Motion random

Real environmental transport processes contain all these phenomena that we encountered in the dining car. For instance, large ocean currents, such as the Gulf Stream, act as the dining car. Within these currents there are parcels of water called turbulent eddies which move relative to each other. In addition, small plants and animals carried in the current move relative to the surrounding water, take nutrients up, and mix them within their bodies. 

In fact, wherever we look at a compartment of the environment, we usually find the simultaneous actions of advective and diffusive motion. In this chapter we focus on the latter. 

Random motion is ubiquitous. At the molecular level, the thermal motions of atoms and molecules are random. Further, motions in macroscopic systems are often described by random processes. For example, the motion of stirred coffee is a turbulent flow that can be characterized by random velocity components. Randomness means that the movement of an individual portion of the medium (i.e., a molecule, a water parcel, etc.) cannot be described deterministically. However, if we analyze the average effect of many individual random motions, we often end up with a simple macroscopic law that depicts the mean motion of the random system (see Box 18.1). 

We can analyze the connection between randomness at small scales and order at large scales by an infinite one-dimensional array of discrete boxes which are positioned along the x-axis (Csanady, 1973). The boxes are numbered by m = 0, 1, 2. where the box m = 0 is situated at x = 0 (Fig. 18.1). Let us assume that at time t = 0 an object (molecule, particle, etc.) begins its random walk at box m = 0 (Fig. 18.1, top line). At fixed times t = At, 2At, 3At... the object jumps randomly either to the left or to the right. The path marked by A represents the path of an object which jumps twice to the left, then once to the right and once to the left again and finally twice to the right. At time t = 6At, the object happens to end up in the same box (m = 0) from which it started. 

Another path marked by B shows the extreme case of an object which jumps six times to the right. Its end position represents the largest distance which an object can travel within six time steps. Obviously, this distance grows with increasing number of time steps, n. 

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Molecules are in continuous random motion, and as a result of this, small volume elements within the liquid continuously experience compression or rarefaction such that the local density deviates from the macroscopic average value. If we represent by 6p the difference in density between one such domain and the average, then it is apparent that, averaged over all such fluctuations, 6p = 0 Equal contributions of positive and negative 6 s occur. However, if we consider the average value of 6p, this quantity has a nonzero value. Of these domains of density fluctuation, the following statements can be made ... 

ThermalJostling. The thermally driven random motion of molecules jostles particles to provide a one-dimensional translational energy which averages kT 12. over several seconds. However, it is conventional to use tiTHERMAL measure of thermal energy. At 298 K,... 

Ordinary diffusion involves molecular mixing caused by the random motion of molecules. It is much more pronounced in gases and Hquids than in soHds. The effects of diffusion in fluids are also greatly affected by convection or turbulence. These phenomena are involved in mass-transfer processes, and therefore in separation processes (see Mass transfer Separation systems synthesis). In chemical engineering, the term diffusional unit operations normally refers to the separation processes in which mass is transferred from one phase to another, often across a fluid interface, and in which diffusion is considered to be the rate-controlling mechanism. Thus, the standard unit operations such as distillation (qv), drying (qv), and the sorption processes, as well as the less conventional separation processes, are usually classified under this heading (see Absorption Adsorption Adsorption, gas separation Adsorption, liquid separation). 

For example, the measured pressure exerted by an enclosed gas can be thought of as a time-averaged manifestation of the individual molecules random motions. When one considers an individual molecule, however, statistical thermodynamics would propose its random motion or pressure could be quite different from that measured by even the most sensitive gauge which acts to average a distribution of individual molecule pressures. The particulate nature of matter is fundamental to statistical thermodynamics as opposed to classical thermodynamics, which assumes matter is continuous. Further, these elementary particles and their complex substmctures exhibit wave properties even though intra- and interparticle energy transfers are quantized, ie, not continuous. Statistical thermodynamics holds that the impression of continuity of properties, and even the soHdity of matter is an effect of scale. 

Static temperature is the temperature of the flowing fluid. Like static pressure, it arises because of the random motion of the fluid molecules. Static temperature is in most practical instaUations impossible to measure since it can be measured only by a thermometer or thermocouple at rest relative to the flowing fluid that is moving with the fluid. Static temperature will increase in a diffuser and decrease in a nozzle. 

