Macroscopic principle

J. A. Roels, Application of macroscopic principles to microbial metabolism, Biotechnol. Bioeng. 1980, 22, 2457-2514. 

The utility of these systems in nanochemical engineering is still far in the future but their study has contributed to identifying some of the perils and promise of extrapolating macroscopic principles to the molecular level, and has provided conceptual insight into the operation of biological motors. 

We have shown how a pointwise DE can be derived by application of the macroscopic principle of mass conservation to a material (control) volume of fluid. In this section, we consider the derivation of differential equations of motion by application of Newton s second law of motion, and its generalization from linear to angular momentum, to the same material control volume. It may be noted that introductory chemical engineering courses in transport phenomena often approach the derivation of these same equations of motion as an application of the conservation of linear and angular momentum applied to a fixed control volume. In my view, this obscures the fact that the equations of motion in fluid mechanics are nothing more than the familiar laws of Newtonian mechanics that are generally introduced in freshman physics. 

So far, we have seen that the basic macroscopic principles of continuum mechanics lead to a set of five scalar DEs sometimes called the field equations of continuum mechanics -namely, (2 5) or (2 20), (2 32), and (2-51) or (2 52). On the other hand, we have identified many more unknown variables, u, T, 9,p, and q, plus various fluid or material properties such as p, Cp (or Cv), (dp/d())p, [or (dp/d0)p], which generally require additional equations of state to be determined from p and 9 if the latter are adopted as the thermodynamic state variables. Let us focus just on the independent variables u, T,9, p, and q. Taking account of the symmetry of T, these comprise 14 unknown scalar variables for which we have so far obtained only the five independent field equations that were just listed. It is evident that we require additional equations relating the various unknown variables if we are to achieve a well-posed problem from a mathematical point of view. Where are these equations to come from Why is it that the fundamental macroscopic principles of continuum physics do not, in themselves, lead to a mathematical problem with a closed set of equations ... 

The third type of boundary condition at the surface S involving the bulk-phase velocities is known as the dynamic condition. It specifies a relationship between the tangential components of velocity, [u - (u n)n] and [u - (u n)n]. However, unlike the kinematic and thermal boundary conditions, there is no fundamental macroscopic principle on which to base this relationship. The most common assumption is that the tangential velocities are continuous across S, i.e.,... 

Shortly after Clausius publication in 1850, Maxwell invented his famous demon (2). Through superior information the demon was supposed to be able to circumvent the Second Law of Thermodynamics. The essential feature of the demon was his (or her) ability to function at the micro level and use information obtained at this level to thwart macroscopic principles. 

THE MACROSCOPIC PRINCIPLE with CELLS as "BLACK BOXES"... 

Figure 2.3. The macroscopic principle applied to bioprocessing expressed with pseudohomogeneous observable process variables in the liquid phase (L) biomass X, substrates S-, oxygen 0, products Pj, carbon dioxide C, and volumetric heat Hy. Pseudohomogeneity is checked by considering a series of mass transfer steps L film at the gas phase-L interface (1), L bulk (2), L film at the L-Solid phase (S) interface (3), cell wall and membranes (4), resp. S-phase cell mass with cytoplasm.

The working concept of the macroscopic principle, recently adapted from chemical engineering (see Roels, 1980 Roels and Kossen, 1978), operates with macroscopically observable variables that are thought to be closely related to significant phenomena of the process (cf. Table 2.1). 

Figure 2.14. Strategies in bioprocess design based on the macroscopic principle and using the formal kinetic concept as part of an integrating strategy. The strategy incorporates the spatial change of mass flux through area Vnl (kg/m h) and the rate of consumption or formation (kg/m h). rds, rate-determining step qss, quasi-steady-state. (From Moser, 1983b.)...

Figure 6.51. Diagrammatic representation of a steady-state bioprocess in balance area (reactor) following the macroscopic principle by analyzing elemental composition of significant process variables (substrate, nitrogen source, biomass, product, O2, CO2, H2O). (Adapted from Roels, 1980a.)...

In the Lewis and Gibson statement of the third law, the notion of a perfect crystalline substance , while understandable, strays far from the macroscopic logic of classical thennodynamics and some scientists have been reluctant to place this statement in the same category as the first and second laws of thennodynamics. Fowler and Guggenheim (1939), noting drat the first and second laws both state universal limitations on processes that are experunentally possible, have pointed out that the principle of the unattainability of absolute zero, first enunciated by Nemst (1912) expresses a similar universal limitation ... 

I be second important practical consideration when calculating the band structure of a malericil is that, in principle, the calculation needs to be performed for all k vectors in the Brillouin zone. This would seem to suggest that for a macroscopic solid an infinite number of ectors k would be needed to generate the band structure. However, in practice a discrete saaipling over the BriUouin zone is used. This is possible because the wavefunctions at points... 

Now, in principle, the angle of contact between a liquid and a solid surface can have a value anywhere between 0° and 180°, the actual value depending on the particular system. In practice 6 is very difficult to determine with accuracy even for a macroscopic system such as a liquid droplet resting on a plate, and for a liquid present in a pore having dimensions in the mesopore range is virtually impossible of direct measurement. In applications of the Kelvin equation, therefore, it is almost invariably assumed, mainly on grounds of simplicity, that 0 = 0 (cos 6 = 1). In view of the arbitrary nature of this assumption it is not surprising that the subject has attracted attention from theoreticians. 

