sizes must be negligible compared to the volume of the container, and the energies generated by forces between molecules must be negligible compared to molecular kinetic energies. The behavior of any real gas departs somewhat from ideality because real molecules occupy volume and exert forces on one another. Nevertheless, departures from ideality are small enough to neglect under many circumstances. We consider departures from ideal gas behavior in Chapter if. 

Scheme 13 may look unfavorable on the face of it, but in fact the second two reactions are thermally allowed 10- and 14-electron electrocyclic reactions, respectively. The aromatic character of the transition states for these reactions is the major reason why the benzidine rearrangement is so fast in the first place.261 The second bimolecular reaction is faster than the first rearrangement (bi-molecular kinetics were not observed) it is downhill energetically because the reaction products are all aromatic, and formation of three molecules from two overcomes the entropy factor involved in orienting the two species for reaction. 

T. Ohta (2002) On the molecular kinetics of acoustic cavitation and the nuclear emission. Int. J. Hydrogen Energy, 27 in printing... 

At Harvard, Theodore William Richards, like Noyes, inherited a course in theoretical chemistry. He renamed it physical chemistry. However, he cautioned students that the molecular kinetic hypotheses might prove ephemeral, and, to the young Lewis s consternation, Richards showed contempt for the notion of chemical bonds. "Twaddle about bonds A very crude method of representing certain known facts about chemical reactions. A mode of representation] not an explanation. "68 It was not so much that Richards sided with energeticists against kinetic and mechanical representations, but he did have a distrust of mathematical formulations too far removed from the laboratory. When J. Robert Oppenheimer enrolled in Richards s course in physical chemistry in 1925, he pronounced it "a great disappointment,. .. a very meager hick course.. . . Richards was afraid of even rudimentary mathematics."69 Thus, physical chemistry by no means necessarily meant mathematical chemistry. 

If you feel the need of a hot cup of coffee, you are going to need to boil some water. Even though there are no obvious physical forces to overcome in this case, you still need a source of energy in order to increase the kinetic energy of the molecules of water. Temperature is a measure of molecular kinetic energy. So you turn on your electric kettle or light your stove burner or whatever to provide the necessary energy. 

Einstein A. (1905) The motion of small particles suspended in static liquids required by the molecular kinetic theory of heat. Ann. Phys. 17, 549-560. 

Based on a molecular kinetic model, Schweikert derives equations which. describe both the process of detonation in a condensed expl and that of the burning of a colloidal powder These processes are shown to differ primarily in the magnitude of the collision efficiency. Relations are derived which relate the max deton vel and pressure with molecular props... 

Soc 34, 985-89(1938) (Deton Expln arising out of thermal decompn) 21) H. Muraour, TrFaradSoc 34, 989-92(1938) (Theory of expl reactions) 22) A. Schmidt, SS 33, 121-25 (1938) (Expln deton considered from a molecular-kinetic standpoint) 23) Ibid,... 

Thermodynamic treatments in physical chemistry were effectively identical with the theory of the subjectin the nineteenth century. No oneunderstoodelectron transfer at interfaces at that time (J. J. Thompson did not discover the electron until 1897). But whereas the molecular kinetic approach gradually seeped into many parts of chemistry by the 1930s, the chemistry of electrode processes remained reluctantly bound up with the older thermodynamic viewpoint. The Faraday Society meeting in Manchester, U.K. in 1947 was a turning point in the application of a molecular-level concepts and even of quantum mechanics. By the mid-1950s, research papers in electrode process chemistry (except for those dealing with electroanalytica] themes)10 were fully kinetic. 

These measurements have been carried out in collaboration with de Maeyek[4]). The rate constant was found to be (1 3 0-2)-10n litres/ mol-sec thus the neutralization reaction is the fastest known bi-molecular reaction in aqueous solution. Molecular-kinetic considerations show that the velocity of recombination is solely determined by the collision frequency of the ions. Furthermore, the effective cross section of the proton is so large that the reaction already proceeds spontaneously when ions approach each other within a distance of two to three H-bonds. This means that the motion of the proton within the hydration complex (the diameter of which corresponds to about two to three H-bonds) proceeds rapidly compared to the actual movement of the ions towards each other. 

T,he stoichiometric characterization of detergent-protein complexes has been the object of many studies over the past 30 years (6). Recent studies have placed more emphasis upon developing a molecular-kinetic description of the complex (2, 8). The importance of such descriptions lies in the fact that detergent-protein complexes can be considered as lipoprotein model systems. Indeed, virtually all conceptions of the microscopic nature of lipid-protein interactions are based on the properties of detergent-protein complexes (3). 

Bodies of water at the same temperature have the same average molecular kinetic energies. The volume of the water has nothing to do with its temperature. 

In a series of papers published from 1905 to 1908, Einstein successfully incorporated the suspended particles into the molecular-kinetic theory of heat. He treated the suspended particles as being in every way identical to the suspending molecules except for the vast difference of their size. Tie set forth several relationships that were capable of experimental verification and he invited experimentalists to solve the problem. 

Internal energy is stored as molecular kinetic and potential energy. 

D is a scalar with respect to transformations of the three-dimensional space, but changes sign when t is replaced by —t. In molecular-kinetic theory this property of D is related to the well-known problems of irreversibility and entropy increase, the more so since by the dissipation theorem,... 

From the technology of combustion we move to the molecular mechanism of flame propagation. We shall give a molecular-kinetic expression for the heat release rate by calculating the frequency v of collisions of fuel molecules with other molecules (v is proportional to the molecular velocity and inversely proportional to the mean free path), further taking into account that only a small (1/j/) part of all collisions are effective. The quantity 1/v—the probability of reaction taken with respect to a single collision— depends on the activation heat of an elementary reaction event, as well as on the fraction of all molecules comprised of those radicals or atoms by means of which the reaction occurs. The molecular-kinetic expression for the coefficient of thermal conductivity follows from formulas (1.2.4) and (1.2.3). 

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