• ### Diffusion Equation Fick s Laws of Diffusion

This assumes that D i is a constant which is only true for dilute solutions This is usually a good assumption for diffusion in solids diffusion of chemicals in a dilute solution water or other typical liquid solvents and diffusion of dilute trace species in

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• ### Solutions to the Diffusion Equation

Estimation of diffusion distance from x 4Dt Superposition of point source solutions to get solutions for arbitrary initial conditions c x 0 Method of Laplace transforms Useful for constant flux boundary conditions time dependent boundary conditions Numerical methods Useful for complex geometries D = D c time

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• ### Diffusionuseful equations

Diffusionuseful equations Diffusion coefficient D D = 1/f kT ffrictional coefficient k T Boltzman constant absolute temperature f = 6p h r hviscosity rradius of sphere The value for f calculated for a sphere is a minimal value asymmetric shape of molecule or non elastic interaction with solvent e g hydration will increase f

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• ### Rotational Diffusion Constantan overview

The rotational diffusion constant is then estimated using the Stokes formula for a sphere D domain R = 3/4 D domain T /R S 2 with the Stokes–Einstein radius R S = k B T 6πηD domain T and is taken to be identical for all domains

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• ### Lecture 4 Diffusion Fick s second law

7 Considering boundary conditions c x = 0 = c s constant fixed c x = ∞ = c 0 corresponding to the original concentration of carbon existing in the phase c 0 remains constant in the far bulk phase at x = ∞ c x t = c s c s –c 0 erf Dt x 2 the concentration profile shown above follows this diffusion equation

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• ### Diffusion Coefficient 2b Rowan University

2 Stokes Einstein Relation n For free diffusion q Assumes a spherical molecule n i e not valid for a long chain protein n k = Boltzman Constant q 1 38 x 10 23 J/K n = solvent viscosity kg/ms n T is temperature K n r is solute molecule radius q related to molecular weight prh kT D 6 =

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• ### 2 10 Carrier diffusionUniversity of Colorado Boulder

Using the definition of the diffusion constant we then obtain the following expressions which are often refered to as the Einstein relations dif13 dif14 2 10 3 Total current The total electron current is obtained by adding the current due to diffusion to the drift current yielding dif8

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• ### Diffusionumich edu

Because 1 mol of A reacts under conditions of constant temperature and pressure to form 1 mol of B we have Equimolar Counter Diffusion EMCD at constant total molar concentration Section 11 2 1A and therefore 12 7 where C A is the number of moles of A per dm3 of open pore volume i e vol ume of gas as opposed to mol/vol of gas and

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• ### Graham s LawDiffusion and EffusionDefinition Formula

Graham s Law of diffusion Graham s Law of diffusion and effusion in chemistry is proposed by Scottish physical chemist Thomas Graham in 1948 to study the rate of diffusion and effusion for gases and liquid molecules According to Graham s law At constant temperature and pressure the rates of diffusion or effusion of different gases are inversely proportional to the square root of their

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• ### Lecture 3 Diffusion Fick s first law

were heat diffusion molecular diffusion and Brownian motion Their mathematical description was elaborated by Joseph Fourier in 1822 Adolf Fick in 1855 and by Albert Einstein in 1905 Specifically atomic diffusion is a diffusion process whereby the random thermally activated movement of atoms in a solid results in the net transport of atoms

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• ### Lecture 3 Diffusion Fick s first law

were heat diffusion molecular diffusion and Brownian motion Their mathematical description was elaborated by Joseph Fourier in 1822 Adolf Fick in 1855 and by Albert Einstein in 1905 Specifically atomic diffusion is a diffusion process whereby the random thermally activated movement of atoms in a solid results in the net transport of atoms

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• ### Diffusion Measurement By NMR

attenuated due to diffusion and whether or not the spectrum phase is constant When finished click return NOTE at this point You may analyze your data one of several ways 1 Topspin s T1/T2 module This analyzes one peak at a time 2 Bruker s Dynamic Center software for DOSY This gives a 2D DOSY plot To do so follow the

