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Transport Phenomena Co07 Diffusion—Macroscopic View Co07

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Transport Phenomena Co07 Diffusion—Macroscopic View Co07 1/18 2/18 Transport phenomena co07 Diffusion—macroscopic view co07 Transport (kinetic) phenomena: : Flux ~J of compound (units: mol m 2 s 1) First Fick Law − − diffusion, electric conductivity, viscosity, heat conduction . For mass ~ J = D∇~c concentration Flux of mass, charge, momentum, heat, . − in kg m 3, ~J = amount (of quantity) transported per unit area (perpendicular is proportional to the concentration gradient − the flux is in to the vector of flux) within time unit ∂ ∂ ∂ ∂c ∂c ∂c 2 1 2 1 2 kg m s Units: energy/heat flux: J m s = W m , ∇~c = grad c = , , c = , , − − − − − ∂ ∂y ∂z ∂ ∂y ∂z current density: A m 2 − 2 1 D = diffusion coefficient (diffusivity) of molecules , unit: m s Cause = (generalized, thermodynamic) force − ~ = gradient of a potential Example. A U-shaped pipe of lenght = 20 cm and cross section F − 0.3 cm2. One end is in Coca-Cola (11 wt.% of sugar), other (chemical potential/concentration, electric potential, tempera- A = ture) end in pure water. How much sugar is transported by diffusion in 6 2 1 mg 0.74 one day? Dsucrose(25 C) = 5.2 10 cm s . Small forces—linearity ◦ × − − × × × × × × ~ ~ A − − − − J const − = = = kg 10 4 . 7 s 60 24 m 10 3 . 0 s m kg 10 56 . 2 t J = m × 7 2 2 4 1 2 F 7 − − − = · = grad s m kg 10 56 . 2 c D J × × 1 2 7 − − − − = In gases we use the kinetic theory: molecules (simplest: hard spheres) fly through = s m 10 2 . 5 s cm 10 2 . 5 D 1 2 10 1 2 6 − grad 5 gm kg 550 = / c = c space and sometimes collide 4 − − = = gm kg 110 dm g 110 c L: 1 in sucrose g 110 3 3 3/18 [blend -g che/sucrose] 4/18 Diffusion—microscopic view co07 Einstein–Stokes equation co07 Flux is given by the mean velocity of molecules ~: Colloid particles or large spherical molecules of radius R in a solvent of viscosity η it holds (Stokes formula) ƒ ~J = ~c Difference of chemical poten- Arrhenius law for vis- Thermodynamic force = grad ~ = 6πηR. ~ tials = reversible work needed F cosity (decreases with of the chemical potential:− to move a particle (mole) from Einstein–Stokes equation: increasing T), diffusiv- ⇒ μ kBT one state to another ity (increases with in- ~ = ∇~ = ∇~c kBT kBT F − NA − c D = D = creasing T ƒ 6πηR st ⇒ where formula μ = μ e + RT ln(c/c ) for infinity dillution was used. Opposite reasoning—hydrodynamic (Stokes) radius defined as: Friction force acting against molecule moving by velocity ~ through a medium is: kBT ~fr ~ R = = ƒ 6πηD F − where ƒ is the friction coeficient. Both forces are in equilibrium: effective molecule size (incl. solvation shell) ~ ≈ fr fr J kBT Example. Estimate the size of the sucrose molecule. Wa- ~ + = 0 i.e. ~ = ƒ~ = ƒ = = ∇~c F F F c F c ter viscosity is 0.891 10 3 m 1 kg s 1 at 25 C. − − − − − ◦ = nm 47 . 0 × R kBT On comparing with ~J = D∇~c we get the Einstein equation: D = − ƒ (also Einstein–Smoluchowski equation) 5/18 [plot/cukr.sh] 6/18 Second Fick Law co07 Second Fick Law co07 Non-stationary phenomenon (c changes with time). Example. Coca-Cola in a cylinder (height 10 cm) + pure water (10 cm). What time The amount of substance increases within is needed until the surface concentration = half of bottom concentration? months 4 time dτ in volume dV = ddydz: Fourier method: 2 [J() J( + d)] dydz ∂c ∂ c c0 < /2 = D c(, 0) = ,y,z − ∂τ ∂2 0 > /2 X § ∂J 2 J {J d} dydz c0 2c0 π π = [ ( ) ( ) + )] c , τ cos exp Dτ ,y,z − ∂ ( ) = + 2 X 2 π − ! ∂J = ddydz = ∇~ ~J dV = ∇~ ( D∇~c) dV − ,y,z ∂ − · − · − 2 2 2 2 X 1 3π 3 π 1 5π 5 π ∂2 ∂2 ∂2 cos exp Dτ + cos exp Dτ 2 3 2 5 2 = D∇~ c dV = D + + c dV − − ! − ! ··· ∂2 ∂y2 ∂z2 ! This type of equation is called ∂c 2 “equation of heat conduction”. = D∇ c ∂τ It is a parrabolic partial differ- ential equation [traj/brown.sh] 7/18 [show/galton.sh] 8/18 Diffusion and the Brownian motion co07 Brownian motion as a random walk + co07 Instead of for c(r,~ τ), let us solve the 2nd Fick (Smoluchowski, Einstein) law for the probability of finding a particle, start- 0.5 t=0 t=1 within time Δτ, a particle moves randomly ing from origin at τ = 0. We get the Gaussian 0.4 t=2 – by Δ with probability 1/2 with half-width ∝ distribution t=3 – by Δ with probability 1/2 2 0.3 t=4 − 1/2 c(x,t) t=5 1D: c(, τ) = (4πDτ)− exp − 4Dτ! 0.2 Using the central limit theorem: 2 2 2 in one step: Var Δ r 0.1 = = 3D: c r,~ τ 4πDτ 3/2 exp 〈 〉 ( ) = ( )− in n steps (in time τ nΔτ): Var nΔ2 − 4Dτ! = = 0.0 2 −3 −2 −1 0 1 2 3 Gaussian normal distribution with σ = nΔ = τ/ΔτΔ: x ⇒ p p 2 1 2 2 1 pΔτ Δτ 2 200 e /2σ exp 1D: = 2Dτ − = − 2 〈 〉 p2πσ p2πτ Δ − 2τ Δ Last example – order-of-magnintude which is for 2D Δ2/Δτ the same as c , τ 2 y y 0 = ( ) τ /2D = 4 months ≈ (for = 0.1 m) 0 def. 2 2 NB: Var = ( ) , for = 0, then Var = 〈 − 〈 〉 〉 〈 〉 〈 〉 2 1/6 3D: r = 6Dτ 0 0 Example. Calculate Var , where is a random number from interval ( 1, 1) 〈 〉 x x − 9/18 10/18 Electric conductivity co07 Electric conductivity co07 Ohm Law (here: U = voltage, U = ϕ2 ϕ1): substance κ/ S m 1 − ( − ) U 1 graphene 1 108 R = = U 1/R = conductivity, [1/R] = 1/Ω = S = Siemens × R silver 63 106 × (Specific) conductivity (conductance) κ is 1/resistance of a unit cube sea water 5 1 Ge 2.2 1 = κ A = area, = layer thickness, [κ] = S m R A − tap water 0.005 to 0.05 Si 1.6 10 3 − ~ ~ ~ × Vector notation: j = κ = κ∇ϕ distilled water (contains CO ) 7.5 10 5 E − 2 − × 6 ~j = el. current density, j = / , ~ = el. field intensity, = U/ deionized water 5.5 10 A E E × − glass 1 10 15–1 10 11 × − × − teflon 1 10 25–1 10 23 × − × − 11/18 12/18 Molar conductivity co07 Mobility and molar conductivity co07 Strong electrolytes: conductivity proportional to concentration. Mobility of an ion = averaged velocity in a unit electric field: Molar conductivity λ: = = U/ = el. intensity, U = voltage κ E λ = E c Charges ze of velocity and concentration c cause the current density “κ ” Units: κ S m 1, λ S m2 mol 1. [ ] = − [ ] = − Watch units—best convert c to mol m 3! ! . − j = czF = czF = λc λ = zF = molar conductivity of ion E E ⇒ Example. Conductivity of a 0.1 M solution of HCl o is 4 S m 1. Calculate the molar Ions (in dilute solutions) migrate independently (Kohlrausch law), for electrolyte − − z zA mol .4Sm S 0.04 + 1 conductivity. 2 Cννν⊕ Aννν −: here we define zA > 0 ⊕ j j j λ c λ c λ ννν λ ννν c = A + C = ( A A + C C)E = ( A A + C C) E Mathematically κ λ = = νννλ c X [cd pic;mz grotthuss.gif]13/18 14/18 Nothing is ideal co07 Conductivity of weak electrolytes co07 Limiting molar conductivity = molar conductivity at infinite dilution We count ions only, not unionized acid ∞ λ = lim λ c 0 In the limiting concentration: Departure from the limiting linear behavior→ (cf. Debye–Hückel theory): ∞ ∞ ∞ ∞ def. exptl 0.04 λ = λ(c) = λ constpc nebo λ = λ const c κ = λ cions = λ αc = λ c − − HCl Typical values: Æ λexptl cation λ∞/ S m2 mol 1 anion λ∞/ S m2 mol 1 α = 0.03 ( − ) ( − ) λ∞ + H 0.035 OH 0.020 /mol − 2 Na+ 0.0050 Cl 0.0076 Ostwald’s dilution law: NaOH − 2 exptl 2 0.02 Ca2+ 0.012 SO 2 0.016 c α c (λ ) / S m 4 λ − K = = cst 1 α cst λ∞ λ∞ λexptl Mobility and molar conductivity decreases with the ion size (Cl is slow), solva- ( ) − − − + tion (small Li + 4 H2O is slow) 0.01 AgNO3 H+, OH are fast movie credit: Matt K. Petersen, Wikipedia − CuSO4 CH3COOH 0 0 0.2 0.4 0.6 0.8 1 1.2 c1/2 / (mol/L)1/2 → 15/18 [pic/nernstovavrstva.sh]16/18 Conductivity and the diffusion coefficient co07 Transference numbers co07 Einstein (Nernst–Einstein) equation: microscopically: Transference number (transport number) of an ion is the fraction of the total z e k T k T k T k T RT current that is carried by that ion during migration (electrolysis). B B B B = D D = = = = = kBT = velocity ƒ / ze /( ) ze/ zF t F E E here z is with sign = = + ννν = stechiom. coeff. 2 2 ⊕ z F z + z zFD = RT λ = zF = D Ions move at different speeds under the same field. For Kννν⊕ Aννν − (electroneutrality: RT ⇒ z c = z c ; here z > 0) ⊕ ⊕ ⊕ diffusion: caused by a gradient of concentration/chemical potential c z z D ννν λ t = = = = = D c z + c z + + z D + z D ννν λ + ννν λ ~J = D∇~c = c ∇~μ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ − − RT t ze zFD Properties: t + t = 1, = = , = D, λ = zF ~j = c ∇~μ = c∇~μ ⊕ t E kBT − RT − ⊕ ⊕ Example. Electrolysis of CuSO : t 2+ 40 %, t 2 60 %. 4 Cu = SO4 = migration: caused by el.
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