Unsolved Classical Problems

IV. UNSOLVED CLASSICAL PROBLEMS A. Finite or Infinite Space... 

Simplicity and reliability of operation make AC impedance measurements attractive as a technique in the evaluation of coating integrity. As opposed to classical salt spray test, analysis times are shorter with the AC impedance technique and quantitative data are obtained permitting relevant mechanistic Information to be derived. Impedance test methods are likely to find many applications in the resolution of unsolved practical problems ( .) ... 

But chaos is more than a tool. There are as yet unsolved philosophical problems in its wake. While relativity and quantum mechanics necessitated - and in fact originated from - a careful analysis of the concepts of space, time and measurement, chaos, already on the classical level, forces us to re-think the concepts of determinism and predictability. Thus, classical mechanics could not be further removed from the dusty subject it is usually portrayed as. On the contrary it is at the forefront of modern scientific research. Since path integrals provide a link between classical and quantum mechanics, conceptual and philosophical problems with classical mechanics are bound to manifest themselves on the quantum level. We are only at the beginning of a thorough exploration of these questions. But one fact is established already chaos has a profound in-fiuence on the quantum mechanics of atoms and molecules. This book presents some of the most prominent examples. 

The classical problems of particle transport were studied by well-known physicists in the late nineteenth and early twentieth centuries. Stokc.s, Einstein, and Millikan investigated the motion of smalt spherical particles under applied forces primarily because of application to (at that lime) unsolved probleins in physics. They derived relatively simple relationships for spherical particles that can be considered as ideal cases irregular particles are usually discussed in terms of their deviations from spherical behavior. 

The problem of producing quite satisfactory LF Dynamite remained unsolved until S. Nauckhoff of Sweden published his classical work iii ZAngewChem 18, ppll 53 (1905). In this work he calculated the molecular lowering of the fr p of NG by a great number of compds and formulated the requirements for a satisfactory. antifreeze... 

Theoretical chemistry has two problems that remain unsolved in terms of fundamental quantum theory the physics of chemical interaction and the theoretical basis of molecular structure. The two problems are related but commonly approached from different points of view. The molecular-structure problem has been analyzed particularly well and eloquent arguments have been advanced to show that the classical idea of molecular shape cannot be accommodated within the Hilbert-space formulation of quantum theory [161, 2, 162, 163]. As a corollary it follows that the idea of a chemical bond, with its intimate link to molecular structure, is likewise unidentified within the quantum context [164]. In essence, the problem concerns the classical features of a rigid three-dimensional arrangement of atomic nuclei in a molecule. There is no obvious way to reconcile such a classical shape with the probability densities expected to emerge from the solution of a molecular Hamiltonian problem. The complete molecular eigenstate is spherically symmetrical [165] and resists reduction to lower symmetry, even in the presence of a radiation field. 

The numerical value of kt in (2-3) depends on how activity is defined and on the units in which concentration is expressed (molarity, mole fraction, partial pressure). Measurement of the absolute activity, or chemical potential, of an Individual ion is one of the classical unsolved problems. Since we cannot measure absolute ion activity, we are then necessarily interested in the next best—comparative changes in activities with changing conditions. To obtain comparative values numerically, we measure activity with respect to an arbitrarily chosen standard state under a given set of conditions of temperature and pressure, where the substance is assigned unit activity. The value of ki in (2-3) thus depends on the arbitrary standard state chosen accordingly, the value of the equilibrium constant also depends on the choice of standard states. 

The second method is more efficient but problems occur when passing from one adsorbate to another if the low coverage isosteric heat of adsorption is not accurately determined. If this problem can be overcome, it is possible to obtain a unique characteristic curve. Unfortunately the classical laws relating the characteristic curve to the structural properties of the adsorbent are no longer valid Anyway, the second method has been successfully used for the evaluation of the buoyancy effect on the adsorbed phase which was still an unsolved problem of the high pressure measurements. 

The quantum enhancement by the factor cosh PhcOij/2) appears clearly. Nevertheless, for larger energies (fttoy > kT) no analytical quantum correction is available, and this problem remains unsolved. In this limit of moderate to large energy release, classical correlation function approaches are only qualitatively valid. 

The mechanisms of acid-catalyzed DME formation from methanol and aromatiza-tion of olefins were widely investigated in the years before the discovery of the methanol-to-gasoline reaction. There is a consensus that the intermediate in DME formation from methan.ol over solid acid catalysts is a protonated surface methoxyl, which is subject to nucleophilic attack by methanol [2]. Aroma-tization of olefins is believed to proceed along classical carbenium pathways, with concurrent hydrogen transfer [3]. The mechanism of the crucial step of initial C-C bond formation from MeOH/DME is an unsolved problem, however, and is the subject of ongoing controversy. At last tally, there were some two dozen mechanistic proposals in the literature. It is not possible here to present a comprehensive review of the entire field. However, a number of common themes can be identified. This commonality is discussed and the concepts currently in vogue are critically reviewed. Another issue, whether ethylene is the "first" olefin, has been widely debated [2], but is beyond the scope of this survey. 

The fundamental problem in classical equilibrium statistical mechanics is to evaluate the partition function. Once this is done, we can calculate all the thermodynamic quantities, as these are typically first and second partial derivatives of the partition function. Except for very simple model systems, this is an unsolved problem. In the theory of gases and liquids, the partition function is rarely mentioned. The reason for this is that the evaluation of the partition function can be replaced by the evaluation of the grand canonical correlation functions. Using this approach, and the assumption that the potential energy of the system can be written as a sum of pair potentials, the evaluation of the partition function is equivalent to the calculation of... 

Finally, it will be demonstrated that fracture mechanics provides insight into the preferred path of crack growth, A perplexing problem that, despite considerable speculation over the last 50 years, has remained unsolved is why do cracks follow a particular path For example, for many practical joints, it has been observed that cracks typically proceed in the adhesive rather than along the interface. This phenomenon is so common that, in his classical text on adhesives, Kinloch states [11] ... 

The term "many-electron atoms includes all atoms and atomic ions with more than one extra-nuclear electron. Helium is the simplest many-electron atom. Even in this case the Schrodinger equation cannot be solved analytically. The helium atom, with two electrons and one nucleus, is an example of the Three-body Problem, the equation of motion of which remains unsolved also in Classical Mechanics. The difficulty is that the motion of every particle, in a many-body problem, is coupled to the motion of all the other particles. 

Solution of the black-body problem had an immediate knock-on effect on two other unsolved problems of classical physics, the photoelectric effect and the interpretation of atomic spectra. 

Initially we will examine homogeneous gas phase nucleation. This is the classic test problem of nucleation theory. It is believed that condensation of a gas is the simplest nucleation problem. This problem provides a test of the conceptual usefulness of nucleation theory. The lessons learned in discovering how to solve this as yet unsolved problem will hopefully provide a guide to more complex nucleation problems. 

The first microvalve was introduced by Terry [1], in 1979, which was the first magnetic MEMS microvalve. After that, microvalves were improved in several ways. Around the year 2000 a revolution in fabrication of microvalves happened [2—4] which solved the problems of the various MEMS-based microvalves. However, many problems (such as (i) using moving parts that causes additional problems and difficulties, (ii) external actuation means, (iii) complex fabrication and installation processes, (iv) resistible flow and pressure, (v) considerable dead volume, (vi) long respmise time, (vii) leakage, and (viii) stability) remain unsolved when classical electrokinetic theory is used to design a microvalve. 

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