Kinetic molecular theory quantitative

We can apply the kinetic-molecular theory quantitatively to phase changes by means of a heating-cooling curve, which shows the changes that occur when heat is added to or removed from a particular sample of matter at a constant rate. As an example, the cooling process is depicted in Figure 12.3 for a 2.50-mol sample of gaseous water in a closed container, with the pressure kept at 1 atm and... 

We now have enough information to turn our qualitative ideas about a gas into a quantitative model that can be used to make numerical predictions. The kinetic model ( kinetic molecular theory, KMT) of a gas is based on four assumptions (Fig. 4.23) ... 

Beginning with these assumptions, it s possible not only to understand the behavior of gases but also to derive quantitatively the ideal gas law (though we ll not do so here). For example, let s look at how the individual gas laws follow from the five postulates of kinetic-molecular theory ... 

We have just seen how each of the gas laws conceptually follows from kinetic molecular theory. We can also derive the ideal gas law from the postulates of kinetic molecular theory. In other words, the kinetic molecular theory is a quantitative model that implies PV = nRT. Let s now explore this derivation. 

Kinetic molecular theory is a quantitative model for gases. The theory has three main assumptions (1) the gas partieles are negligibly small, (2) the average kinetic energy of a gas particle is proportional to the temperature in kelvins, and (3) the collision of one gas particle with another is completely elastic (the particles do not stick together). The gas laws all follow from the kinetic molecular theory. 

For a quantitative description of the behavior of gases, we will employ some simple gas laws and a more general expression called the ideal gas equation. These laws will be explained by the kinetic-molecular theory of gases. The topics covered in this chapter extend the discussion of reaction stoichiometry from the previous two chapters and lay some groundwork for use in the following chapter on thermochemistry. The relationships between gases and the other states of matter— liquids and solids—are discussed in Chapter 12. 

It is worth noting at this point that the various scientific theories that quantitatively and mathematically formulate natural phenomena are in fact mathematical models of nature. Such, for example, are the kinetic theory of gases and rubber elasticity, Bohr s atomic model, molecular theories of polymer solutions, and even the equations of transport phenomena cited earlier in this chapter. Not unlike the engineering mathematical models, they contain simplifying assumptions. For example, the transport equations involve the assumption that matter can be viewed as a continuum and that even in fast, irreversible processes, local equilibrium can be achieved. The paramount difference between a mathematical model of a natural process and that of an engineering system is the required level of accuracy and, of course, the generality of the phenomena involved. 

The relation of rate constants to molecular structure is of course the ultimate goal of any molecular theory of kinetics. As we have already seen, the theoretical approach to the problem of correlating preexponential factors with molecular theory has certainly been qualitatively successful, if somewhat lacking from a quantitative point of view. However, the usefulness of even qualitative correlation is considerable in the guide that it provides to the experimentalist in analyzing the details of a complex mechanism. It is unfortunate that at present there is no similarly useful means for correlating activation energies. 

Ultimately physical theories should be expressed in quantitative terms for testing and use, but because of the eomplexity of liquid systems this can only be accomplished by making severe approximations. For example, it is often neeessary to treat the solvent as a continuous homogeneous medium eharaeterized by bulk properties such as dielectric constant and density, whereas we know that the solvent is a molecular assemblage with short-range structure. This is the basis of the current inability of physical theories to account satisfactorily for the full scope of solvent effects on rates, although they certainly can provide valuable insights and they undoubtedly capture some of the essential features and even cause-effect relationships in solution kinetics. Section 8.3 discusses physical theories in more detail. 

In the final section of this chapter, we shall attempt to give a brief rationalization of the regularities and peculiarities of the reactions of non-labile complexes which have been discussed in the previous sections. The theoretical framework in which the discussion will be conducted is that of molecular orbital theory (mot). The MOT is to be preferred to alternative approaches for it allows consideration of all of the semi-quantitative results of crystal field theory without sacrifice of interest in the bonding system in the complex. In this enterprise we note the apt remark d Kinetics is like medicine or linguistics, it is interesting, it js useful, but it is too early to expect to understand much of it . The electronic theory of reactivity remains in a fairly primitive state. However, theoretical considerations may not safely be ignored. They have proved a valuable stimulus to incisive experiment. 

