Diffusion molecular

FIGURE 9.3 Pulse sequence for measurement of diffusion coefficient D by the pulsed field gradient spin echo technique. The gradient G is applied for a period 8 both before and after the 180° pulse, with separation A. 

A can be measured accurately, and the value of G8 need not be known precisely so long as it is the same for the two magnetic field pulses. This technique permits accurate measurement of diffusion coefficients for one substance dissolved in another or for self-diffusion of a single substance. By Fourier transforming the echo, diffusion coefficients for several substances can be determined. In conjunction with spatial localization methods provided by NMR imaging, studies of diffusion can be quite valuable, as we point out in Chapter 14. 

Even though it is difficult to predict reaction rates in marine systems, the concepts of molecular diffusion and mechanisms of reaction underpin much of geochemical research at the air-water and sea floor-ocean boundaries. A basic knowledge of molecular diffusion and chemical kinetics is essential for understanding the processes that control these fluxes. This chapter explores the topics of molecular diffusion, reaction rate mechanisms and reaction rate catalysis. Catalysis is presented in a separate section because nearly all chemical reactions in nature with characteristic life times of more than a few minutes are catal5 ed. 

A simple model demonstrating the essential ideas of the theory of random walk is illustrated in Fig. 9.1 (Csanady, 1973). Let us assume that a particle is located at x = 0 at time t= 0. Its movements, which 

Schematic description of the random walk of a particle along a straight line. The pyramid shows the probabilities of a molecule reaching position m 

The continuum of probabilities, p, of achieving any repeated event, in, in which two possible outcomes are equally possible describes a normal density function and can be expressed by the equation 

A normal distribution (Eq. (9.2)) in which = 0. The X axis is scaled by the standard deviation, a, and the y axis is scaled hy a The shaded area is the integral under the curve for plus or minus one standard deviation, 

Consideration of length and time scales is fundamental as they provide an indication of the main mechanisms at work. The combination of length and time scales with material parameters such as molecular diffusivity and viscosity leads to dimensionless characteristic numbers that provide guides to the relative importance of competing mechanisms. 

The Reynolds number. Re, is the ratio of inertial forces to viscous forces. If Uand L denote the characteristic velocity and length scales, respectively. Re = UL/v, where v is the kinematic viscosity. Small values of Re (i.e. less than 1000) correspond to laminar (viscosity-dominated) flows and large values of Re to turbulent flows. 

The Schmidt number, Sc=v/D, is the ratio between two transport coefficients, where D is the molecular diffusion coefficient. Sc can be interpreted as the ratio of two rates. The rate at which concentration becomes smoothed out by molecular diffusion is proportional to (Dt) where t denotes the time, whereas the rate for motion to spread out or die is proportional to (vt) The ratio of these two rates is Sc. Thus, if Sc S 1, as in the case of liquids, concentration fluctuations survive without being erased by mechanical mixing until late in the process. The kinematic viscosity of water is about 10 m s . The diffusion coefficient of small molecules in water is about 10 m s hence a typical value of Sc for a liquid such as water is about 1000. 

The Pedet number, Pe, is the ratio of transport by advection (or convection) and by molecular diffusion Pe is defined as Pe = UL/D = ReSc. Pe can be interpreted also as 

Molecular diffusion is the ultimate and finally the only process really able to mix components of a fluid on the molecular scale. The time constant for molecular diffusion is the diffusion time defined as 

Oidinaty diffusion can be defined as the transport of a particular species relative to an appropriate reference plane owing to the random motion of molecules in a region of space in which a composition gradient exists. Although the mechanisms which the molecular motion occurs may vary greatly from those in a gas to those in a crystalline solid, die essential fixtures of a random molecular motion in a composition gradient are the same, as will be seen in the simplified derivations discussed here. 

Consider a perfect gas mixture at constant temperature and pressure (therefore the molar concentration C is constant and diere is no bulk motion of the gas) in which a mole ftaction gradient exists in one direction as shown in Rg. 2.3-1. Molecules within some distance a, whidi is propoitional to the mean fine path of the gas (X), can cross the z = 0 riane. The nd molar flux finm left to r ftt across die z 0 plane would be proportional to the product of the mean molecular speed v, and the difference between the average concentration on the left and on die right. 

The difference in the average mole fiactions can be approximated by The diffiisive flux /a 

The molecular diffusivity (or diffiision coefficient) D, defined as the proportionaiity constam between the diffiisive flux and the negative of the composition gradiem, is therefine proportional to the product of the mean molecular speed and the mean distance between collimons. If the sitiiple kinetic theory ei iressions for the mean molecular speed and the mean flee path of like molecules are used, one finds a modest temperature and pressure dependence of the diffusivity. 

FIGURE 2.3-1 Diffiisitm in a gas mixture owing to random molecular motion in a conqxisition gradient. 

Suppose a chunk of dry ice evaporates at the center of a long tube. The gaseous carbon dioxide molecules initially travel to the left or right with equal probability. Let s oversimplify the problem a bit to begin with assume that, at one specific time, each CO2 molecule moves at a speed s left or right (typically 400 m s-1), and each molecule travels a distance X (typically % 10 nm) before it collides with another gas molecule. Each collision completely randomizes the velocity. 

Under these assumptions, the probability of finding a carbon dioxide molecule MX from its starting point after N collisions is mathematically exactly the same as the coin toss probability of M more heads than tails. The root-mean squared distance traveled 

Of course, real molecules have a distribution of velocities, and they do not all travel the same distance between collisions. However, there is a remarkable theorem which is proven in advanced physics courses which shows that essentially every purely random process gives a normal distribution  

TABLE 4.2 Diffusion Constants for some Gases and Liquids 

Some typical values for gases and liquids are given in Table 4.2. 

