### Contact Info

Stuart Chalk, Ph.D.

Department of Chemistry

University of North Florida

Phone: 1-904-620-1938

Fax: 1-904-620-3535

Email: schalk@unf.edu

Website: @unf

## Diffusion coefficients

### Citations 29

**"Determination Of The Diffusion Coefficient Of 1-(2-pyridylazo)-2-naphthol In Ethanol-water Solutions Using Flow Injection And Nuclear Magnetic Resonance Techniques"**

*Anal. Chim. Acta
1999 Volume 386, Issue 1-2 Pages 137-144*

Lilibeth dlC. Coo, Maruse Sadek, Robert T. C. Brownlee, Terence J. Cardwell, Robert W. Cattrall and Spas D. Kolev

**Abstract:**The values of the diffusion coefficient of PAN (1-(2-pyridylazo)-2-naphthol) in various ethanol-water solutions at 298 K were determined using both flow injection (FI) and nuclear magnetic resonance (NMR) techniques. The transient FI concentration curves monitored by the detector were described by the axially dispersed plug flow model with an axial dispersion coefficient obeying Taylors theory. The diffusion coefficient of PAN determined in 40-100% (v/v) ethanol solutions in water by both curve-fitting and the statistical moments method varied in the range from 2.43 x 10^-10 to 8.31 x 10^-10 m2 s-1. As expected, the diffusion coefficient values obtained by curve-fitting were found to be more reliable. A pulsed field gradient (PFG) spin echo NMR experiment was also used to measure the diffusion coefficient of PAN in the same solutions. The NMR results were found to follow the same trend as the FI results though they were from 5% to 16% lower in value. This deviation was attributed to the association effects facilitated by the experimental conditions under which the NMR measurements were taking place, i.e. in quiet solutions and at considerably higher concentrations than those used by the FI technique. The diffusion coefficient of PAN in pure water was calculated as 2.21 x 10^-10 m2 s-1 by extrapolating the FI results to zero ethanol concentration. The results reported in the present investigation can be used for studying the electrochemical properties of PAN in ethanol-water solutions as well as for elucidating the sensing mechanism of an optode based on immobilization of PAN into Nafion membranes.

**"Laminar-flow Bolus Shapes In Flow Injection Analysis"**

*Anal. Chim. Acta
1986 Volume 179, Issue 1 Pages 119-129*

Joseph T. Vanderslice, A. Gregory Rosenfeld and Gary R. Beecher

**Abstract:**Bolus shapes for diffusion profiles of injected samples have been calculated for various reduced times from 0.002 to 0.704 and are illustrated. The effects of the tubing radius and different diffusion coefficient on the profiles are shown. The implications for merging-zone systems are discussed briefly.

**"Flow Cell And Diffusion Coefficient Effects In Flow Injection Analysis"**

*Anal. Chim. Acta
1986 Volume 179, Issue 1 Pages 427-432*

D. C. Stone and J. F. Tyson

**Abstract:**The effects of flow rates of 0.84, 2.00 and 6.00 mL min-1 were recorded on cells of 0.6, 8.0, 25 and 60 µL. For each flow rate the graphs for the three smaller cells lay close together but for the 60 µL cell a big difference in shape and a reduction in peak height observed. Variation of the dispersion coefficient with flow rate for five different solutes gave a day-to-day precision of 1.95 (mean coefficient) ± 0.012 and over a period of several days 1.90 ± 0.089 (n = 10). Analytical implications of the findings are discussed.

**"A Random Walk Simulation Of Flow Injection Systems With Merging Zones"**

*Anal. Chim. Acta
1987 Volume 194, Issue 1 Pages 49-60*

C. D. Crowe and H. W. Levin, D. Betteridge and A. P. Wade

**Abstract:**A flow-injection system with merging zones is simulated by using a random-walk stochastic (Markovian chain) model. Variables studied include reagent plug size and offset, reagent concentration and flow rate, injection delay time, reaction stoichiometry, rate constant, diffusion constants, viscosity, and temperature. The reaction of Ca(II) with o-cresolphthalein complexone was used to compare simulated and experimental results. In general, there was good agreement among computed and measured results.

