Microscopic theories energy state distributions

As E is decreased one observes a change from the unimodal distribution for subcritical clusters to a bimodal form indicating growth of supercritical clusters. Because the system is adiabatic, the biomodal distributions also represent stationary states in which there are maximum supercritical cluster sizes, which, if exceeded, result in destruction of that supercritical cluster size new bonds formed in the system increase the cluster kinetic energy and decrease the pressure of the monomer gas. In the future it would be desirable to extract from the molecular-dynamics calculation accurate values for the free energy of formation of clusters. Such calculations would resolve the differences between the B - D theory and the Lothe-Pound theory. In the future, molecular-dynamics calculations should make possible development of correct mesoscopic and microscopic theories of homogeneous and even heterogeneous nucleation. 

A microscopic theory may be developed by using a calculational scheme based on following the trajectories (position and velocity) of each molecule in the system. At each molecule-molecule or molecule-wall collision, new trajectories would have to be computed. Such calculations can be performed for limited number of molecules and short periods of time. Such calculations yield the probability distribution of particle velocities or kinetic energies. For example, the temperature of a monoatomic gas could then be computed from the average kinetic energy. Therefore, statistical thermodynamics determine probability distributions and average values of properties when considering all possible states of the molecules consistent with the constraints on the overall system. 

Many applications of probability theory to chemical engineering arise in statistical mechanics, the microscopic theory that imderpins thermodynamics. Consider a system whose state is described by the state vector q, such that the energy in this microstate is E(q). A key result of statistical mechanics is file Boltzmann distribution. For a system closed to its surroundings with respect to the exchange of mass, held at a constant temperature T and volume F,... 

More importantly, a molecular species A can exist in many quantum states in fact the very nature of the required activation energy implies that several excited nuclear states participate. It is intuitively expected that individual vibrational states of the reactant will correspond to different reaction rates, so the appearance of a single macroscopic rate coefficient is not obvious. If such a constant rate is observed experimentally, it may mean that the process is dominated by just one nuclear state, or, more likely, that the observed macroscopic rate coefficient is an average over many microscopic rates. In the latter case k = Piki, where ki are rates associated with individual states and Pi are the corresponding probabilities to be in these states. The rate coefficient k is therefore time-independent provided that the probabilities Pi remain constant during the process. The situation in which the relative populations of individual molecular states remains constant even if the overall population declines is sometimes referred to as a quasi steady state. This can happen when the relaxation process that maintains thermal equilibrium between molecular states is fast relative to the chemical process studied. In this case Pi remain thermal (Boltzmann) probabilities at all times. We have made such assumptions in earlier chapters see Sections 10.3.2 and 12.4.2. We will see below that this is one of the conditions for the validity of the so-called transition state theory of chemical rates. We also show below that this can sometime happen also under conditions where the time-independent probabilities Pi do not correspond to a Boltzmann distribution. 

In classical mechanics It Is assumed that at each Instant of time a particle is at a definite position x. Review of experiments, however, reveals that each of many measurements of position of Identical particles in identical conditions does not yield the same result. In addition, and more importantly, the result of each measurement is unpredictable. Similar remarks can be made about measurement results of properties, such as energy and momentum, of any system. Close scrutiny of the experimental evidence has ruled out the possibility that the unpredictability of microscopic measurement results are due to either inaccuracies in the prescription of initial conditions or errors in measurement. As a result, it has been concluded that this unpredictability reflects objective characteristics inherent to the nature of matter, and that it can be described only by quantum theory. In this theory, measurement results are predicted probabilistically, namely, with ranges of values and a probability distribution over each range. In constrast to statistics, each set of probabilities of quantum mechanics is associated with a state of matter, including a state of a single particle, and not with a model that describes ignorance or faulty experimentation. 

Experimental results are always crucial for any theory which aims to formulate basic physics behind observed phenomenon or property. However, an experiment always cover much wider variety of different influences which have impact on results of experimental observation than any theory can account for, mainly if theory is formulated on microscopic level and some unnecessary approximations and assumptions are usually incorporated. On the other hand, interpretation of many experimental results is based on particular theoretical model. This is also the case of ARPES experiments at reconstruction of Fermi surface for electronic structure determination of high-Tc cuprates. Interpretation of experimental results is based on band structure calculated for particular compound. Methods of band structure calculations are always approximate, with different level of sophistication. Calculated band structure, mainly its topology at EL, is a kind of reference frame for assignment of particular dispersion of energy distribution curve (EDC) or momentum distribution curve (MDC) to particular band of studied compound at interpretation of ARPES. This is in direct relation with theoretical understanding of crucial aspects of SC-state transition in general. 

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