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• ### Electrical Conductivity based on Diffusion Coefficients

In the limit of infinite dilution non interacting ions we obtain from Eq 7a a simple formula that relies on diffusion coefficients 2 EC 0 = ∑ i Λ0 m ici = F 2 RT ∑ i Diz2 i ci E C 0 = ∑ i Λ m i 0 c i = F 2 R T ∑ i D i z i 2 c i In the realistic case of non ideal solutions however

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• ### Lecture 5 Diffusion Coefficient Diffusivity

chemical diffusion coefficient D A is defined as the self or tracer diffusion coefficient D A C denotes diffusion under a concentration gradient D A denotes diffusion of tracer A dilute in uniform concentration In dilute solution γ A = γ H = constant A A d x d ln lnγ = 0 then D A C D A chemical diffusion coefficient D A

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• ### 3 2 4 Rate of Diffusion through a SolutionChemistry

The Rate Constant K d Viscosity and rate of diffusion may be related by the following formula K d = 8 R T 3 n where n is the viscosity of the solution

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• ### Chapter 5 Diffusion in Solids

Processing Using Diffusion magnified image of a computer chip 0 5mm light regions Si atoms light regions Al atoms 2 Heat it 1 Deposit P rich layers on surface silicon Adapted from chapter opening photograph Ch p t er18 lis 7 Chapter 510 Diffusion How do we quantify the amount or rate of diffusion Measured empirically

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• ### Diffusion example 1Iowa State University

EE 432/532 diffusion examples4 An arsenic constant dose diffusion is performed The initial dose is 1014 cm–2 The diffusion temperature is 1100°C and the diffusion time is 2 hr The starting wafer had a p type background doping of 1017 cm–3 Find the concentration of the As at the surface and ﬁnd the junction depth

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• ### THE MATHEMATICS OF DIFFUSION

1 The diffusion equations 1 2 Methods of solution when the diffusion coefficient is constant 11 3 Infinite and sem infinite media 28 4 Diffusion in a plane sheet 44 5 Diffusion in a cylinder 69 6 Diffusion in a sphere 89 7 Concentration dependent diffusion

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• ### CHAPTER 8 DiffusionCity U

The diffusion profile of dopant atoms is dependent on the initial and boundary conditions Solutions for Equation 8 3 have been obtained for various simple conditions including constant surface concentration diffusion and constant total dopant diffusion In the first scenario impurity atoms are transported from a

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• ### Graham s Formula for Diffusion and Effusion

r M ½ = constant In these equations r = rate of diffusion or effusion and M = molar mass Generally this law is used to compare the difference in diffusion and effusion rates between gases often denoted as Gas A and Gas B It assumes that temperature and pressure are constant and equivalent between the two gases

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• ### Lecture 3 Diffusion Reading Chapter 3gatech edu

interstitialcy mechanism diffusion P and B decreasing vacancy mechanism diffusion Since the oxidation rate is time dependent the diffusivity becomes time dependent Where the second term is the oxidation induced diffusion coefficient change x ox is the thickness of the oxide t is time n= 0 3 0 6 for Si and is a proportionality constant

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• ### 3 1 The Physics of DiffusionTU Wien

Diffusion is the kinetic process that leads to the homogenization or uniform mixing of the chemical components in a phase Although mixing in a fluid liquid or gas may occur on many length scales as induced by macroscopic flow diffusive mixing in solids by contrast occurs only on the atomic or molecular level

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• ### Diffusion Equation Fick s Laws of Diffusion

This assumes that D i is a constant which is only true for dilute solutions This is usually a good assumption for diffusion in solids diffusion of chemicals in a dilute solution water or other typical liquid solvents and diffusion of dilute trace species in the gas phase such as carbon dioxide in air

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• ### Diffusionumich edu

Because 1 mol of A reacts under conditions of constant temperature and pressure to form 1 mol of B we have Equimolar Counter Diffusion EMCD at constant total molar concentration Section 11 2 1A and therefore 12 7 where C A is the number of moles of A per dm3 of open pore volume i e vol ume of gas as opposed to mol/vol of gas and