**"Aspects Of Sample Dispersion For Optimizing Flow Injection Analysis Systems"**

*Anal. Chim. Acta
1992 Volume 261, Issue 1-2 Pages 539-548*

Takashi Korenaga

**Abstract:**Details are given of an apparatus with a micro-scale laboratory-made flow-through photometric detector that permits measurements to be carried out without disturbance of laminar flow. This apparatus was used to study the solute dispersion of an injected sample plug in order to develop a hydrodynamic model for designing sensitive and precise flow injection analysis systems. A range of operating conditions was evaluated, and the use of a double-plunger micro-pump with a linear cam mechanism and a short reciprocation time (cf. Korenaga et al., Anal. Sci., 1991, 7, 515) is recommended to achieve smooth mixing and stable pumping with good reproducibility. Solute dispersion of an injected sample plug was studied by using an experimental app. with ideal laminar flow in order to develop a hydrodynamic model for the design of sensitive and precise flow injection analysis systems. The dispersion behavior of the sample slug under different manifold conditions was first studied in detail to evaluate the effects of various operating conditions such as tube radius, tube length, flow-rate, and mol. diffusion coefficient of sample solute. The capillary flow properties were also examined for some commercial micropumps to select the most suitable pumping method. Mixing profiles and baseline stability in short, straight tubes were investigated. A double-plunger micropump having a linear cam mechanism and a fast, short reciprocation time was proposed to obtain smoother mixing and more stable pumping with good reproducibility. Complete mixing and low flow-rate pumping are strongly desired for reliable flow injection methods for industrial process use; biosensing devices for protein and enzyme bioassays require lower consumption of valuable reagents.

**"Mathematical Modeling Of A Flow Injection System With A Membrane Separation Module"**

*Anal. Chim. Acta
1992 Volume 268, Issue 1 Pages 7-27*

Spas D. Kolev*, and Willem E. van der Linden

**Abstract:**The mathematical model developed takes into account the geometrical dimensions and dispersion properties of the main section of the manifold, the mass transfer in the chambers of the separation module and the thickness and diffusion coefficient of the membrane. The model was solved analytically by the Laplace transform technique, in which the equations reduce to ordinary linear differential equations of the second order (details given). Details are given of the experimental flow scheme, which incorporates a dialysis module, and the stimulus - response technique (Levenspiel and Bischoff, Adv. Chem. Eng., 1963, 4, 95) was used to identify the unknown parameters in the model under flow injection conditions. Three experimental series were run: one with a PTFE dialysis membrane impermeable to the KCl tracer; a second series with a Cuprophan membrane which was permeable to KCl; and the third series with water as carrier solution in the acceptor line and 1.6 mM KCl in the donor line. The response curves at the inlet and outlet of each channel were monitored at different flow rates. Applications include the optimization of sensitivity and sample throughput, and characterization and improvement of the membranes. A math. model for a flow injection system with a membrane separation module based on the axially dispersed plug flow model was developed. It takes into account the geometrical dimensions and dispersion properties of the main sections of the manifold, the mass transfer in the channels of the separation module, and the characteristics of the membrane (thickness and diffusion coefficient within it). The model was solved anal. in the Laplace domain. The inverse transformation was found to give satisfactory results for reactor Peclet nos. less than 120. Otherwise a numerical solution based on the implicit alternating-direction finite difference method was preferred. The adequacy of the model was confirmed experimental on a flow injection manifold with a parallel-plate dialysis module. The unknown flow and membrane parameters were determined by curve fitting. The membrane parameters were determined also by steady-state measurements. Fairly good agreement between the dynamic and steady-state results and with results given in the literature was observed, which, together with other experimental results, supported the validity of the model and showed that it can be used successfully for the math. description and optimization of flow injection systems with membrane separation modules. In this connection, the influence of the reactor parameters and the sample volume on the performance of such a system were investigated and conclusions for improving its sensitivity and sample throughput were drawn. Other possible applications of the model are in membrane technol. for characterizing of various membranes and in process engineering for investigating the mass transfer in different dialyzers.