The science of reaction kinetics between molecular species in a homogeneous gas phase was one of the earliest fields to be developed, and a quantitative calculation of the rates of chemical reactions was considerably advanced by the development of the collision theory of gases. According to this approach the rate at which the classic reaction... 

Orbital interaction theory forms a comprehensive model for examining the structures and kinetic and thermodynamic stabilities of molecules. It is not intended to be, nor can it be, a quantitative model. However, it can function effectively in aiding understanding of the fundamental processes in chemistry, and it can be applied in most instances without the use of a computer. The variation known as perturbative molecular orbital (PMO) theory was originally developed from the point of view of weak interactions [4, 5]. However, the interaction of orbitals is more transparently developed, and the relationship to quantitative MO theories is more easily seen by straightforward solution of the Hiickel (independent electron) equations. From this point of view, the theoretical foundations lie in Hartree-Fock theory, described verbally and pictorially in Chapter 2 [57] and more rigorously in Appendix A. 

Analysis of networks in terms of molecular structure relies heavily on the kinetic theory of rubber elasticity. Although the theory is very well established in broad outline, there remain some troublesome questions that plague its use in quantitative applications of the kind required here. The following section reviews these problems as they relate to the subject of entanglement. 

In reality, the data on isothermal contraction for many polymers6 treated according to the free-volume theory show that quantitatively the kinetics of the process does not correspond to the simplified model of a polymer with one average relaxation time. It is therefore necessary to consider the relaxation spectra and relaxation time distribution. Kastner72 made an attempt to link this distribution with the distribution of free-volume. Covacs6 concluded in this connection that, when considering the macroscopic properties of polymers (complex moduli, volume, etc.), the free-volume concept has to be coordinated with changes in molecular mobility and the different types of molecular motion. These processes include the broad distribution of the retardation times, which may be associated with the local distribution of the holes. 

Although possessing certain inherent limitations (Benson, 1960a), transition state theory seems adequate to permit the quantitative computation of kinetic parameters from first principles. As we have seen, however, practical application of the theory is impeded by incomplete information about the molecular properties of the activated complex and, for reactions in solution, the lack of a quantitative description of molecular interactions in condensed phases. It would be highly useful, therefore, to have some other basis on which to assess... 

Despite these difficulties, the kinetic theory in its simple equilibrium approximation and in its more accurate nonequilibrium representation is capable of reproducing physical behavior in a form which is mathematically simple, qualitatively correct in so far as it represents the interdependence of physical variables, and quantitatively correct to within better than an order of magnitude. As such it presents a valuable direct insight into the relations between molecular processes and macroscopic properties and, as we shall see, provides a valuable guide to understanding kinetic behavior. 

The solvated electron is a transient chemical species which exists in many solvents. The domain of existence of the solvated electron starts with the solvation time of the precursor and ends with the time required to complete reactions with other molecules or ions present in the medium. Due to the importance of water in physics, chemistry and biochemistry, the solvated electron in water has attracted much interest in order to determine its structure and excited states. The solvated electrons in other solvents are less quantitatively known, and much remains to be done, particularly with the theory. Likewise, although ultrafast dynamics of the excess electron in liquid water and in a few alcohols have been extensively studied over the past two decades, many questions concerning the mechanisms of localization, thermalization, and solvation of the electron still remain. Indeed, most interpretations of those dynamics correspond to phenomenological and macroscopic approaches leading to many kinetic schemes but providing little insight into microscopic and structural aspects of the electron dynamics. Such information can only be obtained by comparisons between experiments and theoretical models. For that, developments of quantum and molecular dynamics simulations are necessary to get a more detailed picture of the electron solvation process and to unravel the structure of the solvated electron in many solvents. 

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