Number Density or Concentration

The number density or concentration, c, is the number of atoms, molecules, moles, or other entities of component i per unit volume. Therefore, 

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In practical appHcations, diffraction instmments may exhibit certain problems. Eor example, there may be poor resolution for the larger droplets. Also, it is not possible to obtain an absolute measure of droplet number density or concentration. Furthermore, the Fraunhofer diffraction theory cannot be appHed when the droplet number density or optical path length is too large. Errors may also be introduced by vignetting, presence of nonspherical... 

The investigation of the collapse phenomenon have shown that the topological structure of the network plays an essential role in the process of gel collapse [42, 43]. In order to check the influence of the topology of a network on the equilibrium properties of the network-surfactant complexes, a set of experiments with gels differing in the number of crosslinks or in the conditions of synthesis have been performed. It has been shown that tie decrease of crosslink density or concentration of monomers in the polymerization mixture results in a sharper gel collapse. 

The second model, the so-called gradient-flux law, is considered to be more fundamental, although it is based on a more restrictive physical picture. In contrast to the mass transfer model, in which no assumption is made regarding the spatial separation of subsystems A and B, in the gradient-flux law it is assumed that the subsystems and the distance between them, Axa/b, become infinitely small. For very small subsystems the term occupation number loses its meaning and must be replaced by occupation density or concentration. Obviously, the difference in occupation density tends toward zero, as well. Yet the ratio of the two differences, Aoccupa-tion density Axa/b, is equal to the spatial gradient of the occupation density and usually different from zero ... 

Coexistence of binary systems. Coexisting phases are characterized by different figures of the order parameter M. In pure fluids, one identifies M with the density difference of the coexisting phases. In solutions, M is related to some concentration variable, where theory now advocates the number density or the closely related volume fraction [101]. At a quantitative level, these divergences are described by crossover theory [86,87] or by asymptotic scaling laws and corrections to scaling, which are expressed in the form of a so-called Wegner series [104], The two branches of the coexistence curve are described by... 

Brix degree. A measure of the density or concentration of a sugar solution. The number of degrees Brix equals the percentage by weight of sucrose in the solution and is related empirically to the density. 

Tij, = the density or concentration (in number per cm ) of the singly-charged ions of the component j n j = the density of the neutral atoms m — the mass of the electron k = the Boltzmann constant h = Planck s constant Z = the partition function E = the ionization energy... 

Clearly, we can, in each case, transform to other concentration variables such as mole fraction or molality and obtain the appropriate activity coefficient. We shall not elaborate on this since it requires a relatively simple transformation of variables. We stress, however, that the number density (or the molar concentration) is the more natural choice of a concentration scale, and the corresponding standard chemical potentials enjoy some advantages which are not shared by standard chemical potentials based on either the mole fraction or the molality. More details are given in Section 4.11. 

Here c and e are the concentration and charge, respectively, of the species i, and a is the number of components in solution. In the usual preparative concentration range (0.1 to 2 molar) the reduced densities (or concentrations) of solute and solvent are very different. In a 2M NaCl solution for example, the reduced densities of solute and solvent... 

Similar convection-diffusion equations to the Navier-Stokes equation can be formulated for enthalpy or species concentration. In all of these formulations there is always a superposition of diffusive and convective transport of a field quantity, supplemented by source terms describing creation or destruction of the transported quantity. There are two fundamental assumptions on which the Navier-Stokes and other convection-diffusion equations are based. The first and most fundamental is the continuum hypothesis it is assumed that the fluid can be described by a scalar or vector field, such as density or velocity. In fact, the field quantities have to be regarded as local averages over a large number of particles contained in a volume element embracing the point of interest. The second hypothesis relates to the local statistical distribution of the particles in phase space the standard convection-diffusion equations rely on the assumption of local thermal equilibrium. For gas flow, this means that a Maxwell-Boltzmann distribution is assumed for the velocity of the particles in the frame-of-reference co-moving with the fluid. Especially the second assumption may break dovm when gas flow at high temperature or low pressure in micro channels is considered, as will be discussed below. 

Almost all methods of chemical analysis require a series of calibration standards containing different amounts of the analyte in order to convert instrument readings of, for example, optical density or emission intensity into absolute concentrations. These can be as simple as a series of solutions containing a single element at different concentrations, but, more usually, will be a set of multicomponent solutions or solids containing the elements to be measured at known concentrations. It is important to appreciate that the term standard is used for a number of materials fulfilling very different purposes, as explained below. 

A. One Atmosphere Densities. The densities or volume properties of solutions have been studied by a number of methods which are extensively reviewed elsewhere (4,5. 6,7) of all of the methods, only the magnetic float (7-14), the hydrostatic balance (3,15-20), the vibrating flow densimeter (21,22), and dilatometric (23,24,25) methods give data with sufficient precision to study the densities of dilute solutions. For more concentrated... 

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