**"Mathematical Model Of Flow Injection Analysis. First-order Chemical Reaction In A Straight Tube"**

*Anal. Chim. Acta
1993 Volume 278, Issue 2 Pages 307-316*

V. P. Andreev* and M. I. Khidekel

**Abstract:**Analytical expressions for the parameters of sample and chemical reaction product concentration distribution in a straight reaction tube were obtained to analyze the dependence of sample and product peak shapes on their diffusion coefficient, chemical reaction rate constant, tube dimensions and modes of FIA. The model was compared with results available from numerical analysis and experiment.

**"Continuous-flow Determination Of Relative Diffusion Coefficients Of Iron Complexes With Ligands Of The 1,10-phenanthroline Family And With 3-(2-pyridyl)-5,6-diphenyl-1,2,4-triazine In Acetonitrile-water Solutions"**

*Anal. Chim. Acta
1994 Volume 289, Issue 1 Pages 79-85*

Shaofeng Li and Horacio A. Mottola*

**Abstract:**The relative diffusion coefficients for a series of iron(II) complexes with ligands containing the chelating moiety -NCCN- were determined by a continuous-flow procedure. A 4.2 µL volume of 1 mM Fe(II)-complex in carrier solution was injected into the gravity propelled carrier stream (0.3 ml/min) and passed through the coiled tubing manifold (4 m x 0.6 mm i.d.) to the spectrophotometric or amperometric detector. The measurements were carried out at 25±0.1°C and the carrier solution was 36 mM H2SO4 in aqueous 74.6% acetonitrile. The values of the measured diffusion coefficients were related to the power of the mol. wt. and the data was in qualitative agreement with the Stokes-Einstein relationship.

**"Flow Injection Analysis Methods For Determination Of Diffusion Coefficients"**

*Anal. Chim. Acta
1997 Volume 350, Issue 3 Pages 359-363*

Gongwei Zou*, Zhen Liu and Congxiang Wang

**Abstract:**Two flow injection analyzes (FIA) methods for the determination of diffusion coefficients in a straight single tube FIA system were developed. Based on the analytical solution of the convection-diffusion equation, linear relationships of the logarithmic values of the dispersion coefficient (D) and the half-peak width (W-1/2) with the diffusion coefficient (D-m) were obtained. Experiments were designed to verify these methods. For example, for potassium hexacyanoferrate (III) a D-m value of 0.72 x 10(5) cm(2) s-1 was found versus a literature value of 0.76 x 10(5) cm(2) s-1 (error, 5%). For potassium hexacyanoferrate (II) a D-m value of 0.67 x 10(5) cm(2) s-1 was obtained versus a literature Value of 0.63 x 10(5) cm(2) s-1 (error, 6%). The diffusion coefficients of some important biomedical compounds, such as dopamine, epinephrine, norepinephrine and ascorbic acid, were then determined. The values of 10(5) D-m/cm(2) s-1 are 0.60±0.03, 0.44±0.02, 0.60±0.01 and 0.68±0.06, respectively. 18 References

**"Laminar Dispersion In Flow Injection Analysis"**

*Talanta
1981 Volume 28, Issue 1 Pages 11-18*

Joseph T. Vanderslice, Kent K. Stewart, A. Gregory Rosenfeld and Darla J. Higgs

**Abstract:**Simple expressions are given for the dispersion and the travel times of samples in simple flow-injection analysis systems. The sum of these two quantities is the total residence time of the sample in the system. The expressions are based on numerical solutions of the diffusion-convection equation. Preliminary experiments are in agreement with the derived simple expressions, as are peak curve shapes. Diffusion coefficients can be obtained in a straightforward manner.

