Microscopic level, quantity

Let us underline some similarities and differences between a field theory (FT) and a density functional theory (DFT). First, note that for either FT or DFT the standard microscopic-level Hamiltonian is not the relevant quantity. The DFT is based on the existence of a unique functional of ionic densities H[p+(F), p (F)] such that the grand potential Q, of the studied system is the minimum value of the functional Q relative to any variation of the densities, and then the trial density distributions for which the minimum is achieved are the average equihbrium distributions. Only some schemes of approximations exist in order to determine Q. In contrast to FT no functional integrations are involved in the calculations. In FT we construct the effective Hamiltonian p f)] which never reduces to a thermo-... 

Given this context, the use of chemical symbols, formulae and equations can be readily misinterpreted in the classroom, because often the same representations can stand for both the macroscopic and sub-microscopic levels. So H could stand for an atom, or the element hydrogen in an abstract sense H2 could mean a molecule or the substance. One common convention is that a chemical equation represents molar quantities, so in Example 9 in Table 4.1,... 

The substance s molar mass is the mass in grams of the substance that contains one mole of that substance. In the previous chapter, we described the atomic mass of an element in terms of atomic mass units (amu). This was the mass associated with an individual atom. At the microscopic level, we can calculate the mass of a compound by simply adding together the masses in amu s of the individual elements in the compound. However, at the macroscopic level, we use the unit of grams to represent the quantity of a mole. 

The relationship above gives a way of converting from grams to moles to particles, and vice versa. If you have any one of the three quantities, you can calculate the other two. This becomes extremely useful in working with chemical equations, as we will see later, because the coefficients in the balanced chemical equation are not only the number of individual atoms or molecules at the microscopic level, but also the number of moles at the macroscopic level. 

At a microscopic level, even the smallest quantity of a compound is composed of a great number of individual molecules. Hence, the number of nuclei is so high that it is possible to reason statistically. 

To a chemist the entropy of a system is a macroscopic state function, i.e., a function of the thermodynamic variables of the system. In statistical mechanics, entropy is a mesoscopic quantity, i.e., a functional of the probability distribution, viz., the functional given by (V.5.6) and (V.5.7). It is never a microscopic quantity, because on the microscopic level there is no irreversibility. ... 

Perhaps the most remarkable feature of modem chemical theory is the seamless transition it makes from a microscopic level (dealing directly with the properties of atoms) to describe the structure, reactivity and energetics of molecules as complicated as proteins and enzymes. The foundations of this theoretical structure are based on physics and mathematics at a somewhat higher level than is normally found in high school. In particular, calculus provides an indispensable tool for understanding how particles move and interact, except in somewhat artificial limits (such as perfectly constant velocity or acceleration). It also provides a direct connection between some observable quantities, such as force and energy. 

The rearrangement of nuclei in an elementary chemical reaction takes place over a distance of a few angstrom (1 angstrom = 10 10 m) and within a time of about 10-100 femtoseconds (1 femtosecond = 10 15 s a femtosecond is to a second what one second is to 32 million years ), equivalent to atomic speeds of the order of 1 km/s. The challenges in molecular reaction dynamics are (i) to understand and follow in real time the detailed atomic dynamics involved in the elementary processes, (ii) to use this knowledge in the control of these reactions at the microscopic level, e.g., by means of external laser fields, and (iii) to establish the relation between such microscopic processes and macroscopic quantities like the rate constants of the elementary processes. 

It is obvious that such a definition of solvent polarity cannot be measured by an individual physical quantity such as the relative permittivity. Indeed, very often it has been found that there is no correlation between the relative permittivity (or its different functions such as l/sr, (sr — l)/(2er + 1), etc.) and the logarithms of rate or equilibrium constants of solvent-dependent chemical reactions. No single macroscopic physical parameter could possibly account for the multitude of solute/solvent interactions on the molecular-microscopic level. Until now the complexity of solute/solvent interactions has also prevented the derivation of generally applicable mathematical expressions that would allow the calculation of reaction rates or equilibrium constants of reactions carried out in solvents of different polarity. 