Thermodynamics is a deductive science built on the foundation of two fundamental laws that circumscribe the behavior of macroscopic systems the first law of thermodynamics affirms the principle of energy conservation the second law states the principle of entropy increase. In-depth treatments of thermodynamics may be found in References 1—7. 

Semiconductor materials are rather unique and exceptional substances (see Semiconductors). The entire semiconductor crystal is one giant covalent molecule. In benzene molecules, the electron wave functions that describe probabiUty density ate spread over the six ting-carbon atoms in a large dye molecule, an electron might be delocalized over a series of rings, but in semiconductors, the electron wave-functions are delocalized, in principle, over an entire macroscopic crystal. Because of the size of these wave functions, no single atom can have much effect on the electron energies, ie, the electronic excitations in semiconductors are delocalized. 

Thermodynamics is the branch of science that embodies the principles of energy transformation in macroscopic systems. The general restrictions which experience has shown to apply to all such transformations are known as the laws of thermodynamics. These laws are primitive they cannot be derived from anything more basic. 

Macroscopic and Microscopic Balances Three postulates, regarded as laws of physics, are fundamental in fluid mechanics. These are conservation of mass, conservation of momentum, and con-servation of energy. In addition, two other postulates, conservation of moment of momentum (angular momentum) and the entropy inequality (second law of thermodynamics) have occasional use. The conservation principles may be applied either to material systems or to control volumes in space. Most often, control volumes are used. The control volumes may be either of finite or differential size, resulting in either algebraic or differential consei vation equations, respectively. These are often called macroscopic and microscopic balance equations. 

For a removal attempt a molecule is selected irrespective of its orientation. To enhance the efficiency of addition attempts in cases where the system possesses a high degree of orientational order, the orientation of the molecule to be added is selected in a biased way from a distribution function. For a system of linear molecules this distribution, say, g u n ), depends on the unit vector u parallel to the molecule s symmetry axis (the so-called microscopic director [70,71]) and on the macroscopic director h which is a measure of the average orientation in the entire sample [72]. The distribution g can be chosen in various ways, depending on the physical nature of the fluid (see below). However, g u n ) must be normalized to one [73,74]. In other words, an addition is attempted with a preferred orientation of the molecule determined by the macroscopic director n of the entire simulation cell. The position of the center of mass of the molecule is again chosen randomly. According to the principle of detailed balance the probability for a realization of an addition attempt is given by [73]... 

Continuum models of solvation treat the solute microscopically, and the surrounding solvent macroscopically, according to the above principles. The simplest treatment is the Onsager (1936) model, where aspirin in solution would be modelled according to Figure 15.4. The solute is embedded in a spherical cavity, whose radius can be estimated by calculating the molecular volume. A dipole in the solute molecule induces polarization in the solvent continuum, which in turn interacts with the solute dipole, leading to stabilization. 

At present, it is known that the structures of the ECC type (Figs 3 and 21) can be obtained in principle for all linear crystallizable polymers. However, in practice, ECC does not occur although, as follows from the preceding considerations, the formation of linear single crystals of macroscopic size (100% ECC) is not forbidden for any fundamental thermodynamic or thermokinetic reasons60,65). It should be noted that the attained tenacities of rigid- and flexible-chain polymer fibers are almost identical. The reasons for a relatively low tenacity of fibers from rigid-chain polymers and for the adequacy of the model in Fig. 21 a have been analyzed in detail in Ref. 65. 

Chapter 10, the last chapter in this volume, presents the principles and applications of statistical thermodynamics. This chapter, which relates the macroscopic thermodynamic variables to molecular properties, serves as a capstone to the discussion of thermodynamics presented in this volume. It is a most satisfying exercise to calculate the thermodynamic properties of relatively simple gaseous systems where the calculation is often more accurate than the experimental measurement. Useful results can also be obtained for simple atomic solids from the Debye theory. While computer calculations are rapidly approaching the level of sophistication necessary to perform computations of... 

This new three-column format for solutions is designed to enrich the problem-solving experience by helping students to connect the calculation to chemistry concepts and principles, using macroscopic, molecular, and graphical representations. 

The uncertainty principle has negligible practical consequences for macroscopic objects, but it is of profound importance for subatomic particles such as the electrons in atoms and for a scientific understanding of the nature of the world. 

The uncertainty principle is negligible for macroscopic objects. Electronic devices, however, are being manufactured on a smaller and smaller scale, and the properties of nanoparticles, particles with sizes that range from a few to several hundred nanometers, may be different from those of larger particles as a result of quantum mechanical phenomena, (a) Calculate the minimum uncertainty in the speed of an electron confined in a nanoparticle of diameter 200. nm and compare that uncertainty with the uncertainty in speed of an electron confined to a wire of length 1.00 mm. (b) Calculate the minimum uncertainty in the speed of a I.i+ ion confined in a nanoparticle that has a diameter of 200. nm and is composed of a lithium compound through which the lithium ions can move at elevated temperatures (ionic conductor), (c) Which could be measured more accurately in a nanoparticle, the speed of an electron or the speed of a Li+ ion ... 

The molar volumes of CH4 and CD4 are known to differ in the expected direction in the condensed phases at low temperatures (Olusius and Weigand, 1940 Grigor and Steele, 1968). Such a difference would in principle originate in the movement of the molecule as a whole as well as in its carbon-hydrogen oscillations, but a macroscopically available compound would be expected to offer better possibilities for experimental investigation than poorly known transition states. 

---