**"Two Trends Of Sample Dispersion Variation With Carrier Flow-rate In A Single Flow Injection Manifold"**

*Talanta
1995 Volume 42, Issue 12 Pages 2033-2038*

Yonghung Li and Huichang Ma*

**Abstract:**Two trends of sample dispersion variation with carrier flow rate in a single flow injection manifold are completely revealed for the first time and an inflection point in the dispersion coefficient (D) vs. flow rate (q) curve is discovered. With the increase of the flow rate, the value of D increases before the inflection point but decreases after the inflection point. The value of the earlier flow rate at the inflection point (q(m)) is independent of the sample injection volume, the tube length, the tube coil radius and the tube inner diameter. It is only affected by the substance diffusion coefficient (D-m) of the analysis. The value of q(m) decreases as D-m increases. Therefore, the value of D-m for a sample can be estimated according to the D-m vs. q(m) curve. (7 References)

**"Automated Viscometer Based On High-precision Flow Injection Analysis. 2. Measurement Of Viscosity And Diffusion Coefficients"**

*Analyst
1983 Volume 108, Issue 1282 Pages 17-32*

D. Betteridge, W. C. Cheng, E. L. Dagless, P. David, T. B. Goad, D. R. Deans, D. A. Newton and T. B. Pierce

**Abstract:**The use of the apparatus described in Part I (Ibid, 108, 1983, 1) as a viscometer and for the measurement of diffusion coefficient has been evaluated. The effects of radius and length of the tube, sample volume and flow rate were studied. The viscosities of 15 solvents were measured. Diffusion coefficient for KMnO4 in water, ethanol and acetone were obtained.

**"Determination Of Diffusion Coefficients By Flow Injection Analysis"**

*Anal. Chem.
1982 Volume 54, Issue 14 Pages 2618-2620*

Greg Gerhardt and Ralph N. Adams

**Abstract:**A simple, versatile flow-injection anal. procedure is described for the precise and accurate determination of the diffusion coefficients of e.g. biogenic amine neurotransmitters, their metabolites, and related compounds of neurochem. interest. The procedure uses the baseline-to-baseline peak dispersion equation of J. F. Vanderslice et al. (1981), and the app. consists of a pump, autoinjector, flow-module system, recorder, and either a UV/visible spectrophotometer or a fixed-wavelength, liquid chromatography detector. The internal standards are 1-5 mM solutions of K3Fe(CN)6 and K4Fe(CN)6 in 1 or 2 M KCl. Diffusion coefficients are given for such compounds as dopamine, epinephrine, homovanillic acid, leu-enkephalin, amphetamine, and others.

**"Dispersion And Diffusion Coefficients In Flow Injection Analysis"**

*Anal. Chem.
1984 Volume 56, Issue 2 Pages 292-293*

Joseph T. Vanderslice, Gary R. Beecher, and A. Gregory Rosenfeld

**Abstract:**Papers on flow-injection analysis by G. Gerhardt and R. N. Adams (1982) and P. W. Alexander and A. Thalib (1983) make use of theoretically derived expressions for peak appearance and base-line/base-line times. The restrictive conditions under which the expressions were derived were not made sufficiently clear in the by V. et al. (1981). Thus, effort is made to clarify these restrictions and show how they impose certain restraints on the design of flow-injection systems when molecular parameters are to be determined. Experiments with DOPAC and Na fluorescence are reported.

**"Dispersion Coefficient And Moment Analysis Of Flow Injection Analysis Peaks"**

*Anal. Chem.
1988 Volume 60, Issue 24 Pages 2737-2744*

Stephen H. Brooks, Daniel V. Leff, Maria A. Hernandez Torres, and John G. Dorsey

**Abstract:**The dispersion coefficient (D) is the most popular peak descriptor in flow injection analysis (FIA). Yet this concept of dispersion yields no direct information describing peak shape and no information in the time domain. Using an exponentially modified Gaussian peak shape model and previously derived equations, we examine the second moment (variance) of single-line flow injection peaks and use this as a fundamental descriptor of the FIA response curves. Unlike the dispersion coefficient, the second moment is shown to obey a linear relationship with respect to flow rate and to yield valuable information in the presence of a chemical reaction. Reporting descriptors of an FIA response curve as a variance offers several advantages over the classical dispersion coefficient: peak width (in units of time or volume) is immediately obtainable from the variance, yielding a more direct measure of sample throughput; various FIA manifolds can be readily compared from their variance values, and the individual contributions to the total peak variance (including the contribution of the chemical reaction to the total variance) can be easily obtained through the additivity of variances.