Because of the random motion of the sohds, some abrasion of the surface occurs. This is generally quite small, usually amounting to about 0.25 to 1 percent of the solids per day. 

In addition to the effect of the upward velocity on a setthng particle, there is also the random motion of the micro-scale environment, which does not affect large particles veiy much but is a major factor in the concentration and uniformity of particles in the transition and micro-scale size range. 

Mixing Mechanisms There are several basic mechanisms by which solid particles are mixed. These include small-scale random motion (diffusion), large-scale random motion (convec tion), and shear. 

Wind speed has velocity components in all directions so that there are vertical motions as well as horizontal ones. These random motions of widely different scales and periods are essentially responsible for the movement and diffusion of pollutants about the mean downwind path. These motions can be considered atmospheric turbulence. If the scale of a turbulent motion (i.e., the size of an eddy) is larger than the size of the pollutant plume in its vicinity, the eddy will move that portion of the plume. If an eddy is smaller than the plume, its effect will be to difhise or spread out the plume. This diffusion caused by eddy motion is widely variable in the atmosphere, blit even when the effect of this diffusion is least, it is in the vicinity of three orders of magnitude greater than diffusion by molecular action alone. 

The kinetic theory of gases has been used so far, the assumption being that gas molecules are non-interacting particles in a state of random motion. This... 

Neither Table 2-1 nor Table 2-2 lists among the constituents of the air the suspended particulate matter that it always contains. The gases and vapors exist as individual molecules in random motion. Each gas or vapor... 

The way, that the gas temperature scale and the thermodynamic temperature scale are shown to be identical, is based on the microscopic interpretation of temperature, which postulates that the macroscopic measurable quantity called temperature, is a result of the random motions of the microscopic particles that make up a system. 

Flow in the atmospheric boundary layer is turbulent. Turbulence may be described as a random motion superposed on the mean flow. Many aspects of turbulent dispersion are reasonably well-described by a simple model in which turbulence is viewed as a spectrum of eddies of an extended range of length and time scales (Lumley and Panofsky 1964). 

Brownion Movement is o Random Motion Superimposed upon the Grovitotionol Settling Velocity of the Porticle it Becomes Appreciabe tor Particles under 3 Microns Diometer and Becomes Entirely Predominont for Particles Under 0.1 Micron. 

It is easy to see that K = 0 for regular trajectories, while completely random motion yields K = 00. Deterministic chaotic motion, on the other hand, results in K being both finite and positive. 

Heat is a form of energy leicking information. The term heat, as used in this context, is equivalent to, say, uncorrelated photons in a crystal, or the random motion of molecules in a gas. It represents vibrational energy which tends to disorganize, rather than organize, systems. 

The term Brownian motion was originally introduced to refer to the random thermal motion of visible particles. There is no reason why we should not extend its use to the random motion of the molecules and ions themselves. Even if the ion itself were stationary, the solvent molecules in the outer regions of the co-sphere would be continually changing furthermore, the ion itself executes a Brownian motion. We must use the term co-sphere to refer to the molecules which at any time are momentarily in that region of solvent which is appreciably modified by the ion. In this book we are primarily interested in solutions that are so dilute that the co-spheres of the ions do not overlap, and we are little concerned with the size of the co-spheres. In studying any property... 

Although the term Brownian motion, as already mentioned in Chapter 1, was originally introduced to refer to the random motion of particles of visible size, there is no reason why we should not use the... 

It has already been mentioned that in an aqueous solution of KC1 at a concentration of 3.20 X 10-6 mole per liter, the equivalent conductivity was found to have a value, 149.37, that differed appreciably from the value obtained by the extrapolation of a series of measurements to infinite dilution. We may say that, even in this very dilute solution, each ion, in the absence of an electric field, does not execute a random motion that is independent of the presence of other ions the random motion of any ion is somewhat influenced by the forces of attraction and repulsion of other ions that happen to be in its vicinity. At the same time, this distortion of the random motion affects not only the electrical conductivity but also the rate of diffusion of the solute, if this were measured in a solution of this concentration. 