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Thus far we have considered systems where stirring ensured homogeneity witliin tire medium. If molecular diffusion is tire only mechanism for mixing tire chemical species tlien one must adopt a local description where time-dependent concentrations, c r,f), are defined at each point r in space and tire evolution of tliese local concentrations is given by a reaction-diffusion equation... 

Film Theory. Many theories have been put forth to explain and correlate experimentally measured mass transfer coefficients. The classical model has been the film theory (13,26) that proposes to approximate the real situation at the interface by hypothetical "effective" gas and Hquid films. The fluid is assumed to be essentially stagnant within these effective films making a sharp change to totally turbulent flow where the film is in contact with the bulk of the fluid. As a result, mass is transferred through the effective films only by steady-state molecular diffusion and it is possible to compute the concentration profile through the films by integrating Fick s law ... 

Neither the penetration nor the surface renewal theory can be used to predict mass transfer coefficients directiy because T and s are not normally known. Each suggests, however, that mass transfer coefficients should vary as the square root of the molecular diffusivity, as opposed to the first power suggested by the film theory. 

The combined effects of Knudsen and molecular diffusion may be estimated approximately from the reciprocal addition rule ... 

For hquid systems v is approximately independent of velocity, so that a plot of JT versus v provides a convenient method of determining both the axial dispersion and mass transfer resistance. For vapor-phase systems at low Reynolds numbers is approximately constant since dispersion is determined mainly by molecular diffusion. It is therefore more convenient to plot H./v versus 1/, which yields as the slope and the mass transfer resistance as the intercept. Examples of such plots are shown in Figure 16. 

The driving force in diffusion involves differences in the concentration of the diffusing substance. The molecular diffusion of a gas into a hquid is dependent on the characteristics of the gas and the hquid, the temperature of the hquid, the concentration deficit, the gas to hquid contact area, and the period of contact. Diffusion may be expressed by Pick s law (13,14) ... 

The rate of mass transfer (qv) depends on the interfacial contact area and on the rate of mass transfer per unit interfacial area, ie, the mass flux. The mass flux very close to the Hquid—Hquid interface is determined by molecular diffusion in accordance with Pick s first law ... 

Although molecular diffusion itself is very slow, its effect is nearly always enhanced by turbulent eddies and convection currents. These provide almost perfect mixing in the bulk of each Hquid phase, but the effect is damped out in the vicinity of the interface. Thus the concentration profiles at each... 

Circulation of fluid is promoted by surface tension gradients but inhibited by viscosity, which slows the flow, and by molecular diffusion, which tends to even out the concentration differences. The onset of instabibty is described by a critical Marangoni number (Mo), an analogue of the Rayleigh... 

Gases and vapors permeate FEP resin at a rate that is considerably lower than that of most plastics. Because FEP resins are melt processed, they are void-free and permeation occurs only by molecular diffusion. Variation in crystallinity and density is limited, except in unusual melt-processing conditions. 

For weU-defined reaction zones and irreversible, first-order reactions, the relative reaction and transport rates are expressed as the Hatta number, Ha (16). Ha equals (k- / l ) where k- = reaction rate constant, = molecular diffusivity of reactant, and k- = mass-transfer coefficient. Reaction... 

Since the infinite dilution values D°g and Dba. re generally unequal, even a thermodynamically ideal solution hke Ya = Ys = 1 will exhibit concentration dependence of the diffusivity. In addition, nonideal solutions require a thermodynamic correction factor to retain the true driving force for molecular diffusion, or the gradient of the chemical potential rather than the composition gradient. That correction factor is ... 

Binary Electrolyte Mixtures When electrolytes are added to a solvent, they dissociate to a certain degree. It would appear that the solution contains at least three components solvent, anions, and cations, if the solution is to remain neutral in charge at each point (assuming the absence of any applied electric potential field), the anions and cations diffuse effectively as a single component, as for molecular diffusion. The diffusion or the anionic and cationic species in the solvent can thus be treated as a binary mixture. 

Ruthven (gen. refs.) summarizes methods for the measurement of effective pore diffusivities that can be used to obtain tortuosity factors by comparison with the estimated pore diffusion coefficient of the adsorbate. Molecular diffusivities can be estimated with the methods in Sec. 6. 

Neglecting flow nonuniformities, the contributions of molecular diffusion and turbulent mixing arising from stream sphtting and recombination around the sorbent particles can be considered additive [Langer et al., Int. ]. Heat and Mass Transfer, 21, 751 (1978)] thus, the axial dispersion coefficient is given by ... 

The first term in Eqs. (16-79) and (16-80) accounts for molecular diffusion, and the second term accounts for mixing. For the first term,... 

Concentration and temperature differences are reduced by bulk flow or circulation in a vessel. Fluid regions of different composition or temperature are reduced in thickness by bulk motion in which velocity gradients exist. This process is called bulk diffusion or Taylor diffusion (Brodkey, in Uhl and Gray, op. cit., vol. 1, p. 48). The turbulent and molecular diffusion reduces the difference between these regions. In laminar flow, Taylor diffusion and molecular diffusion are the mechanisms of concentration- and temperature-difference reduction. 

For turbines at Reynolds numbers less than 100, toroidal stagnant zones exist above and below the turbine periphery. Interchange of hq-uid between these regions and the rest of the vessel is principally by molecular diffusion. 

Dispersion The movement of aggregates of molecules under the influence of a gradient of concentration, temperature, and so on. The effect is represented hy Tick s law with a dispersion coefficient substituted for molecular diffusivity. Thus, rate of transfer = —Dj3C/3p). 

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