**"Dynamic Surface Tension Detection By Optically Probing A Repeating Drop Rate"**

*Anal. Chem.
1994 Volume 66, Issue 8 Pages 1209-1216*

Lawrence R. III Lima, Darren R. Dunphy, and Robert E. Synovec

**Abstract:**The output (5 mW at 633 nm) from a He:Ne laser incorporated in a dynamic surface tension detector (DSTD; construction described) was focused 2 cm below the end of a suspended glass capillary (0.33 mm o.d., 0.2 mm i.d.) and provided measurements of the repeating drop rate associated with linked FIA or HPLC separation systems. Surface-active analytes (SAA) caused a significant decrease in drop volume which was readily measured. A calibration graph for aqueous PEG 1470 (molecular mass 1470 g/mol, nominal), at a flow rate of 66 µL/min, was linear for 4 (detection limit) to 20 ppm of PEG 1470. ppm. The linear calibration range was extended to 8-40 ppm with a flow rate of 133 µL/min but the sensitivity was decreased 2-fold. The dependency of DSTD signal and flow rate (13-266 µL/min) was examined with a series of PEG and a good correlation between analyte transitional diffusion coefficient and detector signal was observed. The selectivity of DSTD for SAA was illustrated with 0.04% PEG 8650 separated (66 µL/min) from 0.04% ethylene glycol (EG) by size-exclusion chromatography. PEG 8650 was detected with a signal-to-noise ratio of 25 but no signal was observed with EG. The RI responses for the two analytes were similar.

**"Electrocatalysis Of NADH Oxidation With Electropolymerized Films Of 3,4-dihydroxybenzaldehyde"**

*Anal. Chem.
1994 Volume 66, Issue 23 Pages 4337-4344*

F. Pariente, E. Lorenzo, and H. D. Abruna

**Abstract:**The oxidation of 3,4-dihydroxybenzaldehyde (3,4-DHB) on glassy carbon electrodes gives rise to stable redox-active electropolymerized films containing a quinone moiety. The redox response of the films is that anticipated for a surface-immobilized redox couple, and the pH dependence of the redox activity of these films is 60 mV/pH unit, which is very close to the anticipated Nernstian dependence of 59 mV/pH unit. We have characterized the dependence of the growth of these films on the solution concentration of 3,4-DHB, the potential for deposition, the time of deposition, and the pH. In addition, we have measured their permeability and apparent diffusion coefficient (Dapp) of these films. These films exhibit potent and persistent electrocatalytic behavior toward NADH oxidation. In a flow injection analysis determination, the limit of detection is estimated to be in the submicromolar regime. Copyright 1994, American Chemical Society.

**"Studies On Peak Width Measurement-based Flow Injection Analysis Acid - Base Determinations"**

*Microchim. Acta
1985 Volume 87, Issue 1-2 Pages 49-64*

Jae-Seong Rhee and Purnendu K. Dasgupta

**Abstract:**Peak width measurement was used to determine strong acids or bases by injection into an aqueous indicator flow stream and spectrophotometric measurement in a flow-through cell (8 or 75 µL volume) at 615 nm (bandwidth 4 nm). The parameters investigated included concentration. and pK of the indicator, dimensions of the dispersion tube, signal level used for peak width measurement, sample volume, carrier flow rate, viscosity of the carrier stream and the diffusion coefficient of the analyte. As an example, the use of bromothymol blue is illustrated.

