Proportionality physical basis

This would seem to indicate that the proportionality assumption has no physical basis. However, in many cases the bulk modulus is much larger than the shear modulus (except perhaps near / = 0) so that, for practical purposes, we can take V = Y- In other words, K(t) is so much larger than G(t) that it does not matter what shape we assume for K(t). This is the case for amorphous polymers at temperatures well above their glass transition temperatures. Also for many rigid plastics, for example, amorphous polymers in the glassy state, v is constant but less than j-, typically having values in the range 0.35-0.41 [Schapery (1974)]. In such cases, the proportionality assumption would seem to have approximate validity. In summary, therefore, this assumption, while motivated primarily by the need for mathematical simplicity, is a reasonable approximation for many materials. 

The requirement of dimensional consistency places a number of constraints on the form of the functional relation between variables in a problem and forms the basis of the technique of dimensional analysis which enables the variables in a problem to be grouped into the form of dimensionless groups. Since the dimensions of the physical quantities may be expressed in terms of a number of fundamentals, usually mass, length, and time, and sometimes temperature and thermal energy, the requirement of dimensional consistency must be satisfied in respect of each of the fundamentals. Dimensional analysis gives no information about the form of the functions, nor does it provide any means of evaluating numerical proportionality constants. 

The effects of miniaturization on the performance of an analytical separation system are often discussed in terms of a reduction of a characteristic length parameter (e.g., column diameter dc or particle diameter dp) and the associated consequences for lateral zone dispersion processes and their interplay with longitudinal (axial) zone dispersion. A rigid discussion of the physical-chemical basis is beyond the scope of this chapter. For a general account in terms of scaling laws and proportionality considerations, the reader is referred to the references [12,14]. A comprehensive and detailed description with emphasis on the underlying physical-chemical processes can be found in the book by Giddings [20]. 

We therefore find that A < 0. However, we already know that A = 0 is only possible if/(r) is constant. We have therefore obtained the result that if a nonzero density variation is proportional to the potential that generates it, i.e., 8n = A 8v, then the constant of proportionality A is negative. This is exactly what one would expect on the basis of physical considerations. In actual calculations one indeed finds that the eigenvalues of x are negative. Moreover, it is found that there is a infinite number of negative eigenvalues arbitrarily close to zero, which causes considerable numerical difficulties when one tries to obtain the potential variation that is responsible for a given density variation. We finally note the invertability proof for the static response function can be extended to the time-dependent case. For a recent review we refer to Ref. [15]. 

Irving Langmuir, the Nobel prize-winning industrial physical chemist who worked at General Electric, built an elegant structure upon this foundation in kinetic theory. He reasoned that not every molecule would adsorb, but only some would do so. Furthermore, one reason for this was that to be adsorbed there should be a site for adsorption to occur. It stands to reason then that on the basis of mass action, the rate of adsorption should be proportional to the concentration of molecules in the gas phase and to the number of sites available on the surface. Additionally, the rate should be related at any time to the number of sites not covered at that time rather than to the total number of sites present per unit area. Conversely, and again by the principle of mass action, the rate of desorption should be proportional to the number of sites currently occupied at that time. Using ka and kd as the proportionality constants (that we will call the rate constant for adsorption and desorption, respectively), we can write the net rate of adsorption for gas phase species i as the difference between the rate of adsorption and the rate of desorption ... 

P is the gas pressure, Vis the volume, n is the number of moles of gas present, and T is the temperature. P is a proportionality constant, called the universal gas constant, because its value (0.08206 L atm moE is the same for any gas. This familiar equation will form the basis of much of our discussion in this chapter, so it will be important that we have a thorough understanding of the terms that appear in it. Let s start by taking a close look at the meaning and physical origin of gas pressure. 

Note carfully that Equation (10) has been derived on the basis of the Debye-Huckel free energy of interaction. No divalent ions are bound to the polyion all small ions are in the diffuse ionic atmosphere surrounding the line charge. Yet Equation (10) has the formal structure of a mass action law the left side would represent the number of divalent ions bound to the polyion per polyion charged group the right side is proportional to the Mg" concentration. Such an interpretation, however, is purely formal and devoid of any physical content. In line with this observation, note that the proportionality constant < / Naci the right side of Equation (10) is independent of the divalent ion species. 

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