The charge system that we are concerned with is one in which the charges are the sub-atomic charged particles, electrons and protons, which make up dielectric materials. It is clear that the quantities D, E, and p must now be regarded as inaccessible to simple macroscopic measurements. Indeed we shall regard D, E, and p as variables at a microscopic level and our immediate task is to find a means of relating them to corre ond-ing macroscopic quantities. 

As thermodynamics required postulates or laws, so does statistical mechanics. Gibbs postulates which define statistical mechanics are (1) Thermodynamic quantities can be mapped onto averages over all possible microstates consistent with the few macrosopic parameters required to specify the state of the system (here, NVE). (2) We construct the averages using an ensemble . An ensemble is a collection of systems identical on the macroscopic level but different on the microscopic level. (3) The ensemble members obey the principle of equal a priori probability . That is, no one ensemble member is more important or probable than another. 

The calculation of the electric properties of individual molecules as found in an infinitely dilute gas has for long been of great interest to quantum chemists. This curiosity has been spurred in recent decades by the increasing importance of the communications industry in the world and the parallel need for materials having specific properties for electronic, optical, and other devices. In particular, the nonlinear-optical quantities, defined at the microscopic level as hyperpolarizabilities and at the macroscopic level as nonlinear susceptibilities, have played a... 

An ideal gas is simply a model of the way that particles (molecules or atoms) behave at the microscopic level. The behavior of the individual particles can be inferred from the macroscopic behavior of samples of real gases. We can easily measure temperature, volume, pressure, and quantity (mass) of real gases. Similarly, when we systematically change one of these properties, we can determine the effect on each of the others. For example, putting more molecules in a balloon (the act of blowing up a balloon) causes its volume to increase in a predictable way. In fact, careful measurements show a direct proportionality between the quantity of molecules and the volume of the balloon, an observation made by Amadeo Avo-gadro more than 200 years ago. 

The macroscopic quantities characteristic of conduction and the various conduction mechanisms at microscopic level are outlined in section 4.2. 

In the previous description using electrochemical potentials, it is assumed that movements by migration and diffusion have identical mechanisms at the microscopic level. This hypothesis may lead to errors if ever the charge carriers are not sufficiently well identified, which is especially the case when large quantities of neutral ion pairs are involved. In fact, in this instance, the ion pairs play a part in the diffusion without contributing to migration. 

The heat capacity of a substance can be defined as the amount of heat required to change its temperature by one degree. A more useful quantity is specific heat capacity, which is the amount of heat required to change the temperature of one unit mass of a material by one degree. Heat capacity is a fundamental property of any material. It is a macroscopic parameter that can be linked to molecular structure and vibrational motions at microscopic level [1]. 

Much of chemistry deals with the properties of matter at the microscopic level of atoms and molecules. We live, however, in a world that is much larger than atoms and molecules. We live in a macroscopic world in which we experience matter in bulk quantities that we can see, touch, taste, and smell. Thermodynamics describes just such a world. The word thermodynamics means heat and work, or the conversions of heat energy into work and vice versa. 

The formal development given above constitutes a complete thermodyrramic description of a general qrrantum system. Adopting a similar development for classical systems, one finds that the final results are identical in form to the quantum results. The only difference between the quantum and classical results lies at the microscopic level in the statistical mechanical expressions for the thermodynamic quantities. 

We should always bear in mind that the Classical Theory of Nucleation is, in essence, a macroscopic theory. But, at the microscopic level, such a level of description is not adequate. In the end, all observable quantities should be expressed as functions of material properties that are, themselves, unambiguously observable. 

On the continuum level of gas flow, the Navier-Stokes equation forms the basic mathematical model, in which dependent variables are macroscopic properties such as the velocity, density, pressure, and temperature in spatial and time spaces instead of nf in the multi-dimensional phase space formed by the combination of physical space and velocity space in the microscopic model. As long as there are a sufficient number of gas molecules within the smallest significant volume of a flow, the macroscopic properties are equivalent to the average values of the appropriate molecular quantities at any location in a flow, and the Navier-Stokes equation is valid. However, when gradients of the macroscopic properties become so steep that their scale length is of the same order as the mean free path of gas molecules,, the Navier-Stokes model fails because conservation equations do not form a closed set in such situations. 

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