**"Amperometric Response Of Mediating Layers On Electrode Surfaces To Gaussian Concentration Profiles In Flowing Streams"**

*Electroanalysis
1992 Volume 4, Issue 8 Pages 751-756*

John F. Cassidy, William Breen, Anthony McGee, Johannes G. Vos, Michael E. G. Lyons

**Abstract:**A model is presented for a semi-porous layer containing catalytic sites, coated on an electrode and used for the mediation of analytes. The coated electrodes are applicable as detectors in flow injection analysis or HPLC. Thin layers with high catalyst loading and high rates of electron exchange led to rectilinear graphs of peak current vs. concentration. for chemically modified electrodes in flowing streams. A model for a layer, containing catalytic centers, coated on an electrode and used for the mediation of analyte in a flowing stream is proposed and solved. The model applies to a situation of a Gaussian concentration. profile of analyte impinging on the outer edge of the mediating layer. The important parameters that limit the current magnitude are the diffusion coefficient of the analyte through the layer, the effective electron diffusion coefficient through the layer, and the rate of reaction between the mediator and the analyte. Literature data showed that layers, high catalyst loading, and high rates of electron exchange lead to linear peak current height vs. concentration. plots.

**"Flow Injection Techniques. Analytical Chemistry At The Interface"**

*Anal. Proc.
1981 Volume 18, Issue 1 Pages 26-31*

D. Betteridge, E. L. Dagless, B. Fields, P. Sweet and D. R. Deans

**Abstract:**In our work we have been concerned exclusively with exploiting the chemical and physical changes that take place across the interfacial region. Insofar as we are dealing with fast reactions in flow systems, two practical problems arise in pursuing this investigation. The first is an instrumental one, resulting from the need to measure concentration profiles across the sample and to process the data obtained. This problem requires the development of sensitive flow detectors that are compatible with digital computers. The second problem is in sorting out and taking advantage of the chemistry that occurs in the few seconds which elapse between the points of injection and detection. This paper presents a progress report of this long-term study.

**"The Electrochemistry Of Neurotransmitters At Conducting Organic Polymer Electrodes Electrocatalysis And Analytical Applications"**

*Bioelectrochem. Bioenerg.
1995 Volume 38, Issue 2 Pages 229-245*

Harry B. Mark, Jr., N. Attaa, Y. L. Ma, K. L. Petticrew, H. Zimmer, Y. Shia, S. K. Lunsford, J. F. Rubinson and Ahmed Galal

**Abstract:**The electrooxidation of catechols, catecholamines and NADH at conventional electrode materials is generally characterized by high degrees of irreversibility as well as strong adsorption and, hence, fouling by reactants and/or products of the reactions. On the contrary, the rates of the electron transfer are highly catalyzed by the use of conducting polymer films, such as poly(3-methylthiophene), polyphenylene, polyanaline and polypyrrole, as described here. Furthermore, the usual fouling problems are eliminated. Even interference from electroinactive large proteins, such as haemoglobin, and other surfactants are substantially reduced. Also, electron spectroscopy for chemical analysis, energy-dispersive analysis of X-rays, theoretical diffusion coefficient calculations, metal ion coordination, solution diffusion analyzes of cyclic voltammograms etc. show that the electron transfer occurs at the polymer-solution interface and not at the inert electrode substrate surface after diffusion through the polymer matrix or through pores. The analytical application of these polymer electrodes as amperometric detectors for flow injection analysis and high performance liquid chromatography are given. In addition, selective potentiometric electrodes for catecholamines based on conducting polymer films of crown ethers, such as binaphthyl-20-crown-6, dibenzo-18-crown-6, etc., have been developed and characterized. These potentiometric detectors significantly decrease the usual interferences of ascorbic acid, uric acid and acetaminophen found in amperometric detection. (60 References)