The Mechanism of Electrical Conduction. Let us first give some description of electrical conduction in terms of this random motion that must exist in the absence of an electric field. Since in electrolytic conduction the drift of ions of either sign is quite similar to the drift of electrons in metallic conduction, we may first briefly discuss the latter, where we have to deal with only one species of moving particle. Consider, for example, a metallic bar whose cross section is 1 cm2, and along which a small steady uniform electric current is flowing, because of the presence of a weak electric field along the axis of the bar. Let the bar be vertical and in Fig. 16 let AB represent any plane perpendicular to the axis of the bar, that is to say, perpendicular to the direction of the cuirent. 

If we reduce the electric field to a value near zero, the current will fall to a negligible value. The situation will now be as follows the number of electrons which, in their random motion, cross the plane AB in one direction does not differ appreciably from the number which cross the... 

If we were to forget that the flow of current is due to a random motion which was already present before the field was applied—if we were to disregard the random motion entirely and assume that each and every electron, in the uniform field X, moves with the same steady velocity, the distance traveled by each electron in unit time would be the distance v used in the construction of Fig. 16 this is the value which would lead to a current density j under these assumptions, since all electrons initially within a distance v of the plane AB on one side would cross AB in unit time, and no others would cross. Further, in a field of unit intensity, the uniform velocity ascribed to every electron would be the u of (34) this quantity is known as the mobility of the charged particle. (If the mobility is given in centimeters per second, the value will depend on whether electrostatic units or volts per centimeter are used for expressing the field strength.)... 

Let us now consider the situation when this balance has been upset by the presence of a weak electric field perpendicular to AB. The motion of the ions will no longer be completely random, but a tendency to drift will be superimposed on the random motion. If in unit time there has been an appreciable excess flow of negative ions across AB in one direction, we can be certain that there has been an appreciable excess flow of positive ions across AB in the opposite direction. These two separate contributions will together constitute the electric current. 

Consider now the observed values of the equivalent conductivity for the various species of ions given in Table 2 [disregarding the ions (OH)-and H+, which need special consideration]. If we ask, from this point of view, why such a wide variety of values is found, this must be ascribed to the wide variety in the character of the random motion executed by different species of ions in the absence of an electric field. We shall not go into the details of Einstein s theory of the Brownian motion but the liveliness of the motion for any species of particle may be expressed by assigning a value to a certain parameter for a charged particle in an... 

Consider, for example, a dilute aqueous solution of KC1, in which a field of 1 millivolt/cm is maintained. From the mobilities given in Table 3 we calculate that, when, for example, -is second has elapsed, the average drift in either direction for the K+ and the Cl- ions will have been less than (0.0007 X 10 3)/25 cm, that is to say, less than 3 X 10- cm (which is the diameter of one water molecule). Clearly, this distance is nothing but an average drift of the ions for during the 5 5 second, the ions in their (almost) random motion will, of course, have moved in all directions. As mentioned above, periods of molecular vibration usually lie between 10"1 - and 10- 5 sec and in 3V second each ion may have shifted its position many thousand times. Owing to the presence of the applied field the motion of the ions will not be quite random as a result of their drift the solution will appear to carry a steady current. 

With rise of temperature any solvent becomes less viscous. For visible particles the Brownian motion is observed to become more lively and in the same way we should expect a solute particle to execute a more lively random motion. As a result, the mobility of each species of ion should increase with rise of temperature. 

Electrical Conductivity in Non-aqueous Solvents. Let us now discuss the random motion of an atomic ion dissolved in methanol or ethanol. It will be seen from Table 41 that the value of the dipole moment on the OH group of these molecules differs little from that of the... 

Turning now to the Brownian motion of the ion, we must ask to what extent adjacent solvent molecules will tend to accompany the ion in its random motion, as a result of the mutual attraction. It appears that the strength of this mutual attraction will be similar in the three solvents. But we notice that the size of the solvent molecules that tend to accompany the ion is considerably larger in methanol than in water and will be still larger in ethanol. This fact must be taken into account, if we attempt to predict the relative mobilities of the ion in the three solvents. 

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