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• ### Diffusion and Fluid FlowUniversity of Florida

1 Diffusion Diffusion refers to the transport of substance against a concentration gradient Mass transfer movement of mass from one place to another Diffusion movement of mass from region of high concentration to low concentration J = D Flux of mass D diffusion coefficient 2 Diffusion is an important process in chromatography in determining

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• ### DIFFUSIONPennsylvania State University

concentration to drive the net diffusion of a chemical and formulated the law dz dx j c D C Cw z w Cw c w = molar density of water dx C /dz= molar gradient of C in z direction j Cw z = molar flux of C in z direction D= Diffusion constant fitted parameter

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• ### Electrical Conductivity based on Diffusion Coefficients

In the limit of infinite dilution non interacting ions we obtain from Eq 7a a simple formula that relies on diffusion coefficients 2 EC 0 = ∑ i Λ0 m ici = F 2 RT ∑ i Diz2 i ci E C 0 = ∑ i Λ m i 0 c i = F 2 R T ∑ i D i z i 2 c i In the realistic case of non ideal solutions however

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• ### DiffusionUMD

k = Boltzmann s constant = 1 38 x10 23 J/K T = absolute temperature Dt product is the measure of driving force in the diffusion –D is proportional to Temp –Time t –Increase either of these or both and you will change the diffusion parameters At high concentrations n i diffusion constant becomes dependant on concentration

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• ### Appendix A 9 Henry s law constant and diffusion

Appendix A 9 Henry s law constant and diffusion coefficients of contaminants in air and water for T = 0 to 25 oC abstracted from Crawford 1976 except where noted RReid et al 1977 PPerry and Chilton 1973 SMachay et al 1981 VVargaftik 1975 MMackay and Yeun 1983 substance

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• ### Diffusion Measurement By NMR

attenuated due to diffusion and whether or not the spectrum phase is constant When finished click return NOTE at this point You may analyze your data one of several ways 1 Topspin s T1/T2 module This analyzes one peak at a time 2 Bruker s Dynamic Center software for DOSY This gives a 2D DOSY plot To do so follow the

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• ### DIFFUSION COEFFICIENTthermopedia

Diffusion coefficient is the proportionality factor D in Fick s law see Diffusion by which the mass of a substance dM diffusing in time dt through the surface dF normal to the diffusion direction is proportional to the concentration gradient grad c of this substance dM = −D grad c dF dt Hence physically the diffusion coefficient implies that the mass of the substance diffuses through a unit surface in a unit time at a

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• ### Diffusion Measurement By NMR

Adjust the following parameters that are specific to the diffusion experiment Δ = D20 the time over which diffusion occurs Set to 0 05 2 sec δ =P30 2 length of gradient pulse Set to 15 msec Thus P30 is half the total δ Enter the value of rg determined in the 1D spectrum above

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• ### Chapter 4 Permeability Diffusivity and Solubility of

Constant Diffusion Coefficient In his seminal work Crank proposed a classical Fickian diffusion model which links the mass of diffusant M with time t for a specific thickness of film l 12 If the concentration of the diffusant is assumed to be initially

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• ### DIFFUSIONPennsylvania State University

D= Diffusion constant fitted parameter Fick s First Law Fick recognized that there must be a difference in concentration to drive the net diffusion of a chemical and formulated the law dz dx = atomic diffusion volume from formula and tabulated values cm3 1/2 1/3 2 1/3

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• ### Diffusion Coefficient and Laws Fick s Laws Metallurgy

The diffusion couple provides one method of experimentally determining the diffusion coefficient Casehardening Diffusion with Constant Concentration Casehardening is a process in which one element usually in gaseous form is diffused into another a solid the diffusing being limited to a small region near the surface

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• ### Chapter 5 Diffusion

where D0 is a pre exponential constant Q is the activation energy for diffusion R is gas constant Boltzmann s constant and T is absolute temperature From the above equation it can be inferred that large activation energy means relatively small diffusion coefficient

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