**"Diffusivity Determination From The Response To Injection Into Laminar Liquid Flow In A Capillary"**

*Chem. Eng. Commun.
1979 Volume 3, Issue 3 Pages 155-163*

V. Hancil; V. Rod; M. Rosenbaum

**Abstract:**A method for the determination of diffusivities in liquids is described; the method is based on measurements of the response to injection in a laminar flow of liquid in a capillary. By injection of δ-type, the component whose diffusivity is to be measured is introduced into the inlet part of a capillary of internal diameter 1.6 mm. The response to the injection is measured at the end of the capillary by a flow-through refractometer, the output signal of which is recorded on tape at 16-s intervals. The output signal is compared with a theoretical model of the time dependence of the response, based on Taylor's analysis of the dispersion of an injected component in laminar flow. The diffusivity is calculated from one of the model parameters, which were evaluated by the method of maximum likelihood. The calibration of the instrument was performed by measuring the diffusivity of KC1 in aqueous solution. The following diffusivities at 25°C were determined: n-butanol 971 µm2/s, dioxane 1093 µm2/s, caprolactame 948 µm2/s, Na2SO4, 1126 µm2/s, salicylaldoxime 895 µm2/s, cyclohexanone 960 µm2/s, acetone 1316 µm2/s, oxine 820.5 µm2/s, all in water; caprolactame in trichlorethylene 1958 µm2/s, caprolactame in 10% aqueous (NH44)2,SO4 solution 1148 µm2/s, salicylaldoxime in chloroform 1878 µm2/s.

**"Holding Time Distributions Of The Gaussian Type"**

*Chem. Eng. Sci.
1956 Volume 5, Issue 6 Pages 258-270*

A. Klinkenberg and F. Sjenitzer

**Abstract:**Holding-time distributions are of great importance in a considerable number of separation and conversion processes. In the majority of cases it is desired that the distribution be as narrow as possible and there are accordingly many examples where a peak signal at the entrance gives rise to a comparatively narrow band in the effluent, which has then Gaussian shape. Several mechanisms have been proposed to account for such a phenomenon. In the present study, these mechanisms are analyzed on a statistical basis in terms of their basic variables. It is also shown that various mechanisms may be combined by adding variances, diffusion coefficients or heights corresponding to a theoretical plate. It is obviously not possible to identify the mechanism by observing a single Gaussian elution curve. Neither is this possible by observing a series of such curves for systems or columns of different lengths, in all theories the width of the Gaussian curve increasing in proportion to the square root of the length of the system. In the experiments, variation of flow rate, particle size, and diffusion coefficients is necessary in order to reach conclusions regarding the mechanism or mechanisms responsible for the elution curve.

**"Fluid Dispersion-generalization And Comparison Of Mathematical Models. 1. Generalization Of Models"**

*Chem. Eng. Sci.
1962 Volume 17, Issue 4 Pages 245-255*

K. B. Bischoff and Octave Levenspiel

**Abstract:**Methods for handling mixing have been based on models that use diffusion equations with modified diffusion coefficients. These are called dispersion models. The relation between all models which have been used is summarized, and the associated measurement techniques are generalized. A quantitative evaluation of the error involved with a given model can be made.

**"Determination Of Diffusion Coefficients Of Proteins By Flow Injection Analysis And Its Application To Estimation Of Molecular Masses Of Proteins"**

*Instrum. Sci. Technol.
1998 Volume 26, Issue 4 Pages 333-341*

Mohammed Ibrahim; Zou Gongwei; Zhu Junjie

**Abstract:**A flow injection analysis (FIA) method for determination of molecular diffusion coefficients (Dm) of proteins, in a helically coiled tube, was reported. The coiling effect of the tube that causes secondary flow was avoided through using a Taylor-Aris and Golay's dispersion equation, modified by the velocity profile factor, K, and the modified equation was used for direct determination of diffusion coefficients. The experimental values of Dm were comparable with the literature values. The linear relationship between the Relative Molecular Mass (RMM, Mr) and the Molecular Diffusion Coefficients (Dm) was verified and a two-point calibration FIA method for estimation of Relative Molecular Masses of proteins was suggested. Relative molecular masses of two protein samples, Hb and glucose oxidase were then estimated by this method. The values of RMM were 6.88 x 10^4 for Hb and 16.0 x 10^4 for glucose oxidase.

**"Flow Injection Analysis Estimation Of Diffusion Coefficients Of Paucidisperse And Polydisperse Polymers Such As Polystyrene Sulfonates"**

*J. Appl. Poly. Sci.
1991 Volume 42, Issue 7 Pages 1969-1977*

William A. Boyle, Richard F. Buchholz, John A. Neal, Joseph L. McCarthy

**Abstract:**Flow injection analysis, often used for determination of diffusion coefficients of nonpolymeric substances, has now been applied to the characterization of pauci- and polydisperse polymers in solution. A relative method was found useful for obtaining moderate quality evaluations of diffusion coefficients and related parameters of polymers. The width at half-height W1/2 of the trace peak is found to be proportional to the number average molecular weight n of pauci- and polydisperse polymers, allowing estimation of r and diffusion coefficients. For sodium polystyrene sulfonates at substantially infinite dilution in 1.0 g L-1 Na2SO4, a linear relation has been observed between the logarithms of the molecular weight n and the mean diffusion coefficient D in the n range of 1000-90,000 g mol-1 or the D range of 30 x 10^-7 to 2 x 10^-7 cm2 s-1.

**"Measurement Of Diffusion-coefficients Of Some Indoles And Ascorbic Acid By Flow Injection Analysis"**

*J. Phys. Chem.
1990 Volume 94, Issue 2 Pages 1003-1005*

David Robinson, James E. Anderson, and Jeong Long Lin

**Abstract:**Diffusion coefficients of 5-hydroxyindole, 5-methoxyindole, their derivatives, and ascorbic acid have been determined at 0.5 X and 5 X lo-' M by flow injection analysis. The mobile phase is a 0.1 M phosphate buffer solution at pH 7.4. Results obtained are found to correlate well with the size of the molecule and may be interpreted on the basis of the Stokes-Einstein relation.

**"Determination Of Ternary Diffusion Coefficients By The Taylor Dispersion Method"**

*J. Phys. Chem.
1990 Volume 94, Issue 12 Pages 5180-5183*

Derek G. Leaist

**Abstract:**A new procedure is described for the rapid determination of ternary mutual coefficients using Taylor dispersion. Pulses of liquid are injected into a long capillary tube containing a laminar flow of liquid of different composition. A differential refractometer at the tube outlet monitors the dispersion of the injected samples. The ternary diffusion coefficients are calculated from the temporal moments of the changes in refractive index. To test the procedure, diffusion coefficients are determined for the three-component systems sucrose + potassium chloride + water and sodium chloride + magnesium chloride + water at 25 OC. The measured diffusion coefficients are compared with accurate values obtained by optical interferometry.

**"Dispersion Of Soluble Matter In Solvent Flowing Slowly Through A Tube"**

*Proc. Royal Soc. A
1953 Volume 219, Issue 1137 Pages 186-203*

Taylor, Geoffrey

**Abstract:**When a soluble substance is introduced into a fluid flowing slowly through a small-bore tube it spreads out under the combined action of molecular diffusion and the variation of velocity over the cross section. It is shown analytically that the distribution of concentration produced in this way in centered on a point which moves with the mean speed of flow and is symmetrical about it in spite of the asymmetry of the flow. The dispersion along the tube is governed by a virtual coefficient of diffusivity which can be calculated from observed distributions of concentration. Since the analysis relates the longitudinal diffusivity to the coefficient of molecular diffusion, observations of concentration along a tube provide a new method for measuring diffusion coefficients. The coefficient so obtained was found, with KMnO4, to agree with that measured in other ways. The dispersion in steady flow is due to the combined action of convection parallel to the axis and molecular diffusion in the radial direction. It is of interest to consider, first, dispersion by convection alone,and then to introduce the effect of molecular diffusion. The results may be useful to physiologists who may wish to know how a soluble salt is dispersed in a blood vessel, but they may also be useful to physicists who wish to measure molecular diffusion coefficients. The experimental technique used for KMnO4 is described in detail, and results are compared with earlier measurements.